Local Rapid Learning for Integer Programs

TL;DR

This paper extends Rapid Learning, a hybrid CP/MIP conflict learning technique, to integer programs by applying it at local search nodes, significantly improving MIP solving efficiency especially on complex problems.

Contribution

It introduces a novel approach for applying Rapid Learning repeatedly within the MIP search tree for integer programs, with heuristics to predict success.

Findings

Significant speed-up in MIP search times.

Particularly effective on highly dual degenerate problems.

Heuristics improve the efficiency of local Rapid Learning.

Abstract

Conflict learning algorithms are an important component of modern MIP and CP solvers. But strong conflict information is typically gained by depth-first search. While this is the natural mode for CP solving, it is not for MIP solving. Rapid Learning is a hybrid CP/MIP approach where CP search is applied at the root to learn information to support the remaining MIP solve. This has been demonstrated to be beneficial for binary programs. In this paper, we extend the idea of Rapid Learning to integer programs, where not all variables are restricted to the domain {0,1}, and rather than just running a rapid CP search at the root, we will apply it repeatedly at local search nodes within the MIP search tree. To do so efficiently, we present six heuristic criteria to predict the chance for local \rapidlearning to be successful. Our computational experiments indicate that our extended Rapid Learning algorithm significantly speeds up MIP search and is particularly beneficial on highly dual degenerate problems.

Figures

Tables

Table 1. Table 1 : Settings for Rapid Learning sub- SCIP .

parameter name	value	effect
lp/solvefreq	-1	disable LP
conflict/fuiplevels	1	use 1-UIP
nodeselection/dfs/stdpriority	$\text{INT\_MAX}/4$	use DFS
branching/inference/useweightedsum	FALSE	pure inference, no VSIDS
constraints/disableenfops	TRUE	no extra checks
propagating/pseudoobj/freq	-1	no objective propagation
conflict/maxvarsfac	0.05	only short conflicts
history/valuebased	TRUE	extensive branch. statistics

Table 2. Table 2 : Computational results for every individual heuristic criterion on MMM-IP .

		solved	time	nodes	timeQ	nodesQ
Exp.1	default	304	50.92	2466	–	–
	degeneracy	306	48.02	2084	0.943	0.845
	dualbound	304	50.26	2369	0.987	0.961
	leaves	305	49.31	2347	0.968	0.952
	obj	307	51.23	2399	1.006	0.973
	nsols	304	47.94	2142	0.941	0.869
	sblps	304	50.38	2296	0.989	0.931
	nochecks	299	61.10	1925	1.200	0.781
	onlyroot	307	54.89	2404	1.078	0.975
Exp.2	degeneracy + leaves	306	47.81	2073	0.939	0.841
	leaves + obj	307	49.80	2284	0.978	0.926
	all6criterion	303	48.23	2030	0.947	0.823

Table 3. Table 3 : Performance impact of individual gained information on MMM-IP .

		solved	time	nodes	timeQ	nodesQ
Exp.3	degeneracy	306	48.02	2084	–	–
	variablebounds	305	48.69	2089	1.014	1.002
	conflicts	306	50.11	2225	1.044	1.067
	infervals	305	48.29	2179	1.006	1.046
	primsols	305	52.77	2349	1.099	1.127

Table 4. Table 4 : Detailed computational results on MMM-IP with five random seeds. Relative changes by at least 5 % percent 5 5\% are highlighted in bold and blue ( improvement ) or italic and red ( deterioration ).

	default		degeneracy		dualbound		leaves		obj		nsols		sblps		nochecks		onlyroot
Instance	time	nodes	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ
10teams	2.14	1	1.277	1.000	0.990	1.000	0.994	1.000	0.990	1.000	1.271	1.000	0.994	1.000	1.248	1.000	1.258	1.000
	20.00	1369	0.241	0.072	1.290	0.687	1.236	1.001	0.995	1.000	0.244	0.072	0.963	0.303	0.245	0.072	0.246	0.072
	15.57	287	1.607	2.729	2.062	4.067	0.998	1.000	1.006	1.000	1.308	2.729	2.080	4.238	1.341	0.829	0.672	0.323
	4.16	5	2.721	1.219	1.000	1.000	1.006	1.000	0.996	1.000	2.713	1.219	1.004	1.000	2.709	1.219	2.329	1.219
	4.94	5	0.909	1.000	1.000	1.000	1.002	1.000	1.007	1.000	0.909	1.000	0.995	1.000	0.912	1.000	0.914	1.000
30n20b8	314.13	195	0.605	0.553	1.002	1.000	1.001	1.000	0.990	1.000	0.289	0.346	0.999	1.000	0.285	0.346	0.289	0.346
	157.33	71	0.712	0.596	0.999	1.000	0.998	1.000	0.995	1.000	0.810	0.596	0.999	1.000	0.789	0.596	0.852	0.596
	87.25	4	1.001	1.000	1.000	1.000	1.003	1.000	0.998	1.000	1.191	1.000	0.997	1.000	1.180	0.981	1.188	1.000
	213.06	102	1.208	1.312	0.996	1.000	0.991	1.000	0.978	0.950	1.251	0.896	0.993	1.000	1.047	0.782	1.251	0.896
	470.09	290	0.811	0.723	0.998	1.000	1.003	1.000	0.998	1.000	0.238	0.423	0.996	1.000	0.241	0.421	0.238	0.423
acc-tight5	125.35	172	1.114	1.171	0.387	0.295	0.423	0.302	1.011	0.965	0.651	0.735	0.991	1.000	1.087	1.171	1.115	1.622
	84.38	98	1.310	2.072	1.512	1.286	0.448	0.326	0.856	0.551	1.786	2.650	0.991	1.000	1.792	2.650	0.721	0.952
	50.45	480	0.973	0.929	2.656	2.631	1.753	1.966	1.022	0.993	0.973	0.929	1.002	1.000	0.974	0.929	3.299	6.017
	85.88	87	0.340	0.189	1.590	1.794	0.401	0.290	0.420	0.274	0.339	0.189	1.000	1.000	0.340	0.189	1.464	1.891
	73.38	864	0.417	0.161	1.440	1.091	2.064	1.326	1.488	1.309	0.417	0.161	0.996	1.000	0.419	0.161	1.256	1.088
air03	1.70	1	1.011	1.000	1.015	1.000	1.015	1.000	1.015	1.000	1.004	1.000	1.007	1.000	6.433	1.000	6.515	1.000
	1.71	1	1.004	1.000	1.004	1.000	1.011	1.000	1.004	1.000	0.996	1.000	1.007	1.000	6.590	1.000	6.472	1.000
	1.74	1	0.993	1.000	1.011	1.000	1.004	1.000	0.989	1.000	1.000	1.000	0.985	1.000	6.365	1.000	6.372	1.000
	1.75	1	1.000	1.000	0.996	1.000	1.000	1.000	0.996	1.000	1.000	1.000	1.000	1.000	6.378	1.000	6.335	1.000
	1.75	1	0.996	1.000	0.993	1.000	0.985	1.000	0.985	1.000	0.993	1.000	1.004	1.000	6.491	1.000	6.378	1.000
air04	49.46	110	1.003	1.000	1.007	1.000	1.008	1.000	1.005	1.000	1.003	1.000	1.006	1.000	0.978	0.776	0.911	0.652
	36.09	39	1.008	1.000	1.003	1.000	1.004	1.000	0.996	1.000	1.000	1.000	1.002	1.000	1.172	1.144	1.112	1.158
	52.88	118	1.005	1.000	1.000	1.000	1.002	1.000	0.999	1.000	0.997	1.000	0.990	1.000	0.993	0.665	0.949	0.665
	36.56	31	1.014	1.000	1.007	1.000	1.010	1.000	1.001	1.000	0.995	1.000	1.010	1.000	1.552	1.481	1.735	3.061
	48.35	122	1.006	1.000	1.002	1.000	1.004	1.000	1.001	1.000	1.004	1.000	1.003	1.000	0.803	0.500	0.801	0.500
air05	24.82	189	0.990	1.000	0.996	1.000	0.999	1.000	1.006	1.000	1.003	1.000	0.990	1.000	2.135	1.388	1.270	1.547
	26.54	239	1.004	1.000	1.005	1.000	1.001	1.000	1.008	1.000	1.001	1.000	1.003	1.000	3.006	1.714	1.275	1.419
	31.37	438	0.992	1.000	0.992	1.000	0.997	1.000	0.996	1.000	0.994	1.000	0.997	1.000	1.869	0.690	1.132	0.946
	27.36	283	1.007	1.000	1.006	1.000	1.006	1.000	1.007	1.000	1.006	1.000	0.996	1.000	1.679	1.222	1.175	1.204
	27.14	279	1.003	1.000	1.003	1.000	1.002	1.000	1.002	1.000	1.004	1.000	0.996	1.000	2.084	1.343	1.128	0.934
blend2	0.69	714	1.041	1.000	1.041	1.000	1.036	1.000	1.047	1.000	1.041	1.000	1.047	1.000	1.041	1.000	1.041	1.000
	1.08	973	0.986	1.000	1.005	1.000	0.971	1.000	1.019	1.000	1.005	1.000	0.971	1.000	1.010	1.000	0.986	1.000
	1.05	912	1.005	1.000	0.995	1.000	0.990	1.000	1.000	1.000	0.985	1.000	1.000	1.000	1.000	1.000	0.995	1.000
	0.67	543	0.994	1.000	1.006	1.000	1.012	1.000	1.030	1.000	1.000	1.000	1.012	1.000	1.012	1.000	1.018	1.000
	0.66	709	1.030	1.000	1.018	1.000	1.018	1.000	1.006	1.000	1.036	1.000	1.042	1.000	1.024	1.000	1.030	1.000
bley_xl1	164.49	1	0.997	1.000	1.002	1.000	0.998	1.000	1.000	1.000	1.008	1.000	0.996	1.000	0.997	1.000	1.003	1.000
	187.04	3	1.012	1.146	1.007	1.000	1.005	1.000	1.002	1.000	1.005	1.000	1.003	1.000	0.906	0.981	0.907	0.981
	182.51	13	0.892	0.903	1.012	1.000	1.015	1.000	1.012	1.000	1.013	1.000	1.001	0.956	0.901	0.903	0.898	0.903
	133.90	1	1.006	1.000	1.008	1.000	1.003	1.000	1.005	1.000	1.005	1.000	1.011	1.000	0.982	1.000	1.118	1.000
	135.56	1	1.006	1.000	1.006	1.000	1.001	1.000	1.000	1.000	1.005	1.000	1.008	1.000	1.014	1.000	1.023	1.000
bnatt350	378.09	89	0.052	0.030	0.431	0.301	0.107	0.054	3.061	2.782	0.053	0.030	0.121	0.041	0.052	0.030	2.109	2.434
	521.76	326	0.033	0.022	0.276	0.183	0.144	0.084	1.081	1.382	0.033	0.022	1.011	1.431	0.033	0.022	0.487	0.420
	397.10	160	0.048	0.026	0.119	0.057	0.131	0.063	0.610	0.445	0.048	0.026	0.301	0.229	0.048	0.026	0.889	0.846
	155.46	4028	0.109	0.067	0.262	0.129	2.325	2.588	1.524	1.482	0.109	0.067	0.389	0.127	0.110	0.067	4.491	3.610
	994.16	2528	0.044	0.021	0.575	0.724	0.329	0.259	0.713	0.670	0.044	0.021	0.156	0.076	0.045	0.021	0.317	0.315
cap6000	2.69	1830	1.003	1.000	1.019	1.000	1.003	1.000	1.019	1.000	1.016	1.000	0.989	1.000	1.412	1.131	1.098	0.866
	2.67	1837	0.986	1.000	0.984	1.000	0.989	1.000	0.986	1.000	0.975	1.000	1.003	1.000	1.455	1.256	1.155	1.142
	2.61	1959	0.992	1.000	1.000	1.000	0.997	1.000	1.006	1.000	0.997	1.000	0.992	1.000	1.280	0.982	1.150	1.066
	3.17	1924	1.007	1.000	0.990	1.000	1.007	1.000	0.993	1.000	0.995	1.000	0.995	1.000	1.422	1.166	1.086	0.894
	2.82	1762	1.008	1.000	1.008	1.000	0.995	1.000	0.987	1.000	0.995	1.000	1.008	1.000	1.387	1.325	1.157	1.258
cov1075	3600.00	1445091	0.989	1.189	1.000	0.981	1.000	0.996	1.000	1.004	1.000	0.991	1.000	0.985	1.000	0.314	0.961	1.189
	3600.00	1787171	1.000	0.975	1.000	0.982	1.000	0.998	1.000	1.004	1.000	1.003	1.000	0.989	1.000	0.303	1.000	0.989
	3600.00	1528146	1.000	1.198	1.000	0.986	1.000	0.998	1.000	0.997	1.000	0.999	1.000	0.956	1.000	0.373	1.000	1.210
	3600.00	1808088	1.000	0.985	1.000	0.995	1.000	1.006	1.000	1.002	1.000	1.006	1.000	0.974	1.000	0.250	0.997	0.995
	3600.00	1201194	1.000	1.508	1.000	0.986	1.000	0.997	1.000	0.999	1.000	1.000	1.000	0.898	1.000	0.460	0.979	1.514
csched010	3600.00	228369	1.000	1.003	1.000	1.001	1.000	0.963	1.000	0.999	0.758	0.644	1.000	1.001	1.000	0.571	0.907	0.925
	3600.00	175223	1.000	1.001	1.000	1.001	1.000	0.864	1.000	1.000	0.972	1.038	1.000	0.999	1.000	0.756	0.850	1.037
	2891.69	214060	1.000	1.000	0.951	0.941	1.147	1.206	0.997	1.000	1.245	1.145	1.002	1.000	1.245	0.689	1.245	1.280
	3600.00	195746	1.000	0.998	1.000	0.996	0.988	0.820	1.000	0.998	1.000	0.960	1.000	0.997	1.000	0.578	1.000	1.028
	3168.37	226703	1.001	1.000	0.999	1.000	1.136	1.303	1.000	1.000	1.136	1.352	1.001	1.000	1.136	0.682	1.136	1.307
eil33-2	69.44	583	1.003	1.000	1.034	1.000	1.005	1.000	1.034	1.000	1.031	1.000	1.008	1.000	1.772	0.909	1.094	0.909
	81.92	739	1.007	1.000	1.007	1.000	1.010	1.000	0.997	1.000	1.009	1.000	1.007	1.000	1.671	0.852	1.028	0.852
	84.87	847	0.989	1.000	0.999	1.000	1.001	1.000	1.001	1.000	1.008	1.000	1.002	1.000	1.904	0.852	1.195	0.852
	73.60	543	0.988	1.000	1.007	1.000	1.000	1.000	1.001	1.000	1.010	1.000	1.003	1.000	2.006	1.509	1.260	2.728
	70.01	583	0.987	1.000	1.007	1.000	1.002	1.000	1.002	1.000	1.007	1.000	1.003	1.000	2.176	1.433	1.602	1.416
eilB101	378.99	10582	0.998	1.000	1.021	1.000	1.002	1.000	1.023	1.000	0.999	1.000	1.009	1.000	1.083	0.724	0.705	0.724
	365.27	8427	0.984	1.000	0.995	1.000	0.993	1.000	0.999	1.000	1.005	1.000	0.997	1.000	1.391	1.150	0.916	1.050
	401.63	9358	0.995	1.000	1.000	1.000	0.998	1.000	0.993	1.000	1.008	1.000	0.997	1.000	1.147	0.780	0.819	0.780
	397.46	11465	0.996	1.000	0.999	1.000	0.998	1.000	0.999	1.000	1.002	1.000	0.995	1.000	1.304	1.025	0.905	1.025
	365.79	10006	0.989	1.000	0.998	1.000	0.989	1.000	0.994	1.000	0.996	1.000	1.001	1.000	1.201	0.894	0.818	0.894
enigma	0.50	809	1.000	0.825	1.020	0.716	1.000	1.000	1.000	0.998	1.000	0.542	1.140	0.686	1.000	0.542	1.000	0.856
	0.50	449	1.000	0.497	1.000	0.814	1.000	1.000	1.000	1.000	1.000	0.470	1.000	0.778	1.000	0.470	1.000	1.616
	0.50	541	1.000	0.254	1.000	0.510	1.000	1.000	1.000	0.977	1.000	0.237	1.060	0.721	1.000	0.237	1.000	0.665
	0.50	679	1.073	0.537	1.000	0.397	1.000	1.000	1.000	1.000	1.000	0.151	1.133	1.113	1.000	0.151	1.000	0.244
	0.50	445	1.020	0.543	1.000	0.833	1.000	1.000	1.000	1.006	1.000	0.543	1.000	0.806	1.013	0.543	1.000	1.675
enlight13	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
enlight14	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
fiber	1.00	2	1.010	1.000	1.010	1.000	0.990	1.000	1.005	1.000	1.000	1.000	1.000	1.000	1.160	1.010	1.075	1.010
	1.42	4	0.988	1.000	1.004	1.000	0.988	1.000	1.000	1.000	0.992	1.000	1.008	1.000	1.384	1.240	0.938	0.990
	1.74	4	1.007	1.000	0.993	1.000	0.996	1.000	0.996	1.000	0.978	1.000	0.982	1.000	1.401	1.010	0.949	0.990
	1.53	4	1.008	1.000	1.000	1.000	0.996	1.000	0.996	1.000	0.988	1.000	1.000	1.000	1.194	0.990	1.024	1.000
	1.00	3	1.000	1.000	1.005	1.000	1.005	1.000	1.000	1.000	0.995	1.000	1.005	1.000	1.080	0.990	1.030	0.990
flugpl	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
gt2	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
harp2	691.56	1491878	1.005	1.000	1.001	1.000	1.001	1.000	1.007	1.000	1.000	1.000	1.003	1.000	5.200	5.260	4.215	4.834
	1159.66	2857677	1.004	1.000	0.998	1.000	0.994	1.000	0.998	1.000	0.993	1.000	0.994	1.000	0.410	0.228	2.318	2.253
	336.73	595854	0.992	1.000	1.007	1.000	1.007	1.000	1.005	1.000	1.002	1.000	0.999	1.000	1.604	1.232	1.348	1.300
	937.54	2046181	0.997	1.000	0.987	1.000	0.992	1.000	1.000	1.000	0.997	1.000	0.999	1.000	0.506	0.321	1.065	1.052
	425.68	923780	1.018	1.000	1.010	1.000	1.009	1.000	1.013	1.000	1.014	1.000	1.021	1.000	2.779	2.192	7.701	8.726
iis-100-0-cov	629.92	103302	0.993	1.000	0.993	1.000	1.014	1.000	0.991	1.000	0.997	1.000	0.992	1.000	0.999	0.999	0.965	0.999
	545.20	92393	0.999	1.000	0.997	1.000	1.001	1.000	1.002	1.000	0.996	1.000	0.998	1.000	1.128	1.046	1.053	1.046
	520.78	81639	0.999	1.000	1.000	1.000	1.002	1.000	1.001	1.000	1.000	1.000	0.997	1.000	1.151	1.128	1.097	1.128
	506.53	84800	1.002	1.000	0.997	1.000	0.995	1.000	0.996	1.000	0.998	1.000	0.998	1.000	1.117	1.080	1.077	1.080
	544.56	92313	0.996	1.000	0.999	1.000	1.004	1.000	1.001	1.000	1.002	1.000	0.998	1.000	1.089	1.052	0.990	1.052
iis-bupa-cov	1915.72	157619	1.001	1.000	1.009	1.000	1.002	1.000	1.000	1.000	1.007	1.000	1.003	1.000	1.704	1.376	1.344	1.376
	1944.83	164079	0.997	1.000	0.995	1.000	0.997	1.000	0.996	1.000	0.994	1.000	0.995	1.000	1.259	1.099	1.117	1.099
	1881.83	156888	0.998	1.000	0.999	1.000	0.993	1.000	0.998	1.000	0.996	1.000	0.996	1.000	1.189	1.093	1.081	1.093
	1959.19	161969	0.989	1.000	1.005	1.000	0.996	1.000	0.996	1.000	0.994	1.000	0.998	1.000	1.398	1.234	1.193	1.234
	1820.52	158284	0.990	1.000	0.997	1.000	0.994	1.000	0.995	1.000	0.995	1.000	0.994	1.000	1.378	1.219	1.241	1.219
iis-pima-cov	293.79	4824	0.995	1.000	0.997	1.000	1.002	1.000	1.002	1.000	0.993	1.000	0.997	1.000	1.072	1.109	1.016	1.109
	437.19	12189	0.995	1.000	0.997	1.000	0.994	1.000	1.001	1.000	1.003	1.000	0.999	1.000	0.746	0.445	0.712	0.445
	246.99	4668	1.000	1.000	1.001	1.000	1.004	1.000	1.006	1.000	1.005	1.000	1.004	1.000	1.326	1.583	1.214	1.583
	284.88	6607	1.002	1.000	1.003	1.000	1.000	1.000	1.001	1.000	0.996	1.000	0.998	1.000	1.225	1.265	1.144	1.265
	556.78	17821	1.003	1.000	0.998	1.000	1.000	1.000	0.999	1.000	0.999	1.000	0.995	1.000	0.594	0.343	0.562	0.343
l152lav	5.51	23	0.994	1.000	0.989	1.000	0.994	1.000	0.994	1.000	0.995	1.000	1.005	1.000	2.278	1.520	1.035	1.244
	1.67	31	0.985	1.000	0.993	1.000	0.985	1.000	0.993	1.000	0.989	1.000	0.966	1.000	2.131	1.122	1.225	1.183
	2.93	50	1.000	1.000	0.995	1.000	0.992	1.000	1.000	1.000	1.003	1.000	1.008	1.000	3.046	1.367	0.962	0.967
	2.34	20	0.985	1.000	0.994	1.000	0.991	1.000	0.991	1.000	0.985	1.000	1.003	1.000	3.713	2.008	1.275	1.342
	3.02	52	0.995	1.000	1.000	1.000	0.995	1.000	0.993	1.000	0.998	1.000	1.002	1.000	2.167	1.204	0.891	0.921
lectsched-4-obj	27.04	805	0.753	0.215	1.223	0.740	1.004	1.000	1.009	0.924	1.005	1.000	0.730	0.314	0.748	0.215	0.978	1.608
	23.16	868	1.201	0.288	0.813	0.525	1.029	1.080	0.887	0.765	0.988	1.000	0.666	0.311	1.201	0.288	0.995	1.282
	86.08	1061	0.288	0.264	0.416	0.216	0.627	0.461	1.020	1.006	0.385	0.696	0.458	0.146	0.265	0.121	0.384	0.696
	26.83	619	0.327	0.082	0.768	0.367	0.928	0.540	0.926	0.772	0.257	0.082	2.333	0.477	0.330	0.082	0.257	0.082
	22.73	745	0.909	0.253	1.038	0.805	1.005	1.000	0.992	0.954	1.004	1.000	1.484	0.910	0.913	0.253	0.701	0.647
lseu	0.75	268	1.011	1.000	1.011	1.000	1.006	1.000	1.000	1.000	1.011	1.000	1.034	1.000	1.566	1.095	1.006	0.986
	0.50	4	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.010	1.000	1.000
	0.51	205	0.993	1.000	1.000	1.000	0.993	1.000	0.993	1.000	0.993	1.000	1.000	1.000	1.238	0.866	0.993	0.931
	0.50	91	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.738	1.000	0.969
	0.50	191	1.007	1.000	1.000	1.000	1.000	1.000	1.007	1.000	1.007	1.000	1.007	1.000	1.380	0.670	1.000	0.680
m100n500k4r1	3600.00	2166065	1.000	1.118	1.000	0.786	1.000	0.998	1.000	1.001	1.000	1.001	1.000	0.777	1.000	0.780	1.000	1.171
	3600.00	2266017	1.000	1.035	1.000	0.712	1.000	0.995	1.000	0.999	1.000	0.998	1.000	0.701	1.000	0.782	1.000	1.059
	3600.00	2611447	1.000	0.884	1.000	0.646	1.000	0.998	1.000	1.002	1.000	1.001	1.000	0.640	1.000	0.559	1.000	0.930
	3600.00	2305045	1.000	0.937	1.000	0.723	1.000	0.998	1.000	0.999	1.000	0.999	1.000	0.723	1.000	0.640	1.000	0.989
	3600.00	2226238	1.000	0.833	1.000	0.733	1.000	0.991	1.000	0.999	1.000	0.997	1.000	0.732	1.000	0.645	1.000	0.923
macrophage	125.50	5163	1.001	1.000	1.006	1.000	1.001	1.000	1.005	1.000	1.001	1.000	1.001	1.000	1.774	1.158	0.911	0.701
	157.18	7241	0.999	1.000	0.999	1.000	0.998	1.000	1.000	1.000	0.999	1.000	0.998	1.000	1.535	1.184	0.948	0.852
	209.11	9131	1.000	1.000	0.999	1.000	1.002	1.000	0.998	1.000	0.999	1.000	0.999	1.000	1.438	1.250	0.821	1.053
	162.53	7927	1.000	1.000	1.002	1.000	1.002	1.000	1.000	1.000	0.998	1.000	1.000	1.000	1.876	1.088	1.083	0.840
	121.37	5239	0.999	1.000	0.997	1.000	0.998	1.000	1.000	1.000	1.001	1.000	0.997	1.000	1.987	1.355	1.059	1.071
manna81	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.007	1.000	1.020	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.013	1.000	1.013	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.013	1.000	1.033	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.013	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.007	1.000	1.033	1.000
markshare1	3600.00	19010451	1.000	0.931	1.000	0.786	1.000	0.996	1.000	1.000	1.000	0.996	1.000	0.797	1.000	0.924	1.000	1.083
	3600.00	20192578	1.000	0.862	1.000	0.893	1.000	0.999	1.000	0.997	1.000	1.000	1.000	0.890	1.000	0.856	1.000	1.017
	3600.00	19529435	1.000	0.919	1.000	0.907	1.000	1.000	1.000	0.998	1.000	0.997	1.000	0.919	1.000	0.918	1.000	1.219
	3600.00	20381558	1.000	0.898	1.000	0.865	1.000	0.998	1.000	1.001	1.000	1.002	1.000	0.818	1.000	0.903	1.000	1.017
	3600.00	21608999	1.000	0.797	1.000	0.808	1.000	1.004	1.000	1.000	1.000	0.996	1.000	0.806	1.000	0.795	1.000	1.090
markshare2	3600.00	4050504	1.000	0.843	1.000	1.137	1.000	0.995	1.000	0.991	1.000	0.995	1.000	1.143	1.000	0.846	1.000	0.958
	3600.00	5646401	1.000	0.697	1.000	0.783	1.000	1.001	1.000	1.000	1.000	1.000	1.000	0.787	1.000	0.697	1.000	0.636
	3600.00	3589038	1.000	1.026	1.000	1.002	1.000	1.002	1.000	0.996	1.000	0.998	1.000	1.002	1.000	1.026	1.000	1.083
	3600.00	4244067	1.000	1.069	1.000	1.098	1.000	1.001	1.000	1.003	1.000	1.001	1.000	1.106	1.000	1.069	1.000	0.891
	3600.00	5150867	1.000	0.830	1.000	0.737	1.000	1.002	1.000	0.996	1.000	0.999	1.000	0.736	1.000	0.832	1.000	0.948
mcsched	285.92	13689	0.999	1.000	0.997	1.000	0.998	1.000	1.012	1.000	0.998	1.000	1.000	1.000	1.093	0.678	0.820	0.732
	271.88	11373	0.999	1.000	0.998	1.000	1.002	1.000	0.999	1.000	1.001	1.000	1.000	1.000	3.196	1.747	1.459	1.710
	233.97	10059	0.996	1.000	0.997	1.000	1.005	1.000	0.998	1.000	0.997	1.000	1.001	1.000	2.468	1.296	1.287	1.310
	286.25	12496	0.997	1.000	0.998	1.000	0.995	1.000	0.998	1.000	0.998	1.000	0.997	1.000	2.450	1.245	1.055	1.146
	270.69	10373	1.005	1.000	1.000	1.000	1.005	1.000	1.000	1.000	1.001	1.000	1.001	1.000	1.542	0.996	1.007	0.992
mine-166-5	58.69	1569	0.995	1.000	0.995	1.000	0.997	1.000	0.993	1.000	1.020	1.000	1.001	1.000	0.776	1.022	0.851	2.359
	57.78	1191	0.990	1.000	0.992	1.000	0.996	1.000	0.990	1.000	0.990	1.000	0.991	1.000	0.920	0.930	1.110	0.901
	46.45	2244	1.001	1.000	1.004	1.000	1.006	1.000	1.007	1.000	1.001	1.000	1.003	1.000	1.103	1.067	1.132	0.650
	64.93	139	1.006	1.000	1.004	1.000	1.005	1.000	1.001	1.000	1.005	1.000	1.002	1.000	0.711	5.397	0.619	6.515
	42.28	2863	1.000	1.000	1.000	1.000	1.001	1.000	1.000	1.000	1.001	1.000	0.998	1.000	1.165	0.555	0.994	0.873
mine-90-10	276.07	51926	1.002	1.000	1.001	1.000	1.157	1.090	0.999	1.000	1.000	1.000	1.003	1.000	1.033	0.750	0.767	0.596
	125.32	22195	0.998	1.000	1.005	1.000	0.998	1.000	0.999	1.000	1.003	1.000	1.001	1.000	4.652	3.660	1.749	1.438
	173.25	25979	1.001	1.000	1.004	1.000	0.999	1.000	0.999	1.000	1.004	1.000	0.999	1.000	1.687	1.060	2.725	2.264
	500.83	92879	1.003	1.000	1.000	1.000	1.324	1.061	0.999	1.000	0.999	1.000	0.999	1.000	0.809	0.459	0.556	0.457
	236.18	40409	1.005	1.000	1.003	1.000	0.907	1.242	1.004	1.000	1.002	1.000	1.013	1.000	2.162	1.767	0.762	0.592
misc03	1.39	398	1.004	1.000	1.004	1.000	1.000	1.000	1.013	1.000	1.004	1.000	0.983	1.000	0.799	0.267	0.711	0.263
	0.71	15	1.000	1.000	0.994	1.000	1.012	1.000	1.006	1.000	1.076	1.226	1.006	1.000	1.427	1.226	1.094	1.226
	0.70	19	1.018	1.000	0.994	1.000	1.006	1.000	1.018	1.000	1.165	1.487	0.976	1.000	1.512	1.471	1.176	1.487
	1.00	111	0.990	1.000	1.010	1.000	1.010	1.000	1.000	1.000	0.985	1.000	1.000	1.000	1.015	0.763	0.855	0.706
	0.66	15	0.988	1.000	1.000	1.000	0.988	1.000	0.988	1.000	1.187	1.609	0.988	1.000	1.608	1.435	1.151	1.609
misc07	38.05	28289	1.006	1.000	1.027	1.000	0.998	1.000	0.998	1.000	1.089	1.188	1.000	1.000	1.496	0.758	1.087	1.188
	36.66	29782	0.995	1.000	0.996	1.000	0.998	1.000	0.999	1.000	1.001	1.000	1.004	1.000	2.177	1.017	0.946	1.080
	21.56	17632	0.995	1.000	0.993	1.000	0.998	1.000	1.004	1.000	1.001	1.000	0.993	1.000	3.660	1.598	2.274	2.333
	45.95	38975	0.997	1.000	0.994	1.000	0.996	1.000	1.004	1.000	1.005	1.000	1.005	1.000	1.340	0.588	0.812	0.806
	30.94	29476	0.988	1.000	1.001	1.000	0.998	1.000	0.990	1.000	1.082	0.937	0.994	1.000	1.266	0.557	1.084	0.937
mitre	10.20	1	1.390	1.000	0.996	1.000	1.012	1.000	1.054	1.000	0.991	1.000	1.071	1.000	1.381	1.000	1.374	1.000
	10.28	1	1.469	1.000	0.989	1.000	1.006	1.000	0.991	1.000	1.481	1.000	0.989	1.000	1.449	1.000	1.456	1.000
	10.94	1	1.384	1.000	0.955	1.000	0.961	1.000	0.946	1.000	1.357	1.000	0.955	1.000	1.389	1.000	1.447	1.000
	10.45	1	1.412	1.000	0.994	1.000	0.990	1.000	0.987	1.000	0.991	1.000	0.979	1.000	1.397	1.000	1.400	1.000
	13.12	1	1.019	1.000	0.996	1.000	0.984	1.000	0.982	1.000	1.008	1.000	0.994	1.000	1.046	1.000	1.055	1.000
mod008	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.54	2	0.994	1.000	1.006	1.000	1.000	1.000	0.994	1.000	0.994	1.000	1.000	1.000	1.026	1.000	1.032	1.000
	0.52	2	1.013	1.000	1.013	1.000	1.007	1.000	0.993	1.000	0.987	1.000	1.007	1.000	1.026	0.990	1.020	0.990
	0.52	1	0.993	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.020	1.000	1.033	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
mod010	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.460	1.000	1.273	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.193	1.000	1.180	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.233	1.010	1.273	1.010
	0.52	2	1.007	1.000	1.020	1.000	0.993	1.000	0.993	1.000	1.007	1.000	1.007	1.000	1.329	1.000	1.197	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.347	1.010	1.320	1.010
mzzv11	332.43	2022	0.999	1.000	1.000	1.000	0.995	1.000	1.005	1.000	0.995	1.000	0.996	1.000	1.490	0.583	0.932	0.330
	229.41	959	1.002	1.000	0.999	1.000	1.001	1.000	1.005	1.000	1.002	1.000	1.003	1.000	1.453	0.818	1.573	1.220
	310.05	1421	1.004	1.000	1.000	1.000	1.001	1.000	0.995	1.000	0.999	1.000	0.999	1.000	0.910	0.415	1.175	0.525
	327.03	1579	0.987	1.000	1.001	1.000	0.995	1.000	0.995	1.000	0.994	1.000	0.993	1.000	1.000	0.378	0.964	0.392
	186.14	627	1.000	1.000	1.003	1.000	1.003	1.000	1.008	1.000	1.007	1.000	1.007	1.000	1.746	0.791	1.458	0.770
mzzv42z	152.40	265	0.992	1.000	0.996	1.000	0.999	1.000	0.993	1.000	0.991	1.000	0.999	1.000	2.086	1.477	0.984	0.753
	131.25	125	0.999	1.000	0.993	1.000	0.993	1.000	0.983	1.000	0.992	1.000	1.107	1.053	2.096	1.636	1.692	1.667
	197.62	200	1.006	1.000	1.003	1.000	1.006	1.000	1.005	1.000	1.002	1.000	1.016	0.923	2.262	1.597	1.436	1.117
	213.83	456	1.002	1.000	1.001	1.000	1.001	1.000	1.006	1.000	0.996	1.000	0.945	0.736	1.504	0.845	1.677	0.968
	167.65	300	0.999	1.000	0.998	1.000	0.998	1.000	1.000	1.000	1.007	1.000	1.003	1.000	1.347	0.657	1.143	0.695
neos-1109824	37.52	1796	0.996	1.000	1.037	1.000	0.995	1.000	0.988	1.000	1.572	2.657	1.018	1.000	2.154	2.246	1.553	2.657
	46.67	3438	0.998	1.000	1.027	1.072	0.996	1.000	0.998	1.000	0.971	0.962	0.782	0.594	1.320	0.810	0.969	0.962
	35.16	1913	0.995	1.000	1.002	1.000	0.997	1.000	1.002	1.000	0.995	1.000	1.079	0.966	1.478	0.997	1.373	1.906
	77.65	8290	0.996	1.000	1.001	1.000	0.997	1.000	1.000	1.000	0.693	0.534	1.000	1.000	0.984	0.445	0.695	0.534
	39.64	2247	0.990	1.000	0.992	1.000	0.991	1.000	0.994	1.000	1.121	1.077	1.030	0.933	1.926	1.546	1.120	1.077
neos-1337307	3600.00	299554	1.000	1.013	1.000	1.008	1.000	0.990	1.000	1.001	1.000	1.175	1.000	0.993	1.000	0.167	1.000	1.142
	3600.00	268930	1.000	0.999	1.000	1.001	1.000	0.998	1.000	0.999	1.000	1.085	1.000	0.999	1.000	0.169	1.000	0.837
	3600.00	350002	1.000	1.003	1.000	0.996	1.000	1.004	1.000	1.005	1.000	1.013	1.000	0.999	1.000	0.153	1.000	1.101
	3600.00	300182	1.000	1.003	1.000	0.998	1.000	0.999	1.000	1.003	1.000	1.136	1.000	0.998	1.000	0.152	1.000	1.124
	3600.00	367948	1.000	1.015	1.000	1.008	1.000	1.001	1.000	1.005	1.000	0.749	1.000	1.004	1.000	0.112	1.000	0.986
neos-1601936	2034.60	1453	0.999	1.000	1.714	3.615	0.998	1.000	0.999	1.000	1.769	0.679	1.002	1.000	1.769	1.201	1.769	0.679
	3600.00	2861	1.000	1.004	1.000	1.039	1.000	1.004	1.000	0.997	1.000	0.393	1.000	1.002	1.000	0.356	1.000	0.609
	1348.55	2661	0.993	1.000	2.668	5.411	1.001	1.000	1.001	1.000	2.668	0.548	1.000	1.000	2.668	0.541	2.668	0.528
	3600.00	6917	1.000	1.000	1.000	1.168	1.000	1.000	1.000	1.004	1.000	0.429	1.000	1.000	1.000	0.831	0.898	0.158
	3600.00	6139	1.000	1.000	1.000	0.686	1.000	1.000	1.000	1.000	0.419	0.176	1.000	1.000	1.000	0.441	0.417	0.176
neos-686190	83.42	4906	0.998	1.000	0.998	1.000	0.992	1.000	0.998	1.000	1.001	1.000	1.000	1.000	7.579	3.397	1.703	2.367
	86.19	6047	1.004	1.000	1.001	1.000	1.000	1.000	0.999	1.000	1.001	1.000	0.999	1.000	8.670	3.407	4.366	6.493
	114.74	9156	1.005	1.000	1.000	1.000	0.999	1.000	0.998	1.000	0.999	1.000	1.001	1.000	3.742	0.651	0.736	0.632
	116.53	9998	1.006	1.000	1.005	1.000	1.006	1.000	1.000	1.000	0.999	1.000	1.004	1.000	3.255	0.460	0.681	0.564
	121.04	9806	0.998	1.000	1.002	1.000	0.999	1.000	0.998	1.000	1.004	1.000	0.997	1.000	2.484	0.493	0.627	0.572
neos-849702	114.99	1680	1.802	0.595	0.862	0.835	0.866	0.835	1.801	0.595	1.802	0.595	1.731	4.759	1.802	0.595	9.601	11.858
	353.12	77703	2.900	0.364	2.216	4.411	2.214	4.411	2.910	0.364	2.907	0.364	9.463	19.062	2.919	0.364	2.830	2.084
	357.57	9988	2.795	0.359	0.511	0.342	0.514	0.342	2.802	0.359	2.797	0.359	2.369	2.377	2.799	0.359	2.111	0.080
	261.49	16745	1.021	0.048	1.101	1.032	1.100	1.032	1.019	0.048	1.016	0.048	1.557	1.159	1.023	0.048	5.019	1.101
	143.57	25014	1.981	0.335	2.290	4.955	2.283	4.955	1.981	0.335	1.985	0.335	6.512	16.346	1.980	0.335	5.285	5.390
neos-934278	3600.00	1175	1.000	0.993	1.000	0.991	1.000	0.993	1.000	0.993	1.000	0.993	1.000	1.000	1.000	0.335	1.000	0.340
	3600.00	267	1.000	1.005	1.000	1.005	1.000	1.005	1.000	1.005	1.000	1.014	1.000	1.005	1.000	1.101	1.000	1.153
	3600.00	266	1.000	1.000	1.000	1.003	1.000	1.008	1.000	0.989	1.000	1.000	1.000	1.008	1.000	1.124	1.000	1.182
	3600.00	399	1.000	1.020	1.000	0.953	1.000	1.016	1.000	1.014	1.000	1.014	1.000	1.012	1.000	0.835	1.000	0.857
	3600.00	523	1.000	1.002	1.000	0.998	1.000	1.003	1.000	1.000	1.000	1.003	1.000	1.000	1.000	0.680	1.000	0.717
neos18	24.17	1378	0.363	0.121	0.993	1.000	0.993	1.000	0.981	1.047	0.997	1.000	0.575	0.287	0.354	0.121	0.915	0.918
	21.26	1604	1.162	0.653	1.403	1.457	1.028	1.199	0.996	1.000	1.000	1.000	1.429	1.072	0.967	0.467	1.031	1.431
	23.36	1646	1.229	0.684	0.993	1.000	0.994	1.000	0.998	1.000	0.995	1.000	0.840	0.248	1.218	0.684	0.914	0.824
	19.21	996	0.812	0.443	0.840	0.849	0.997	1.000	0.997	1.000	1.021	1.000	0.789	0.366	0.810	0.443	1.059	1.091
	22.07	1049	0.987	0.925	0.855	0.634	1.004	1.000	1.001	1.000	0.997	1.000	0.906	0.657	0.948	0.795	1.174	2.047
ns1208400	2439.01	780	0.385	0.219	0.396	0.178	0.146	0.062	0.994	1.000	0.485	0.271	0.296	0.090	0.485	0.271	0.482	0.404
	1985.63	5751	1.378	1.450	0.948	0.886	1.673	0.733	1.003	1.000	1.395	1.451	0.415	0.213	1.390	1.450	0.486	0.380
	1467.85	1351	0.363	0.355	0.810	0.504	0.415	0.283	1.000	1.000	0.368	0.355	0.713	0.156	0.367	0.355	0.673	0.768
	3600.00	2617	0.071	0.122	0.493	1.157	0.274	0.476	0.309	0.607	0.070	0.122	0.244	0.332	0.071	0.122	0.147	0.400
	2388.79	3984	0.551	0.313	0.915	1.198	0.649	0.464	0.999	1.000	0.054	0.069	0.434	0.229	0.051	0.068	0.533	0.656
ns1688347	55.09	818	0.939	0.975	0.998	1.013	1.209	1.666	1.084	1.283	1.040	0.935	1.000	1.000	0.999	1.103	1.038	0.935
	45.64	422	0.931	0.518	0.996	1.000	1.062	0.892	0.986	0.969	1.391	1.591	0.998	1.000	1.058	0.839	1.396	1.591
	106.90	1690	1.509	1.626	1.000	1.000	0.999	0.911	1.152	0.855	1.478	1.240	1.001	1.000	0.884	0.571	1.492	1.240
	139.57	1219	0.957	1.024	1.018	1.095	0.994	1.000	0.945	1.088	1.130	0.999	0.998	1.000	0.682	0.329	0.771	0.500
	55.82	414	0.924	0.591	0.999	1.000	0.719	0.608	0.996	1.000	1.169	1.244	1.002	1.000	0.850	0.550	1.169	1.244
ns1830653	132.81	5720	1.002	1.000	1.000	1.000	0.810	0.829	1.001	1.000	1.130	1.028	0.998	1.000	1.142	0.663	1.132	1.028
	123.49	5512	1.003	1.000	1.002	1.000	1.248	1.054	1.001	1.000	1.221	1.285	1.003	1.000	1.548	1.133	1.223	1.285
	161.13	10467	0.999	1.000	1.002	1.000	0.953	1.090	1.000	1.000	1.078	0.985	0.803	0.886	1.229	0.479	1.093	0.985
	143.05	7519	1.001	1.000	0.826	0.713	1.065	1.011	0.999	1.000	1.108	1.177	0.785	0.589	1.193	0.833	1.104	1.177
	146.20	5836	0.999	1.000	0.999	1.000	0.779	0.639	0.999	1.000	1.158	0.738	0.851	0.746	1.028	0.406	1.162	0.738
nsrand-ipx	422.05	50124	0.995	1.000	1.011	1.000	0.999	1.000	0.996	1.000	1.002	1.000	1.302	0.946	6.327	0.938	1.897	1.892
	643.65	86869	0.992	1.000	1.002	1.000	0.996	1.000	0.998	1.000	0.996	1.000	1.275	0.923	4.263	0.891	0.503	0.378
	568.28	56575	1.002	1.000	1.011	1.000	1.001	1.000	1.000	1.000	1.002	1.000	1.063	0.996	6.326	0.715	1.101	1.016
	582.50	61056	1.000	1.000	0.998	1.000	1.001	1.000	1.003	1.000	1.008	1.000	2.283	0.885	5.237	1.122	0.898	0.893
	367.03	38012	0.991	1.000	0.995	1.000	0.993	1.000	0.999	1.000	0.995	1.000	1.142	1.032	9.405	1.958	1.867	2.088
opt1217	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
p0033	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
p0201	0.55	5	0.994	1.000	1.006	1.000	0.987	1.000	1.000	1.000	0.981	1.000	1.000	1.000	0.968	1.057	0.968	1.057
	0.71	9	1.018	1.000	1.006	1.000	1.012	1.000	1.012	1.000	1.000	1.000	1.023	1.000	0.924	0.982	0.924	0.982
	1.59	5	1.019	1.000	1.008	1.000	1.000	1.000	1.000	1.000	1.015	1.000	1.023	1.000	0.749	1.210	0.819	2.419
	0.90	13	1.000	1.000	0.995	1.000	1.011	1.000	1.005	1.000	1.005	1.000	0.989	1.000	1.058	1.088	0.911	1.088
	0.91	59	0.990	1.000	0.990	1.000	1.000	1.000	0.990	1.000	0.995	1.000	0.995	1.000	0.890	0.761	0.812	0.761
p0282	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	3	1.000	1.000	1.000	1.000	1.000	1.000	1.007	1.000	1.000	1.000	1.013	1.000	1.000	0.990	1.033	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
p0548	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
p2756	2.46	5	1.014	1.000	1.014	1.000	1.000	1.000	1.006	1.000	1.012	1.000	1.014	1.000	0.942	1.067	0.861	0.990
	3.33	5	1.000	1.000	1.005	1.000	1.005	1.000	1.005	1.000	1.002	1.000	1.012	1.000	1.102	1.010	0.998	0.990
	2.16	3	0.991	1.000	0.987	1.000	0.994	1.000	0.981	1.000	0.991	1.000	0.978	1.000	1.073	1.010	0.981	1.000
	2.60	3	0.983	1.000	0.994	1.000	0.994	1.000	0.983	1.000	0.983	1.000	0.978	1.000	0.969	1.010	0.964	1.019
	1.22	2	1.027	1.000	1.041	1.000	1.023	1.000	1.027	1.000	1.023	1.000	1.014	1.000	1.077	1.010	1.063	1.010
protfold	3600.00	4825	1.000	0.685	1.000	1.007	1.000	1.009	1.000	0.990	1.000	1.008	1.000	1.002	1.000	0.575	1.000	0.530
	3600.00	8262	1.000	0.315	1.000	1.001	1.000	1.011	1.000	0.994	1.000	0.997	1.000	1.001	1.000	0.303	1.000	0.289
	3600.00	5818	1.000	0.481	1.000	0.999	1.000	0.997	1.000	0.999	1.000	0.995	1.000	0.998	1.000	0.558	1.000	0.358
	3600.00	7269	1.000	0.262	1.000	1.002	1.000	1.001	1.000	1.001	1.000	1.006	1.000	1.001	1.000	0.264	1.000	0.231
	3600.00	4748	1.000	0.344	1.000	0.998	1.000	0.848	1.000	1.000	1.000	1.000	1.000	1.003	1.000	0.415	1.000	0.299
pw-myciel4	3600.00	519342	1.000	0.695	1.000	0.895	1.000	0.999	0.662	0.587	1.000	1.000	1.000	0.858	1.000	1.087	1.000	0.868
	3600.00	442617	1.000	0.826	0.800	0.789	1.000	0.999	0.932	0.954	1.000	1.000	1.000	0.858	1.000	0.636	1.000	1.040
	3600.00	389396	1.000	0.826	1.000	1.146	1.000	0.994	1.000	0.974	1.000	0.992	1.000	1.053	1.000	0.780	1.000	0.910
	2920.77	395762	1.232	0.905	1.232	0.938	1.006	1.000	1.232	0.872	1.000	1.000	1.232	0.943	1.232	1.125	1.232	0.984
	3600.00	405310	0.848	0.934	1.000	0.991	1.000	1.000	0.939	1.044	1.000	1.001	1.000	1.031	0.971	0.834	1.000	1.204
qnet1	4.52	33	1.004	1.000	1.005	1.000	1.013	1.000	0.998	1.000	1.005	1.000	1.007	1.000	0.455	0.797	0.455	0.797
	3.09	3	1.007	1.000	1.005	1.000	1.005	1.000	1.010	1.000	1.010	1.000	0.998	1.000	1.086	0.990	0.804	1.000
	3.94	29	0.996	1.000	1.002	1.000	0.994	1.000	1.004	1.000	1.000	1.000	1.002	1.000	1.138	1.000	0.949	1.000
	2.23	3	1.009	1.000	1.000	1.000	1.006	1.000	1.003	1.000	1.009	1.000	1.000	1.000	1.161	1.087	1.071	1.010
	3.69	34	1.004	1.000	0.989	1.000	0.998	1.000	0.994	1.000	0.989	1.000	1.000	1.000	1.198	0.970	0.887	1.022
qnet1_o	3.93	27	0.988	1.000	0.992	1.000	0.996	1.000	1.002	1.000	0.996	1.000	0.988	1.000	0.919	0.929	0.836	0.929
	3.35	24	1.007	1.000	1.005	1.000	1.000	1.000	1.002	1.000	0.998	1.000	1.009	1.000	1.094	0.992	0.897	0.992
	3.37	15	1.005	1.000	1.009	1.000	0.995	1.000	1.011	1.000	1.014	1.000	1.002	1.000	0.712	0.904	0.616	0.887
	1.66	2	1.004	1.000	0.996	1.000	1.000	1.000	0.992	1.000	1.004	1.000	1.008	1.000	1.143	1.000	0.996	1.000
	1.83	3	1.004	1.000	1.007	1.000	1.007	1.000	1.000	1.000	1.007	1.000	1.004	1.000	1.085	1.000	1.018	1.000
reblock67	222.59	49530	1.002	1.000	0.999	1.000	1.007	1.000	0.999	1.000	0.997	1.000	0.995	1.000	2.701	1.225	0.803	0.598
	161.89	40625	1.000	1.000	0.997	1.000	1.000	1.000	0.996	1.000	0.998	1.000	1.000	1.000	2.270	0.943	1.272	1.214
	194.41	42542	1.001	1.000	1.002	1.000	0.857	0.853	0.997	1.000	1.003	1.000	1.001	1.000	2.191	0.951	0.902	0.876
	199.06	52613	1.006	1.000	1.003	1.000	0.992	1.032	1.006	1.000	1.005	1.000	1.001	1.000	3.355	1.638	0.868	0.767
	239.84	57566	0.997	1.000	0.997	1.000	0.994	1.000	0.997	1.000	0.996	1.000	0.997	1.000	1.707	1.170	0.716	0.795
rmine6	971.86	111047	1.000	1.000	1.006	1.000	1.002	1.000	1.000	1.000	1.001	1.000	0.999	1.000	1.724	1.068	1.441	0.906
	883.48	89213	0.999	1.000	1.000	1.000	0.990	1.000	0.999	1.000	1.002	1.000	1.002	1.000	2.249	1.574	0.945	0.921
	775.32	91530	1.002	1.000	1.002	1.000	1.000	1.000	1.001	1.000	1.002	1.000	1.000	1.000	2.090	1.268	1.037	1.014
	740.58	85429	1.002	1.000	1.000	1.000	0.999	1.000	1.005	1.000	0.999	1.000	0.992	1.000	1.817	1.249	1.310	1.072
	771.48	76017	1.002	1.000	1.001	1.000	1.002	1.000	1.001	1.000	0.998	1.000	1.001	1.000	1.581	1.168	1.134	1.144
rococoC10-001000	601.83	46983	1.003	1.000	1.000	1.000	0.998	1.000	0.998	1.000	0.998	1.000	1.000	1.000	2.365	1.898	2.192	1.817
	833.40	75156	1.003	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.002	1.000	2.401	1.509	1.679	1.377
	517.62	37178	0.998	1.000	1.002	1.000	1.003	1.000	1.092	1.105	1.002	1.000	1.001	1.000	1.872	1.710	1.428	1.458
	892.04	51410	1.001	1.000	1.002	1.000	0.999	1.000	1.001	1.000	1.003	1.000	1.003	1.000	1.737	1.994	1.076	1.344
	1047.39	66016	1.001	1.000	1.000	1.000	1.001	1.000	1.001	1.000	1.003	1.000	1.003	1.000	2.255	1.863	1.367	1.434
seymour	3600.00	102303	1.000	0.996	1.000	0.993	1.000	0.997	1.000	0.999	1.000	0.997	1.000	1.001	1.000	0.717	1.000	0.768
	3600.00	120875	1.000	0.999	1.000	1.001	1.000	1.005	1.000	0.997	1.000	1.005	1.000	1.001	1.000	0.626	1.000	0.656
	3600.00	123371	1.000	1.006	1.000	1.006	1.000	1.008	1.000	1.010	1.000	1.010	1.000	1.006	1.000	0.737	1.000	0.791
	3600.00	107935	1.000	1.005	1.000	1.003	1.000	1.004	1.000	1.003	1.000	1.002	1.000	1.011	1.000	0.836	1.000	0.943
	3600.00	97288	1.000	1.003	1.000	1.001	1.000	1.006	1.000	1.001	1.000	1.003	1.000	1.002	1.000	1.002	1.000	1.069
sp98ir	57.90	3948	1.003	1.000	1.009	1.000	1.003	1.000	1.020	1.000	1.010	1.000	1.008	1.000	2.980	1.355	1.637	1.467
	65.87	4178	1.001	1.000	1.003	1.000	1.003	1.000	1.003	1.000	1.004	1.000	1.007	1.000	2.428	1.260	1.115	1.144
	67.64	4735	1.004	1.000	1.002	1.000	1.001	1.000	1.003	1.000	1.001	1.000	1.008	1.000	3.056	1.357	1.306	1.249
	72.68	4603	0.997	1.000	1.002	1.000	1.003	1.000	0.999	1.000	1.001	1.000	1.005	1.000	2.275	0.892	0.890	0.882
	70.05	4780	0.997	1.000	0.997	1.000	0.995	1.000	0.999	1.000	1.005	1.000	1.004	1.000	2.109	1.134	1.088	1.248
stein27	1.12	2749	1.024	1.000	0.995	1.000	1.028	1.000	1.024	1.000	1.033	1.000	0.991	1.000	1.491	1.081	1.094	1.081
	1.19	2699	1.018	1.000	1.009	1.000	1.005	1.000	1.009	1.000	1.027	1.000	1.023	1.000	1.502	1.025	1.014	1.009
	1.25	2889	0.973	1.000	0.991	1.000	0.982	1.000	1.000	1.000	0.991	1.000	0.964	1.000	1.382	0.990	0.978	0.990
	1.14	2715	0.995	1.056	0.972	1.000	0.967	1.000	0.981	1.000	0.972	1.000	0.981	1.000	1.486	1.056	0.958	1.056
	1.13	2911	0.977	0.972	1.000	1.000	0.986	1.000	1.000	1.000	0.981	1.000	0.977	1.000	1.394	0.951	0.977	0.972
stein45	12.80	40723	0.985	0.962	0.995	1.000	0.981	1.000	0.986	1.000	0.979	1.000	0.981	1.000	1.630	0.971	0.964	0.916
	12.94	39012	0.968	0.995	0.996	1.000	0.984	1.000	0.972	1.000	0.989	1.000	0.972	1.000	1.698	0.966	0.935	0.976
	13.04	41930	1.000	0.985	0.999	1.000	0.988	1.000	0.994	1.000	0.999	1.000	1.006	1.000	1.628	0.931	0.994	0.975
	12.20	37543	1.022	1.015	0.993	1.000	1.001	1.000	0.994	1.000	0.992	1.000	0.992	1.000	1.665	1.055	1.047	1.015
	13.29	40503	0.981	1.006	0.994	1.000	0.978	1.000	0.992	1.000	0.992	1.000	0.999	1.000	1.666	0.984	0.985	1.006
tanglegram2	4.09	3	1.004	1.000	1.010	1.000	1.010	1.000	1.012	1.000	1.006	1.000	1.014	1.000	1.026	1.000	1.020	1.000
	3.85	3	1.010	1.000	1.008	1.000	1.010	1.000	1.008	1.000	1.008	1.000	1.002	1.000	1.029	1.000	1.023	1.000
	4.25	3	0.990	1.000	1.000	1.000	0.998	1.000	1.008	1.000	1.008	1.000	1.008	1.000	1.011	1.000	1.023	1.000
	4.11	3	1.002	1.000	1.002	1.000	1.000	1.000	0.992	1.000	0.992	1.000	0.998	1.000	1.016	1.000	1.010	1.000
	4.23	3	0.994	1.000	1.000	1.000	0.985	1.000	0.990	1.000	0.998	1.000	0.994	1.000	1.010	1.000	1.011	1.000

Table 5. Table 5 : Detailed computational results on MMM-IP with five random seeds. Relative changes by at least 5 % percent 5 5\% are highlighted in bold and blue ( improvement ) or italic and red ( deterioration ).

	default		degeneracy + leaves		leaves + obj		all6criterion
Instance	time	nodes	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ
10teams	2.14	1	1.293	1.000	1.022	1.000	1.248	1.000
	20.00	1367	0.244	0.072	1.230	1.001	0.246	0.072
	15.57	287	1.636	2.729	0.998	1.000	1.508	1.835
	4.16	5	2.723	1.219	1.006	1.000	2.725	1.219
	4.94	5	0.912	1.000	1.012	1.000	0.911	1.000
30n20b8	314.13	195	0.604	0.553	0.992	1.000	0.344	0.346
	157.33	71	0.711	0.596	1.000	1.000	0.808	0.596
	87.25	4	1.001	1.000	0.997	1.000	1.193	1.000
	213.06	102	1.202	1.312	0.976	0.950	1.048	0.782
	470.09	290	0.813	0.723	1.004	1.000	0.243	0.421
acc-tight5	125.35	802	1.093	1.171	0.426	0.302	1.087	1.171
	84.38	508	1.786	2.650	0.426	0.252	1.790	2.650
	50.45	195	0.974	0.929	1.740	1.966	0.978	0.929
	85.88	545	0.339	0.189	0.420	0.274	0.343	0.189
	73.38	627	0.418	0.161	1.969	2.048	0.418	0.161
air03	1.70	1	1.037	1.000	1.037	1.000	1.007	1.000
	1.71	1	1.004	1.000	1.007	1.000	1.004	1.000
	1.74	1	0.996	1.000	1.004	1.000	0.993	1.000
	1.75	1	1.000	1.000	0.993	1.000	1.002	1.000
	1.75	1	1.000	1.000	1.000	1.000	0.996	1.000
air04	49.46	110	1.015	1.000	1.003	1.000	1.007	1.000
	36.09	39	1.005	1.000	0.999	1.000	1.003	1.000
	52.88	118	0.997	1.000	0.999	1.000	0.989	1.000
	36.56	31	1.010	1.000	1.015	1.000	1.001	1.000
	48.35	122	0.998	1.000	1.007	1.000	1.001	1.000
air05	24.82	189	0.997	1.000	1.001	1.000	0.991	1.000
	26.54	239	1.004	1.000	1.001	1.000	1.009	1.000
	31.37	438	1.001	1.000	1.000	1.000	0.997	1.000
	27.36	283	1.005	1.000	1.004	1.000	1.010	1.000
	27.14	279	1.001	1.000	0.999	1.000	1.000	1.000
blend2	0.69	714	1.036	1.000	1.047	1.000	1.036	1.000
	1.08	973	0.995	1.000	1.010	1.000	1.010	1.000
	1.05	912	1.000	1.000	0.985	1.000	0.990	1.000
	0.67	543	0.982	1.000	1.000	1.000	1.006	1.000
	0.66	709	1.042	1.000	1.048	1.000	1.042	1.000
bley_xl1	164.49	1	1.009	1.000	1.000	1.000	1.007	1.000
	187.04	3	1.013	1.146	1.007	1.000	1.017	1.146
	182.51	13	0.887	0.903	1.006	1.000	0.896	0.903
	133.90	1	1.002	1.000	1.011	1.000	1.004	1.000
	135.56	1	1.012	1.000	1.008	1.000	1.007	1.000
bnatt350	378.09	3383	0.052	0.030	0.107	0.054	0.052	0.030
	521.76	4947	0.033	0.022	0.292	0.147	0.033	0.022
	397.10	4029	0.047	0.026	0.123	0.053	0.048	0.026
	155.46	1495	0.109	0.067	2.608	2.974	0.109	0.067
	994.16	10057	0.044	0.021	0.330	0.259	0.044	0.021
cap6000	2.69	1830	1.011	1.000	1.011	1.000	0.997	1.000
	2.67	1837	0.995	1.000	0.995	1.000	1.000	1.000
	2.61	1959	0.994	1.000	1.017	1.000	1.014	1.000
	3.17	1924	1.005	1.000	1.000	1.000	0.998	1.000
	2.82	1762	1.008	1.000	1.005	1.000	0.997	1.000
cov1075	3600.00	1450527	0.991	1.189	1.000	0.996	0.991	1.189
	3600.00	1790225	1.000	0.974	1.000	1.002	1.000	0.978
	3600.00	1531529	1.000	1.179	1.000	1.000	1.000	1.151
	3600.00	1798178	1.000	0.984	1.000	1.008	1.000	0.984
	3600.00	1204894	1.000	1.507	1.000	0.998	1.000	1.505
csched010	3600.00	237077	1.000	0.961	1.000	0.963	0.812	0.726
	3600.00	202876	1.000	0.863	1.000	0.861	1.000	1.145
	2891.69	177527	1.145	1.206	1.144	1.206	1.245	1.177
	3600.00	238611	0.989	0.820	0.986	0.820	1.000	0.927
	3168.37	173932	1.136	1.300	1.136	1.304	1.136	1.369
eil33-2	69.44	583	1.013	1.000	1.007	1.000	1.006	1.000
	81.92	739	1.000	1.000	0.998	1.000	1.003	1.000
	84.87	847	1.004	1.000	1.005	1.000	1.001	1.000
	73.60	543	1.005	1.000	1.003	1.000	1.005	1.000
	70.01	583	1.001	1.000	1.001	1.000	0.999	1.000
eilB101	378.99	10582	1.002	1.000	1.015	1.000	0.993	1.000
	365.27	8427	0.994	1.000	0.994	1.000	0.996	1.000
	401.63	9358	1.004	1.000	0.996	1.000	1.004	1.000
	397.46	11465	1.000	1.000	1.000	1.000	1.001	1.000
	365.79	10006	1.005	1.000	0.997	1.000	0.999	1.000
enigma	0.50	809	1.000	0.825	1.000	0.998	1.000	0.542
	0.50	449	1.000	0.497	1.000	1.000	1.000	0.470
	0.50	541	1.000	0.254	1.000	0.977	1.000	0.237
	0.50	679	1.060	0.537	1.000	1.000	1.000	0.151
	0.50	445	1.000	0.543	1.000	1.006	1.033	0.543
enlight13	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
enlight14	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
fiber	1.00	2	1.005	1.000	1.025	1.000	1.005	1.000
	1.42	4	1.000	1.000	0.988	1.000	1.000	1.000
	1.74	4	0.996	1.000	1.004	1.000	1.007	1.000
	1.53	4	1.000	1.000	1.008	1.000	1.000	1.000
	1.00	3	1.000	1.000	0.990	1.000	0.990	1.000
flugpl	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
gt2	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
harp2	691.56	1491878	1.008	1.000	0.999	1.000	1.005	1.000
	1159.66	2857677	0.998	1.000	0.998	1.000	0.997	1.000
	336.73	595854	0.999	1.000	1.006	1.000	0.999	1.000
	937.54	2046181	0.999	1.000	1.002	1.000	1.002	1.000
	425.68	923780	1.013	1.000	1.007	1.000	1.008	1.000
iis-100-0-cov	629.92	103302	0.998	1.000	1.021	1.000	0.992	1.000
	545.20	92393	0.995	1.000	0.997	1.000	1.001	1.000
	520.78	81639	1.001	1.000	1.000	1.000	1.000	1.000
	506.53	84800	0.999	1.000	0.995	1.000	1.000	1.000
	544.56	92313	1.007	1.000	1.003	1.000	1.004	1.000
iis-bupa-cov	1915.72	157619	1.000	1.000	1.003	1.000	1.001	1.000
	1944.83	164079	0.999	1.000	0.996	1.000	0.999	1.000
	1881.83	156888	0.991	1.000	0.992	1.000	1.008	1.000
	1959.19	161969	0.994	1.000	0.999	1.000	0.997	1.000
	1820.52	158284	0.994	1.000	0.993	1.000	1.014	1.000
iis-pima-cov	293.79	4824	1.004	1.000	0.999	1.000	0.999	1.000
	437.19	12189	1.007	1.000	1.001	1.000	1.013	1.000
	246.99	4668	1.002	1.000	1.000	1.000	1.002	1.000
	284.88	6607	1.002	1.000	1.000	1.000	1.003	1.000
	556.78	17821	0.997	1.000	0.997	1.000	1.001	1.000
l152lav	5.51	23	1.011	1.000	1.014	1.000	0.992	1.000
	1.67	31	0.989	1.000	0.993	1.000	1.000	1.000
	2.93	50	0.995	1.000	1.003	1.000	1.000	1.000
	2.34	20	0.985	1.000	0.994	1.000	0.991	1.000
	3.02	52	1.000	1.000	1.012	1.000	1.002	1.000
lectsched-4-obj	27.04	805	0.750	0.215	1.052	0.838	0.749	0.215
	23.16	796	1.193	0.288	1.008	0.823	1.198	0.288
	86.08	2419	0.285	0.206	0.626	0.461	0.265	0.121
	26.83	1231	0.327	0.082	0.933	0.540	0.331	0.082
	22.73	745	0.903	0.253	0.992	0.954	0.911	0.253
lseu	0.75	268	0.994	1.000	1.029	1.000	1.006	1.000
	0.50	4	1.000	1.000	1.000	1.000	1.000	1.000
	0.51	205	0.993	1.000	0.993	1.000	1.000	1.000
	0.50	91	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	191	1.000	1.000	1.000	1.000	1.000	1.000
m100n500k4r1	3600.00	2170459	1.000	1.118	1.000	1.002	1.000	0.781
	3600.00	2278212	1.000	1.035	1.000	0.996	1.000	0.783
	3600.00	2617105	1.000	0.884	1.000	1.000	1.000	0.560
	3600.00	2309660	1.000	0.935	1.000	0.998	1.000	0.638
	3600.00	2245563	1.000	0.834	1.000	0.996	1.000	0.647
macrophage	125.50	5163	1.000	1.000	1.006	1.000	1.001	1.000
	157.18	7241	0.998	1.000	0.998	1.000	0.999	1.000
	209.11	9131	1.000	1.000	1.000	1.000	1.001	1.000
	162.53	7927	0.998	1.000	1.000	1.000	1.002	1.000
	121.37	5239	0.995	1.000	0.996	1.000	0.997	1.000
manna81	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
markshare1	3600.00	19093205	1.000	0.927	1.000	0.997	1.000	0.924
	3600.00	20203360	1.000	0.859	1.000	0.998	1.000	0.856
	3600.00	19520584	1.000	0.919	1.000	0.999	1.000	0.918
	3600.00	20427204	1.000	0.901	1.000	1.000	1.000	0.903
	3600.00	21513467	1.000	0.795	1.000	1.000	1.000	0.796
markshare2	3600.00	4072360	1.000	0.843	1.000	0.995	1.000	0.847
	3600.00	5640455	1.000	0.699	1.000	1.000	1.000	0.699
	3600.00	3582521	1.000	1.025	1.000	0.998	1.000	1.023
	3600.00	4239467	1.000	1.067	1.000	1.006	1.000	1.070
	3600.00	5142521	1.000	0.828	1.000	1.004	1.000	0.836
mcsched	285.92	13689	0.999	1.000	1.000	1.000	0.997	1.000
	271.88	11373	0.999	1.000	0.999	1.000	1.001	1.000
	233.97	10059	1.003	1.000	0.995	1.000	1.000	1.000
	286.25	12496	0.995	1.000	1.000	1.000	0.997	1.000
	270.69	10373	1.004	1.000	0.999	1.000	1.000	1.000
mine-166-5	58.69	1569	0.995	1.000	1.001	1.000	0.998	1.000
	57.78	1191	0.993	1.000	0.994	1.000	0.992	1.000
	46.45	2244	0.997	1.000	1.004	1.000	1.005	1.000
	64.93	139	0.998	1.000	1.007	1.000	1.000	1.000
	42.28	2863	1.001	1.000	1.001	1.000	1.000	1.000
mine-90-10	276.07	47643	1.162	1.090	1.162	1.090	1.157	1.090
	125.32	22195	0.999	1.000	0.997	1.000	1.002	1.000
	173.25	25979	1.001	1.000	0.999	1.000	1.000	1.000
	500.83	87496	1.322	1.061	1.321	1.061	1.322	1.061
	236.18	32520	0.907	1.242	0.908	1.242	0.904	1.242
misc03	1.39	398	1.008	1.000	0.996	1.000	0.992	1.000
	0.71	15	0.994	1.000	1.018	1.000	1.094	1.226
	0.70	19	1.006	1.000	1.006	1.000	1.200	1.370
	1.00	111	1.000	1.000	0.990	1.000	0.995	1.000
	0.66	15	1.018	1.000	0.994	1.000	1.193	1.609
misc07	38.05	28289	0.998	1.000	1.026	1.000	1.090	1.188
	36.66	29782	0.999	1.000	0.999	1.000	0.999	1.000
	21.56	17632	0.999	1.000	0.996	1.000	0.994	1.000
	45.95	38975	1.006	1.000	1.007	1.000	1.004	1.000
	30.94	29476	0.993	1.000	0.996	1.000	1.088	0.937
mitre	10.20	1	1.376	1.000	0.992	1.000	1.354	1.000
	10.28	1	1.444	1.000	1.009	1.000	1.473	1.000
	10.94	1	1.380	1.000	0.960	1.000	1.391	1.000
	10.45	1	1.401	1.000	0.994	1.000	1.399	1.000
	13.12	1	1.038	1.000	0.991	1.000	1.051	1.000
mod008	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.54	2	1.000	1.000	1.000	1.000	1.000	1.000
	0.52	2	1.000	1.000	0.993	1.000	1.000	1.000
	0.52	1	0.993	1.000	0.993	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
mod010	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.52	2	1.000	1.000	0.993	1.000	0.993	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
mzzv11	332.43	2022	0.999	1.000	0.994	1.000	1.000	1.000
	229.41	959	1.002	1.000	1.003	1.000	1.001	1.000
	310.05	1421	0.998	1.000	1.000	1.000	0.998	1.000
	327.03	1579	0.994	1.000	0.995	1.000	0.994	1.000
	186.14	627	1.001	1.000	1.002	1.000	1.000	1.000
mzzv42z	152.40	265	0.994	1.000	0.992	1.000	0.993	1.000
	131.25	125	0.993	1.000	0.993	1.000	1.105	1.053
	197.62	200	1.004	1.000	1.006	1.000	1.024	0.923
	213.83	456	0.999	1.000	1.001	1.000	0.933	0.736
	167.65	300	1.000	1.000	1.000	1.000	1.004	1.000
neos-1109824	37.52	1796	0.998	1.000	0.995	1.000	1.662	2.859
	46.67	3438	1.000	1.000	0.998	1.000	1.009	0.969
	35.16	1913	0.999	1.000	0.998	1.000	1.088	0.966
	77.65	8290	1.003	1.000	1.003	1.000	0.696	0.534
	39.64	2247	0.993	1.000	0.991	1.000	1.156	1.084
neos-1337307	3600.00	302483	1.000	1.004	1.000	1.006	1.000	1.178
	3600.00	269510	1.000	1.001	1.000	1.003	1.000	1.084
	3600.00	348643	1.000	1.002	1.000	1.000	1.000	1.012
	3600.00	300587	1.000	1.011	1.000	0.993	1.000	1.131
	3600.00	367746	1.000	1.002	1.000	1.007	1.000	0.748
neos-1601936	2034.60	1453	0.996	1.000	1.009	1.000	1.769	2.100
	3600.00	2849	1.000	1.003	1.000	1.000	1.000	0.377
	1348.55	2661	1.000	1.000	1.000	1.000	2.668	0.581
	3600.00	6919	1.000	0.999	1.000	0.999	1.000	0.584
	3600.00	6139	1.000	1.000	1.000	1.010	0.993	0.519
neos-686190	83.42	4906	1.003	1.000	0.997	1.000	0.995	1.000
	86.19	6047	1.001	1.000	1.000	1.000	1.002	1.000
	114.74	9156	1.000	1.000	1.001	1.000	1.000	1.000
	116.53	9998	1.008	1.000	1.002	1.000	0.999	1.000
	121.04	9806	1.004	1.000	1.003	1.000	0.998	1.000
neos-849702	114.99	2031	1.820	0.595	1.817	0.595	1.804	0.595
	353.12	17540	2.901	0.364	2.924	0.364	2.903	0.364
	357.57	29405	2.792	0.359	2.795	0.359	2.803	0.359
	261.49	16226	1.025	0.048	1.027	0.048	1.025	0.048
	143.57	4968	1.988	0.335	1.991	0.335	1.994	0.335
neos-934278	3600.00	1184	1.000	0.991	1.000	0.993	1.000	0.993
	3600.00	265	1.000	1.014	1.000	1.011	1.000	1.003
	3600.00	263	1.000	1.000	1.000	1.000	1.000	0.989
	3600.00	391	1.000	1.014	1.000	1.014	1.000	0.963
	3600.00	521	1.000	1.002	1.000	0.997	1.000	0.998
neos18	24.17	1378	0.360	0.121	1.004	1.047	0.359	0.121
	21.26	1321	1.154	0.653	1.027	1.199	0.971	0.467
	23.36	1646	1.227	0.684	1.000	1.000	1.229	0.684
	19.21	996	0.808	0.443	1.003	1.000	0.805	0.443
	22.07	1049	0.989	0.925	0.997	1.000	0.951	0.795
ns1208400	2439.01	14168	0.485	0.271	0.146	0.062	0.484	0.271
	1985.63	7882	1.390	1.450	1.673	0.733	1.392	1.450
	1467.85	5026	0.367	0.355	0.416	0.283	0.367	0.355
	3600.00	5611	0.071	0.122	0.274	0.476	0.071	0.122
	2388.79	8711	0.050	0.068	0.650	0.464	0.051	0.068
ns1688347	55.09	451	0.942	0.975	1.215	1.666	0.940	0.975
	45.64	485	0.930	0.518	1.058	0.892	0.929	0.518
	106.90	1865	1.530	1.438	1.160	0.638	1.330	1.366
	139.57	1219	0.956	1.024	0.942	1.088	1.205	1.089
	55.82	745	0.858	0.527	0.724	0.608	0.862	0.527
ns1830653	132.81	6918	0.813	0.829	0.811	0.829	0.980	0.788
	123.49	5226	1.250	1.054	1.251	1.054	1.515	1.580
	161.13	9597	0.948	1.090	0.946	1.090	0.863	0.656
	143.05	7435	1.065	1.011	1.069	1.011	0.901	0.816
	146.20	9194	0.783	0.639	0.782	0.639	1.160	0.740
nsrand-ipx	422.05	50124	0.998	1.000	1.003	1.000	1.297	0.946
	643.65	86869	0.999	1.000	0.999	1.000	1.274	0.923
	568.28	56575	1.004	1.000	1.006	1.000	1.058	0.996
	582.50	61056	0.996	1.000	1.006	1.000	2.291	0.885
	367.03	38012	0.995	1.000	0.994	1.000	1.147	1.032
opt1217	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
p0033	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
p0201	0.55	5	0.981	1.000	0.987	1.000	0.987	1.000
	0.71	9	1.018	1.000	1.023	1.000	0.994	1.000
	1.59	5	1.015	1.000	1.015	1.000	1.004	1.000
	0.90	13	1.005	1.000	1.000	1.000	1.005	1.000
	0.91	59	0.984	1.000	0.984	1.000	0.984	1.000
p0282	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	3	1.007	1.000	1.000	1.000	1.007	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000
p0548	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000
p2756	2.46	5	1.006	1.000	1.012	1.000	1.000	1.000
	3.33	5	1.005	1.000	1.018	1.000	0.995	1.000
	2.16	3	0.994	1.000	0.987	1.000	0.994	1.000
	2.60	3	0.994	1.000	0.992	1.000	0.975	1.000
	1.22	2	1.036	1.000	1.027	1.000	1.041	1.000
protfold	3600.00	4783	1.000	0.685	1.000	1.008	1.000	0.685
	3600.00	8172	1.000	0.314	1.000	1.003	1.000	0.315
	3600.00	5838	1.000	0.478	1.000	1.004	1.000	0.480
	3600.00	7263	1.000	0.262	1.000	1.000	1.000	0.263
	3600.00	5620	1.000	0.348	1.000	0.844	1.000	0.345
pw-myciel4	3600.00	519843	1.000	0.696	0.656	0.587	1.000	0.901
	3600.00	443116	1.000	0.824	0.930	0.954	1.000	0.688
	3600.00	391642	1.000	0.830	1.000	0.968	1.000	0.969
	2920.77	395762	1.232	0.907	1.232	0.872	1.232	1.067
	3600.00	405235	0.847	0.934	0.938	1.044	1.000	0.856
qnet1	4.52	33	1.004	1.000	1.009	1.000	1.007	1.000
	3.09	3	1.000	1.000	1.002	1.000	1.000	1.000
	3.94	29	1.002	1.000	1.004	1.000	1.000	1.000
	2.23	3	0.994	1.000	0.994	1.000	1.006	1.000
	3.69	34	0.998	1.000	0.998	1.000	0.989	1.000
qnet1_o	3.93	27	1.008	1.000	0.986	1.000	1.004	1.000
	3.35	24	1.009	1.000	1.005	1.000	1.002	1.000
	3.37	15	1.000	1.000	1.007	1.000	1.000	1.000
	1.66	2	1.008	1.000	1.004	1.000	0.992	1.000
	1.83	3	1.004	1.000	1.000	1.000	1.004	1.000
reblock67	222.59	49530	0.997	1.000	0.997	1.000	1.001	1.000
	161.89	40625	0.993	1.000	1.002	1.000	1.000	1.000
	194.41	49882	0.858	0.853	0.866	0.853	0.859	0.853
	199.06	50996	0.991	1.032	0.994	1.032	0.998	1.032
	239.84	57566	0.993	1.000	0.996	1.000	0.997	1.000
rmine6	971.86	111047	1.004	1.000	1.000	1.000	0.999	1.000
	883.48	89213	1.001	1.000	0.990	1.000	1.001	1.000
	775.32	91530	1.005	1.000	0.999	1.000	1.011	1.000
	740.58	85429	1.003	1.000	0.993	1.000	1.006	1.000
	771.48	76017	0.999	1.000	1.000	1.000	1.004	1.000
rococoC10-001000	601.83	46983	0.999	1.000	0.999	1.000	0.997	1.000
	833.40	75156	1.008	1.000	1.004	1.000	0.996	1.000
	517.62	37178	1.003	1.000	1.090	1.105	1.089	1.105
	892.04	51410	1.002	1.000	1.002	1.000	1.002	1.000
	1047.39	66016	1.000	1.000	1.001	1.000	1.001	1.000
seymour	3600.00	102645	1.000	0.996	1.000	0.999	1.000	0.997
	3600.00	120322	1.000	0.999	1.000	0.997	1.000	0.998
	3600.00	122337	1.000	1.006	1.000	1.004	1.000	1.007
	3600.00	107537	1.000	0.998	1.000	1.000	1.000	0.996
	3600.00	96726	1.000	1.007	1.000	1.001	1.000	0.999
sp98ir	57.90	3948	1.004	1.000	1.006	1.000	1.001	1.000
	65.87	4178	1.002	1.000	1.004	1.000	0.996	1.000
	67.64	4735	1.005	1.000	1.003	1.000	1.001	1.000
	72.68	4603	0.995	1.000	0.995	1.000	1.002	1.000
	70.05	4780	1.004	1.000	1.000	1.000	1.000	1.000
stein27	1.12	2749	1.009	1.000	1.024	1.000	1.024	1.000
	1.19	2699	1.009	1.000	0.991	1.000	1.000	1.000
	1.25	2889	0.991	1.000	0.982	1.000	0.996	1.000
	1.14	2715	0.977	1.056	0.963	1.000	0.991	1.056
	1.13	2911	0.981	0.972	0.991	1.000	1.023	0.972
stein45	12.80	40723	0.967	0.962	0.987	1.000	1.041	1.002
	12.94	39012	0.963	0.995	0.999	1.000	0.991	0.995
	13.04	41930	1.010	0.985	0.998	1.000	1.015	0.985
	12.20	37543	1.038	1.015	0.998	1.000	1.044	1.015
	13.29	40503	0.984	1.006	0.985	1.000	1.003	1.006
tanglegram2	4.09	3	1.012	1.000	1.006	1.000	1.010	1.000
	3.85	3	1.000	1.000	1.006	1.000	0.996	1.000
	4.25	3	0.998	1.000	0.996	1.000	1.050	1.000
	4.11	3	0.994	1.000	0.990	1.000	1.004	1.000
	4.23	3	0.990	1.000	0.992	1.000	0.998	1.000

Table 6. Table 6 : Detailed computational results on MMM-IP with five random seeds. Relative changes by at least 5 % percent 5 5\% are highlighted in bold and blue ( improvement ) or italic and red ( deterioration ).

	degeneracy		variablebounds		conflicts		infervals		primsols
Instance	time	nodes	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ	timeQ	nodesQ
10teams	3.01	1	0.990	1.000	0.825	1.000	0.995	1.000	0.988	1.000
	4.07	5	1.006	1.000	1.266	1.000	1.162	1.000	1.026	1.000
	25.62	956	1.012	1.000	0.668	0.159	1.091	1.195	0.441	0.118
	13.04	28	0.998	1.000	0.331	0.820	0.428	0.820	1.002	1.000
	4.40	5	1.004	1.000	4.026	2.943	1.278	1.000	1.007	1.000
30n20b8	189.53	63	1.521	1.245	1.573	1.607	1.458	1.503	1.000	1.000
	111.78	2	2.089	1.873	1.975	1.676	1.023	1.000	0.998	1.000
	87.35	4	1.001	1.000	0.993	1.000	0.999	1.000	1.001	1.000
	257.61	165	0.841	0.762	0.994	1.000	1.009	1.000	1.004	1.000
	381.26	182	0.982	0.926	1.061	0.858	1.005	1.000	0.999	1.000
acc-tight5	139.72	956	1.284	1.562	0.205	0.116	0.433	0.267	1.560	1.788
	110.82	1160	0.516	0.318	1.515	1.136	0.513	0.239	1.140	1.150
	49.06	174	2.796	4.310	1.596	2.558	0.623	0.442	1.712	2.423
	28.52	22	2.037	1.828	2.899	4.975	1.400	1.434	5.993	11.180
	30.00	17	1.019	1.000	0.936	1.325	2.597	3.556	1.012	1.103
air03	1.73	1	1.000	1.000	1.007	1.000	0.993	1.000	1.011	1.000
	1.72	1	1.015	1.000	1.000	1.000	1.011	1.000	0.996	1.000
	1.72	1	1.011	1.000	1.007	1.000	1.000	1.000	1.007	1.000
	1.75	1	1.000	1.000	0.993	1.000	0.993	1.000	1.004	1.000
	1.74	1	1.026	1.000	1.007	1.000	0.989	1.000	1.000	1.000
air04	49.60	110	1.009	1.000	1.008	1.000	1.008	1.000	1.008	1.000
	36.38	39	1.001	1.000	0.995	1.000	0.997	1.000	0.995	1.000
	53.17	118	0.999	1.000	0.988	1.000	0.998	1.000	1.002	1.000
	37.09	31	0.995	1.000	0.993	1.000	0.992	1.000	0.996	1.000
	48.66	122	0.996	1.000	0.994	1.000	1.002	1.000	0.999	1.000
air05	24.56	189	1.002	1.000	1.002	1.000	1.001	1.000	1.002	1.000
	26.64	239	1.000	1.000	0.998	1.000	0.998	1.000	1.009	1.000
	31.10	438	1.002	1.000	1.005	1.000	1.046	1.000	1.003	1.000
	27.57	283	1.000	1.000	1.000	1.000	0.999	1.000	0.998	1.000
	27.23	279	1.000	1.000	0.996	1.000	0.998	1.000	0.993	1.000
blend2	0.76	714	0.989	1.000	0.994	1.000	0.972	1.000	1.006	1.000
	1.05	973	1.015	1.000	0.980	1.000	1.005	1.000	1.005	1.000
	1.06	912	0.985	1.000	0.985	1.000	0.990	1.000	1.005	1.000
	0.66	543	1.018	1.000	0.994	1.000	0.994	1.000	1.012	1.000
	0.71	709	0.994	1.000	0.988	1.000	1.000	1.000	1.000	1.000
bley_xl1	163.94	1	0.996	1.000	1.004	1.000	1.005	1.000	1.004	1.000
	189.31	18	0.999	1.000	1.002	1.000	1.052	1.314	1.107	1.458
	162.64	2	0.984	1.000	0.999	1.000	1.000	1.000	1.003	1.000
	134.67	1	0.997	1.000	1.001	1.000	0.998	1.000	0.999	1.000
	136.43	1	1.003	1.000	1.001	1.000	1.003	1.000	1.003	1.000
bnatt350	18.90	6	0.897	1.000	6.188	7.047	2.522	1.330	19.530	40.189
	16.29	10	1.098	1.000	27.910	46.845	2.398	1.309	13.634	15.082
	18.02	6	0.834	1.000	15.298	20.085	3.298	3.075	13.075	20.151
	16.08	7	0.999	1.000	12.361	12.720	1.444	1.047	17.894	26.383
	43.21	112	0.732	0.623	9.481	21.113	2.919	3.425	10.242	27.854
cap6000	2.70	1830	1.008	1.000	1.000	1.000	1.003	1.000	1.005	1.000
	2.62	1837	0.992	1.000	1.017	1.000	0.978	1.000	1.003	1.000
	2.58	1959	1.025	1.000	1.028	1.000	1.011	1.000	1.014	1.000
	3.20	1924	0.976	1.000	0.983	1.000	0.974	1.000	0.993	1.000
	2.85	1762	0.982	1.000	0.992	1.000	0.992	1.000	0.984	1.000
cov1075	3560.96	1725170	1.001	1.000	1.001	1.000	1.011	0.827	1.001	1.000
	3600.00	1745291	1.000	1.000	1.000	1.006	1.000	1.009	1.000	1.003
	3600.00	1834965	1.000	0.998	1.000	1.001	1.000	0.823	1.000	0.990
	3600.00	1771419	1.000	1.000	1.000	0.999	1.000	1.007	1.000	1.000
	3600.00	1817274	1.000	0.999	1.000	0.999	1.000	0.657	1.000	0.999
csched010	3600.00	237694	1.000	0.997	1.000	0.997	1.000	0.996	1.000	0.997
	3600.00	203138	1.000	1.000	1.000	1.002	1.000	0.999	1.000	1.000
	2893.02	177527	0.998	1.000	1.001	1.000	1.000	1.000	0.999	1.000
	3600.00	238053	1.000	1.003	1.000	1.001	1.000	0.999	1.000	0.998
	3171.41	173932	0.999	1.000	0.999	1.000	0.998	1.000	1.000	1.000
eil33-2	69.62	583	1.004	1.000	1.001	1.000	0.996	1.000	1.003	1.000
	82.52	739	0.994	1.000	0.994	1.000	0.999	1.000	0.981	1.000
	83.95	847	1.012	1.000	1.013	1.000	1.016	1.000	1.010	1.000
	72.71	543	1.015	1.000	1.019	1.000	1.016	1.000	1.005	1.000
	69.08	583	1.016	1.000	1.016	1.000	1.015	1.000	1.017	1.000
eilB101	378.34	10582	0.995	1.000	0.990	1.000	1.012	1.000	1.000	1.000
	359.35	8427	1.009	1.000	1.001	1.000	1.017	1.000	1.011	1.000
	399.77	9358	1.003	1.000	0.999	1.000	0.999	1.000	1.005	1.000
	396.04	11465	1.006	1.000	0.995	1.000	1.012	1.000	0.997	1.000
	361.93	10006	1.009	1.000	1.011	1.000	1.013	1.000	1.006	1.000
enigma	0.50	650	1.000	0.997	1.033	768.000	1.113	1.135	1.000	1.033
	0.50	173	1.000	1.681	1.000	1.905	1.000	1.495	1.000	1.000
	0.50	63	1.000	1.374	1.000	1.123	1.027	4.491	1.000	1.000
	0.61	318	0.932	0.289	0.932	1.122	1.031	1.218	0.975	1.000
	0.53	196	1.013	1.588	0.980	0.405	0.980	1.493	1.000	1.000
enlight13	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
enlight14	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
fiber	1.02	2	0.995	1.000	0.980	1.000	1.000	1.000	1.000	1.000
	1.39	4	1.004	1.000	1.008	1.000	1.013	1.000	1.004	1.000
	1.76	4	0.982	1.000	0.996	1.000	0.978	1.000	0.993	1.000
	1.55	4	1.000	1.000	0.988	1.000	0.988	1.000	0.988	1.000
	1.00	3	0.995	1.000	1.010	1.000	1.005	1.000	1.005	1.000
flugpl	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
gt2	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
harp2	694.94	1491878	1.003	1.000	0.999	1.000	0.996	1.000	0.989	1.000
	1163.82	2857677	0.992	1.000	0.989	1.000	0.993	1.000	0.993	1.000
	333.99	595854	1.008	1.000	1.002	1.000	1.011	1.000	1.009	1.000
	934.82	2046181	1.006	1.000	0.999	1.000	1.001	1.000	1.003	1.000
	433.24	923780	0.998	1.000	0.985	1.000	0.991	1.000	0.996	1.000
iis-100-0-cov	625.29	103302	0.999	1.000	1.002	1.000	0.998	1.000	0.998	1.000
	544.56	92393	1.001	1.000	1.025	1.000	0.999	1.000	1.001	1.000
	520.20	81639	1.022	1.000	1.018	1.000	1.069	1.000	1.007	1.000
	507.53	84800	0.995	1.000	1.008	1.000	1.001	1.000	1.025	1.000
	542.59	92313	1.055	1.000	1.036	1.000	1.011	1.000	1.016	1.000
iis-bupa-cov	1917.51	157619	0.991	1.000	0.994	1.000	0.997	1.000	0.996	1.000
	1939.94	164079	1.010	1.000	1.019	1.000	1.014	1.000	1.020	1.000
	1878.21	156888	1.053	1.000	0.998	1.000	1.024	1.000	1.003	1.000
	1938.34	161969	1.002	1.000	1.022	1.000	1.026	1.000	1.002	1.000
	1802.15	158284	1.030	1.000	1.048	1.000	1.003	1.000	1.030	1.000
iis-pima-cov	292.21	4824	1.006	1.000	1.002	1.000	1.002	1.000	1.006	1.000
	434.93	12189	1.020	1.000	1.000	1.000	1.025	1.000	1.039	1.000
	246.99	4668	1.007	1.000	1.019	1.000	1.009	1.000	0.995	1.000
	285.55	6607	1.005	1.000	1.006	1.000	1.011	1.000	1.015	1.000
	558.43	17821	1.017	1.000	0.998	1.000	1.002	1.000	1.048	1.000
l152lav	5.47	23	0.994	1.000	0.998	1.000	1.009	1.000	1.000	1.000
	1.63	31	1.011	1.000	1.015	1.000	0.996	1.000	1.011	1.000
	2.93	50	0.992	1.000	0.987	1.000	1.000	1.000	1.003	1.000
	2.29	20	1.006	1.000	1.012	1.000	1.009	1.000	1.009	1.000
	3.00	52	1.012	1.000	1.010	1.000	1.015	1.000	1.012	1.000
lectsched-4-obj	20.11	95	0.957	1.113	1.624	3.487	1.008	1.913	2.428	4.513
	28.02	158	1.120	1.093	1.242	2.430	0.989	0.930	2.457	2.988
	24.11	565	0.645	0.415	1.559	1.989	1.552	1.788	2.656	1.714
	8.09	9	1.022	1.000	0.865	1.000	0.818	0.963	9.364	11.716
	20.57	114	1.340	1.178	1.510	3.664	1.381	2.271	7.981	9.061
lseu	0.77	268	1.011	1.000	0.977	1.000	1.000	1.000	0.994	1.000
	0.50	4	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	205	1.000	1.000	1.020	1.000	1.000	1.000	1.000	1.000
	0.50	91	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.51	191	0.993	1.000	1.007	1.000	0.993	1.000	1.020	1.000
m100n500k4r1	3600.00	2426435	1.000	1.032	1.000	0.997	1.000	0.957	1.000	0.883
	3600.00	2358830	1.000	0.997	1.000	0.993	1.000	1.414	1.000	0.897
	3600.00	2313256	1.000	1.028	1.000	0.982	1.000	1.039	1.000	0.982
	3600.00	2163125	1.000	1.040	1.000	0.997	1.000	1.441	1.000	0.950
	3600.00	1871050	1.000	1.054	1.000	0.975	1.000	1.115	1.000	1.032
macrophage	125.68	5163	1.001	1.000	0.999	1.000	1.002	1.000	1.001	1.000
	157.07	7241	1.007	1.000	1.007	1.000	1.007	1.000	1.005	1.000
	209.04	9131	1.010	1.000	0.998	1.000	1.003	1.000	1.006	1.000
	162.46	7927	1.039	1.000	1.002	1.000	1.013	1.000	1.008	1.000
	121.23	5239	0.997	1.000	1.015	1.000	0.998	1.000	1.014	1.000
manna81	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
markshare1	3600.00	17773362	1.000	1.150	1.000	0.990	1.000	1.038	1.000	0.655
	3600.00	17405834	1.000	1.184	1.000	0.998	1.000	1.126	1.000	1.000
	3600.00	17945211	1.000	1.149	1.000	1.001	1.000	0.954	1.000	0.482
	3600.00	18348094	1.000	1.131	1.000	1.003	1.000	0.980	1.000	0.853
	3600.00	17151823	1.000	1.130	1.000	0.997	1.000	1.050	1.000	0.889
markshare2	3600.00	3433363	1.000	1.178	1.000	1.003	1.000	1.352	1.000	1.064
	3600.00	3929356	1.000	0.901	1.000	1.003	1.000	1.127	1.000	0.877
	3600.00	3675025	1.000	1.036	1.000	1.000	1.000	0.975	1.000	1.004
	3600.00	4532415	1.000	0.819	1.000	0.999	1.000	1.032	1.000	0.789
	3600.00	4267105	1.000	0.860	1.000	1.009	1.000	0.887	1.000	1.101
mcsched	285.70	13689	1.000	1.000	0.998	1.000	1.004	1.000	1.000	1.000
	271.59	11373	0.998	1.000	1.015	1.000	1.022	1.000	0.999	1.000
	233.00	10059	1.014	1.000	1.036	1.000	1.006	1.000	1.001	1.000
	285.38	12496	1.011	1.000	1.003	1.000	0.998	1.000	0.996	1.000
	272.13	10373	0.997	1.000	1.009	1.000	1.027	1.000	1.016	1.000
mine-166-5	58.40	1569	0.997	1.000	0.998	1.000	0.997	1.000	0.999	1.000
	57.21	1191	1.007	1.000	1.003	1.000	0.998	1.000	1.005	1.000
	46.49	2244	1.064	1.000	1.010	1.000	1.007	1.000	0.999	1.000
	65.30	139	0.991	1.000	1.003	1.000	1.043	1.000	1.003	1.000
	42.28	2863	1.061	1.000	1.002	1.000	1.021	1.000	1.019	1.000
mine-90-10	276.51	47643	1.001	1.000	0.997	1.000	0.999	1.000	0.997	1.000
	125.11	22195	1.002	1.000	1.001	1.000	1.013	1.000	1.018	1.000
	173.38	25979	1.042	1.000	1.001	1.000	1.010	1.000	1.020	1.000
	502.57	87496	0.998	1.000	0.995	1.000	1.005	1.000	0.997	1.000
	237.48	32520	1.002	1.000	1.054	1.000	0.992	1.000	0.999	1.000
misc03	1.40	398	0.988	1.000	0.992	1.000	0.983	1.000	0.988	1.000
	0.71	15	1.023	1.000	1.018	1.000	1.006	1.000	1.000	1.000
	0.73	19	1.000	1.000	0.983	1.000	0.988	1.000	0.977	1.000
	0.98	111	1.025	1.000	1.015	1.000	0.990	1.000	1.020	1.000
	0.64	15	1.018	1.000	1.000	1.000	1.000	1.000	1.018	1.000
misc07	38.28	28289	0.997	1.000	0.990	1.000	0.982	1.000	0.995	1.000
	36.49	29782	1.007	1.000	1.006	1.000	1.006	1.000	1.007	1.000
	21.45	17632	1.003	1.000	1.010	1.000	1.004	1.000	1.003	1.000
	45.80	38975	1.014	1.000	1.013	1.000	1.004	1.000	1.007	1.000
	30.56	29476	1.010	1.000	1.014	1.000	1.009	1.000	1.009	1.000
mitre	14.57	1	1.006	1.000	0.757	1.000	0.984	1.000	0.988	1.000
	15.57	1	0.992	1.000	0.717	1.000	0.993	1.000	0.999	1.000
	15.52	1	1.005	1.000	0.718	1.000	1.005	1.000	1.001	1.000
	15.17	1	0.988	1.000	0.746	1.000	0.994	1.000	1.001	1.000
	13.39	1	1.037	1.000	0.994	1.000	1.027	1.000	1.028	1.000
mod008	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.53	2	1.000	1.000	0.987	1.000	1.013	1.000	1.007	1.000
	0.54	2	0.987	1.000	0.981	1.000	0.987	1.000	0.994	1.000
	0.51	1	1.000	1.000	1.000	1.000	1.007	1.000	1.013	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
mod010	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.53	2	0.993	1.000	0.980	1.000	1.000	1.000	0.987	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
mzzv11	332.03	2022	0.995	1.000	0.995	1.000	0.998	1.000	0.996	1.000
	229.82	959	1.005	1.000	0.999	1.000	0.991	1.000	0.999	1.000
	311.22	1421	0.987	1.000	1.000	1.000	0.996	1.000	0.988	1.000
	322.67	1579	1.009	1.000	1.014	1.000	1.009	1.000	0.998	1.000
	186.23	627	1.002	1.000	0.999	1.000	1.000	1.000	0.997	1.000
mzzv42z	151.11	265	0.994	1.000	1.003	1.000	1.001	1.000	1.001	1.000
	131.14	125	0.994	1.000	0.995	1.000	0.998	1.000	0.995	1.000
	198.89	200	1.000	1.000	0.996	1.000	0.995	1.000	0.999	1.000
	214.35	456	0.997	1.000	1.005	1.000	0.998	1.000	0.990	1.000
	167.53	300	1.005	1.000	0.998	1.000	1.000	1.000	1.001	1.000
neos-1109824	37.38	1796	0.998	1.000	0.997	1.000	0.996	1.000	0.997	1.000
	46.57	3438	1.012	1.000	1.006	1.000	1.008	1.000	1.035	1.000
	34.97	1913	1.013	1.000	1.009	1.000	1.021	1.000	1.022	1.000
	77.32	8290	1.043	1.000	1.005	1.000	1.055	1.000	1.036	1.000
	39.22	2247	1.008	1.000	1.010	1.000	1.010	1.000	1.038	1.000
neos-1337307	3600.00	306352	1.000	0.991	1.000	0.993	1.000	0.994	1.000	0.993
	3600.00	269284	1.000	0.998	1.000	0.979	1.000	1.002	1.000	0.952
	3600.00	349832	1.000	0.970	1.000	0.996	1.000	0.996	1.000	0.991
	3600.00	301363	1.000	0.989	1.000	0.978	1.000	0.932	1.000	1.006
	3600.00	373217	1.000	0.995	1.000	0.991	1.000	0.983	1.000	0.986
neos-1601936	2033.50	1453	0.999	1.000	0.994	1.000	0.999	1.000	0.996	1.000
	3600.00	2861	1.000	0.996	1.000	1.001	1.000	1.000	1.000	1.000
	1338.93	2661	1.013	1.000	1.006	1.000	1.008	1.000	1.009	1.000
	3600.00	6917	1.000	1.004	1.000	1.000	1.000	0.990	1.000	0.999
	3600.00	6139	1.000	1.000	1.000	0.977	1.000	1.000	1.000	1.010
neos-686190	83.27	4906	0.996	1.000	1.002	1.000	0.995	1.000	0.996	1.000
	86.55	6047	1.000	1.000	1.013	1.000	0.997	1.000	1.000	1.000
	115.34	9156	0.995	1.000	0.994	1.000	1.021	1.000	1.006	1.000
	117.19	9998	0.995	1.000	1.006	1.000	0.992	1.000	0.996	1.000
	120.84	9806	1.008	1.000	0.998	1.000	1.003	1.000	1.007	1.000
neos-849702	207.98	1169	1.979	8.256	6.088	54.068	2.833	25.153	4.876	14.426
	1025.85	6328	0.139	0.049	0.772	1.387	0.158	0.594	0.999	1.000
	1001.35	10499	0.904	0.400	0.181	0.099	0.686	5.053	1.016	1.000
	267.07	683	2.864	2.388	2.815	9.508	0.763	13.476	5.812	84.398
	285.44	1596	1.679	1.538	1.067	0.353	0.470	4.297	5.308	35.870
neos-934278	3600.00	1175	1.000	0.999	1.000	1.000	1.000	1.000	1.000	1.006
	3600.00	267	1.000	0.995	1.000	1.008	1.000	1.000	1.000	0.995
	3600.00	263	1.000	1.000	1.000	1.008	1.000	0.983	1.000	1.008
	3600.00	401	1.000	0.992	1.000	0.994	1.000	0.992	1.000	0.994
	3600.00	522	1.000	0.998	1.000	0.998	1.000	1.000	1.000	0.997
neos18	8.13	79	1.234	1.078	2.113	3.598	1.604	2.369	2.852	4.073
	24.87	828	1.043	1.054	1.347	1.815	1.232	1.531	0.999	1.028
	28.94	1095	1.089	1.000	0.883	0.965	0.680	0.362	0.689	0.585
	15.42	386	1.001	0.965	1.107	1.091	1.161	0.872	2.091	2.175
	21.77	963	0.961	0.949	1.249	1.147	0.928	0.710	1.281	0.871
ns1208400	938.01	3025	0.912	0.753	0.543	0.314	0.763	0.692	1.017	1.032
	2737.49	11477	0.667	0.375	0.269	0.100	0.220	0.081	1.005	1.000
	532.73	1719	1.828	1.341	0.710	0.800	1.579	0.825	2.439	2.687
	253.07	599	2.316	2.724	1.530	1.409	1.318	0.720	2.093	2.482
	1316.58	2660	1.221	1.112	1.259	2.671	0.330	0.303	1.015	1.000
ns1688347	51.67	437	1.239	1.255	1.025	1.220	1.103	0.719	0.998	1.000
	42.42	203	1.423	2.165	1.332	2.370	1.406	2.670	1.886	4.353
	161.82	3095	1.050	0.752	0.833	0.552	1.111	0.556	1.003	1.000
	133.50	1251	0.878	0.597	0.964	0.638	1.049	1.245	1.008	1.000
	51.48	399	0.948	0.890	1.378	2.481	0.953	1.156	1.274	1.691
ns1830653	133.08	6918	0.993	1.000	0.998	1.000	0.994	1.000	0.995	1.000
	123.87	5226	0.999	1.000	0.997	1.000	1.061	1.000	1.007	1.000
	161.00	9597	1.004	1.000	1.001	1.000	1.001	1.000	1.007	1.000
	143.21	7435	1.054	1.000	1.005	1.000	0.995	1.000	1.002	1.000
	146.08	9194	1.000	1.000	1.004	1.000	1.007	1.000	1.002	1.000
nsrand-ipx	419.85	50124	1.006	1.000	1.005	1.000	1.007	1.000	1.001	1.000
	638.54	86869	1.002	1.000	1.005	1.000	0.999	1.000	1.005	1.000
	569.66	56575	0.994	1.000	1.003	1.000	1.005	1.000	1.004	1.000
	582.34	61056	1.004	1.000	1.001	1.000	1.000	1.000	1.002	1.000
	363.56	38012	1.007	1.000	1.002	1.000	1.002	1.000	1.000	1.000
opt1217	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
p0033	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
p0201	0.54	5	1.006	1.000	1.013	1.000	1.019	1.000	1.000	1.000
	0.74	9	0.989	1.000	1.000	1.000	0.983	1.000	1.011	1.000
	1.64	5	1.000	1.000	1.004	1.000	0.985	1.000	0.996	1.000
	0.90	13	1.000	1.000	1.000	1.000	1.000	1.000	1.011	1.000
	0.89	59	1.005	1.000	1.005	1.000	1.011	1.000	1.016	1.000
p0282	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	3	1.000	1.000	1.000	1.000	1.007	1.000	1.000	1.000
	0.50	2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
p0548	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	0.50	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
p2756	2.51	5	0.991	1.000	0.989	1.000	1.000	1.000	0.994	1.000
	3.33	5	1.009	1.000	1.005	1.000	1.009	1.000	1.007	1.000
	2.13	3	0.994	1.000	0.990	1.000	0.987	1.000	1.006	1.000
	2.54	3	1.000	1.000	1.008	1.000	1.006	1.000	1.014	1.000
	1.28	2	1.000	1.000	0.996	1.000	1.004	1.000	1.009	1.000
protfold	3600.00	3244	1.000	0.823	1.000	0.632	1.000	2.305	1.000	1.328
	3600.00	2504	1.000	0.715	1.000	0.988	1.000	2.045	1.000	0.950
	3600.00	2755	1.000	0.858	1.000	1.160	1.000	1.907	1.000	0.733
	3600.00	1831	1.000	1.278	1.000	1.551	1.000	2.081	1.000	1.256
	3600.00	1867	1.000	0.814	1.000	1.325	1.000	1.584	1.000	0.831
pw-myciel4	3600.00	361131	1.000	1.253	0.383	0.439	1.000	1.320	1.000	1.077
	3600.00	365865	1.000	1.039	1.000	0.895	1.000	1.253	1.000	1.054
	3600.00	323450	1.000	1.199	1.000	1.102	1.000	1.294	1.000	1.161
	3600.00	358151	1.000	1.020	1.000	0.856	1.000	1.395	0.925	1.097
	3054.26	378549	1.179	0.843	1.179	1.128	1.179	1.019	1.179	0.893
qnet1	4.54	33	1.002	1.000	0.998	1.000	1.002	1.000	1.005	1.000
	3.12	3	0.995	1.000	1.007	1.000	1.002	1.000	0.993	1.000
	3.92	29	1.002	1.000	0.996	1.000	0.996	1.000	1.006	1.000
	2.26	3	0.991	1.000	0.991	1.000	0.994	1.000	0.991	1.000
	3.71	34	0.994	1.000	1.002	1.000	0.996	1.000	0.987	1.000
qnet1_o	3.87	27	1.012	1.000	1.002	1.000	1.010	1.000	1.008	1.000
	3.38	24	1.000	1.000	0.995	1.000	0.995	1.000	1.002	1.000
	3.39	15	0.995	1.000	1.000	1.000	0.993	1.000	1.000	1.000
	1.67	2	1.000	1.000	1.000	1.000	0.993	1.000	1.004	1.000
	1.84	3	1.000	1.000	1.007	1.000	1.000	1.000	1.007	1.000
reblock67	223.13	49530	0.998	1.000	0.995	1.000	0.995	1.000	0.989	1.000
	161.91	40625	1.001	1.000	1.004	1.000	1.006	1.000	1.019	1.000
	194.68	49882	1.002	1.000	1.006	1.000	0.999	1.000	1.032	1.000
	200.17	50996	1.006	1.000	1.001	1.000	0.992	1.000	0.995	1.000
	239.14	57566	1.017	1.000	0.999	1.000	1.002	1.000	0.999	1.000
rmine6	971.59	111047	0.997	1.000	1.002	1.000	0.999	1.000	1.003	1.000
	882.29	89213	0.999	1.000	1.040	1.000	1.013	1.000	1.026	1.000
	776.69	91530	0.992	1.000	1.000	1.000	1.004	1.000	1.021	1.000
	742.25	85429	1.010	1.000	1.001	1.000	1.005	1.000	1.000	1.000
	773.39	76017	0.995	1.000	1.002	1.000	1.013	1.000	0.995	1.000
rococoC10-001000	603.64	46983	0.995	1.000	0.997	1.000	0.994	1.000	0.995	1.000
	836.01	75156	0.999	1.000	0.998	1.000	0.998	1.000	1.007	1.000
	516.72	37178	1.052	1.000	1.006	1.000	1.003	1.000	1.006	1.000
	893.05	51410	0.999	1.000	1.006	1.000	1.004	1.000	1.001	1.000
	1048.40	66016	1.004	1.000	1.010	1.000	1.008	1.000	1.003	1.000
seymour	3600.00	102192	1.000	1.003	1.000	1.002	1.000	1.007	1.000	1.000
	3600.00	120186	1.000	1.002	1.000	0.996	1.000	1.000	1.000	1.000
	3600.00	123015	1.000	1.003	1.000	0.997	1.000	0.997	1.000	0.990
	3600.00	108085	1.000	1.001	1.000	0.996	1.000	0.998	1.000	0.995
	3600.00	97046	1.000	1.001	1.000	0.994	1.000	0.999	1.000	1.001
sp98ir	58.06	3948	0.999	1.000	0.997	1.000	1.000	1.000	0.999	1.000
	65.95	4178	1.039	1.000	1.003	1.000	1.012	1.000	1.006	1.000
	67.93	4735	1.000	1.000	1.029	1.000	1.020	1.000	1.014	1.000
	72.47	4603	1.003	1.000	1.005	1.000	1.003	1.000	1.011	1.000
	69.87	4780	1.063	1.000	0.998	1.000	1.001	1.000	1.013	1.000
stein27	1.17	2749	0.972	1.000	1.014	1.000	1.023	1.000	0.995	1.000
	1.23	2699	1.000	1.000	0.969	1.000	0.996	1.000	0.978	1.000
	1.19	2889	0.995	1.000	1.018	1.000	1.014	1.000	1.023	1.000
	1.13	2873	0.953	1.000	0.958	1.000	0.962	0.947	1.014	1.000
	1.08	2827	1.019	1.000	1.010	1.000	1.014	1.029	0.990	1.000
stein45	12.59	39175	0.994	1.000	0.989	1.000	0.996	1.039	0.997	1.000
	12.49	38805	1.004	1.000	1.005	1.000	1.024	1.005	1.003	1.000
	13.04	41279	1.005	1.000	1.024	1.000	0.998	1.016	1.004	1.000
	12.49	38104	1.006	1.000	1.004	1.000	0.980	0.985	1.004	1.000
	13.02	40755	1.008	1.000	0.997	1.000	1.009	0.994	0.994	1.000
tanglegram2	4.11	3	1.002	1.000	1.006	1.000	0.990	1.000	1.000	1.000
	3.90	3	1.022	1.000	0.994	1.000	1.014	1.000	1.016	1.000
	4.20	3	1.012	1.000	1.008	1.000	1.015	1.000	1.008	1.000
	4.12	3	1.020	1.000	1.008	1.000	1.008	1.000	1.016	1.000
	4.20	3	1.000	1.000	1.000	1.000	1.038	1.000	1.000	1.000

Equations

$$\min$$

$$Ax$$

$$l_j\le x_j$$

$$x_j$$

$${\text{raise}-2.20001pt\vee }_{i\in \mathcal{L}}(x_i\ge {\lambda }_i)\vee {\text{raise}-2.20001pt\vee }_{i\in \mathcal{U}}(x_i\le {\mu }_i)$$

$$\sum _{i\in \mathcal{U}}{x}_i-\sum _{i\in \mathcal{L}}{x}_i\le \sum _{i\in \mathcal{U}}{u}_i-\sum _{i\in \mathcal{L}}{l}_i-1.$$

$${\lim }_{\mathrm{node}}=\min \{5000,\max \{500,{\mathrm{iter}}_{\mathrm{LP}}\}\}.$$

$${\log }_{\beta }(d/f)\in \mathbb{Z},$$

Full text

1 Introduction

Constraint programming (CP) and integer programming (IP) are two complementary ways of tackling discrete optimization problems. Hybrid combinations of the two approaches have been used for many years, see, e.g., [2, 9, 10, 17, 37, 42, 22]. Both technologies have incorporated conflict learning capabilities [21, 27, 38, 1, 35] that derive additional valid constraints from the analysis of infeasible subproblems extending methods developed by the SAT community [33].

Conflict learning is a technique that analyzes infeasible subproblems encountered during a tree search algorithm. In a tree search, each subproblem can be identified by its local variable bounds, i.e., by local bound changes that come from branching decisions and propagation at the current node and its ancestors. If propagation detects infeasibility, conflict learning will traverse this chain of decisions and deductions reversely, reconstructing which bound changes led to which other bound changes. In this way, conflict learning identifies explanations for the infeasibility. If it can be shown that a small subset of the bound changes suffices to prove infeasibility, a so-called conflict constraint is generated that can be exploited in the remainder of the search to prune parts of the tree.

In the context of constraint programming, conflict constraints are also referred to as no-goods. For binary programs (BPs), i.e., mixed integer (linear) programs for which all variables have domain $\{0,1\}$, conflict constraints will have the form of set covering constraints. These are linear constraints of the form “sum of variables (or their negated form) is greater than or equal to one”.

Rapid Learning [13] is a heuristic algorithm for BPs that searches for valid conflict constraints, global bound reductions, and primal solutions. It is based on the observation that a CP solver can typically perform an incomplete search on a few thousand nodes in a fraction of the time that a MIP solver needs for processing the root node. In addition, CP solvers make use of depth-first search, as opposed to the hybrid best-first/depth-first search of MIP solvers, which more rapidly generates strong no-goods. Typically CP solvers do not differentiate the root node from other nodes. They apply fast (at least typically) propagation algorithms to infer new information about the possible values variables can take, and then take branching decisions. In contrast, a MIP solver invests a substantial amount of time at the root node to gather global information about the problem and to initialize statistics that can help for the search. A significant portion of root node processing time comes from the computational effort needed to solve the initial LP relaxation from scratch. Further aspects are the LP resolves during cutting plane generation, strong branching [7] for branching statistic evaluation, and primal heuristics, see, e.g., [11].

The idea of Rapid Learning is to apply a fast CP depth-first branch-and-bound search for a few hundred or thousand nodes, generating and collecting valid conflict constraints at the root node of a MIP search. Using this, the MIP solver is already equipped with the valuable information of which bound changes will lead to an infeasibility, and can avoid them by propagating the derived constraints. Just as important, the partial CP search might find primal solutions, thereby acting as a primal heuristic. Furthermore, the knowledge of conflict constraints can be used to initialize branching statistics, just like strong branching. In this paper, we will extend Rapid Learning to integer programs and to nodes beyond the root.

The remainder of the paper is organized as follows. In Section 2, we provide more background on conflict learning for MIPs, in particular the extension to general integer variables, which is important for our extended Rapid Learning algorithm. In Section 3, we describe details of the Rapid Learning algorithm for general integer programs, extending the work of Berthold et al. [13]. In Section 4, we discuss what special considerations have to be taken when applying Rapid Learning repeatedly at local subproblems during the MIP tree search instead of using it as a onetime global procedure. We introduce six criteria to predict the benefit of local Rapid Learning. Section 5 presents our computational study, in which we apply our extended Rapid Learning algorithm to a set of integer programs from the well-known benchmark sets of Miplib 3, Miplib 2003, and Miplib 2010 [28]. The experiments have been conducted with the constraint integer programming solver SCIP [24] and indicate that a significant speed-up can be achieved for (pure) integer programs, when using Rapid Learning locally. In Section 6, we conclude.

2 Conflict Learning in Integer Programming

A mixed integer program is a mathematical optimization problem defined as follows.

Definition 1** (mixed integer program)**

Let $m,n\in\mathbb{Z}_{\geq 0}$. Given a matrix $A\in\mathbb{R}^{m\times n}$, a right-hand-side vector $b\in\mathbb{R}^{m}$, an objective function vector $c\in\mathbb{R}^{n}$, a lower and an upper bound vector $l\in(\mathbb{R}\cup\{-\infty\})^{n}$, $u\in(\mathbb{R}\cup\{+\infty\})^{n}$ and a subset $\mathcal{I}\subseteq\mathcal{N}=\{1,\ldots,n\}$, the corresponding mixed integer program (MIP) is given by

[TABLE]

Mixed integer programs can be categorized by the classes of variables that are part of their formulation:

• •

If $\mathcal{N}=\mathcal{I}$, problem (1) is called a (pure) integer program (IP).

• •

If $\mathcal{N}=\mathcal{I}$, $l_{j}=0,j\in\mathcal{N}$ and $u_{j}=1,j\in\mathcal{N}$, problem (1) is called a (pure) binary program (BP).

• •

If $\mathcal{I}=\emptyset$, problem (1) is called a linear program (LP).

Conflict analysis techniques were originally developed by the artificial intelligence research community [40] and, later extended by the SAT community [33]; they led to a huge increase in the size of problems modern SAT solvers can handle [31, 33, 43]. The most successful SAT learning approaches use so-called one-level first unique implication point (1-UIP) [43] learning which in some sense captures the conflict constraint “closest” to the infeasibility. Conflict analysis also is successfully used in the CP community [25, 26, 35] (who typically refer to it as no-good learning) and the MIP world [1, 21, 38, 41]. Nowadays, commercial MIP solvers like FICO Xpress [23] employ conflict learning by default.

Constraint programming and mixed integer programming are two complementary ways of tackling discrete optimization problems. Because they have different strengths and weaknesses hybrid combinations are attractive. One notable example, the software SCIP [3], is based on the idea of constraint integer programming (CIP) [2, 6]. CIP is a generalization of MIP that supports the notion of general constraints as in CP. SCIP itself follows the idea of a very low-level integration of CP, SAT, and MIP techniques. All involved algorithms operate on a single search tree and share information and statistics through global storage of, e.g., solutions, variable domains, cuts, conflicts, the LP relaxation and so on. This allows for a very close interaction amongst CP and MIP (and other) techniques.

There is one major difference between BPs and IPs in the context of Rapid Learning: in IP, the problem variables are not necessarily binary. To deal with this, the concept of a conflict graph needs to be extended. A conflict graph gets constructed whenever infeasibility is detected in a local search node; it represents the logic of how the set of branching decisions led to the detection of infeasibility.

More precisely, the conflict graph is a directed acyclic graph in which the vertices111For disambiguation, we will use the term vertex for elements of the conflict graph, as opposed to nodes of the search tree. represent bound changes of variables, e.g., $x_{i}\leq\lambda_{i}$ or $x_{i}\geq\mu_{i}$. The conflict graph is built such that when the solver infers a bound change $v$ as a consequence of a set of existing bound changes $U$, i.e., $U\rightarrow v$, then we have an arc $(u,v)$ from each $u\in U$ to $v$. Bound changes caused by branching decisions are vertices without incoming edges. Finally the conflict graph includes a dummy vertex $\mathit{false}$ representing failure which is added when the solver infers unsatisfiability.

Given a conflict graph, each cut that separates the branching decisions from the artificial infeasibility vertex $\mathit{false}$ gives rise to a valid conflict constraint. A unique implication point (UIP) is an (inner) vertex of the conflict graph which is traversed by all paths from the branching vertices to the conflict vertex. Or, how Zhang et al. [43] describe it: “Intuitively, a UIP is the single reason that implies the conflict at [the] current decision level.” UIPs are natural candidates for finding small cuts in the conflict graph. The 1-UIP is the first cut separating the conflict vertex from the branching decisions when traversing in reverse assignment order.

For integer programs, conflict constraints can be expressed as so-called bound disjunction constraints:

Definition 2

For an IP, let $\mathcal{L}\subseteq\mathcal{I},\mathcal{U}\subseteq\mathcal{I}$ be disjoint index sets of variables, let $\lambda\in\mathbb{Z}^{\mathcal{L}}$ with $l_{i}\leq\lambda_{i}\leq u_{i}$ for all $i\in\mathcal{L}$, and $\mu\in\mathbb{Z}^{\mathcal{U}}$ with $l_{i}\leq\mu_{i}\leq u_{i}$ for all $i\in\mathcal{U}$. Then, a constraint of the form

[TABLE]

is called a bound disjunction constraint.

For details on bound disjunction constraints, see Achterberg [1]. If all involved conflict values $\lambda,\mu$ correspond to global bounds of the variables, the bound disjunction constraint can be equivalently expressed as a knapsack constraint of form

[TABLE]

Note that for BPs all conflicts only involve global bounds.

The power of conflict learning arises because often branch-and-bound based algorithms implicitly repeat the same search in a slightly different context in another part of the tree. Conflict constraints help to avoid redundant work in such situations. As a consequence, the more search is performed by a solver and the earlier conflicts are detected, the greater the chance for conflict learning to be beneficial. Note that conflict generation has a positive interaction with depth-first search. Depth-first search leads to the creation of no-goods that explain why a whole subtree contains no solutions, and hence the no-goods generated by depth-first search are likely to prune more of the subsequent search.

3 Rapid Learning for Integer Programs

The principle motivation for Rapid Learning [13] is the fact that a CP solver can typically search hundreds or thousand of nodes in a fraction of the time that a MIP solver needs for processing the root node of the search tree. Rapid Learning applies a fast CP search222By CP search we mean applying a depth-first search using only propagation for reasoning, no LP relaxation is solved during the search. for a few hundred or thousand nodes, before starting the MIP search. Using this approach, conflict constraints can be learnt before, and not only during, MIP search. Very loosely speaking: while the aim of conflict learning is to avoid making mistakes a second time, Rapid Learning tries to avoid making them the first time (during MIP search).

Rapid Learning is related to large neighborhood search heuristics, such as rins and rens [12, 20]. But, rather than doing an incomplete search on a subproblem using the same (MIP search) algorithm, Rapid Learning performs an incomplete search on the same problem using a much faster algorithm (CP search). Rapid Learning differs from primal heuristics in that it aims at improving the dual bound by collecting information on infeasibility rather than searching for feasible solutions.

Each piece of information collected in a rapid CP search can be used to guide the MIP search or even deduce further reductions during root node processing. Since the CP solver is solving the same problem as the MIP solver

• •

each generated conflict constraint is valid for the MIP search,

• •

each global bound change can be applied at the MIP root node,

• •

each feasible solution can be added to the MIP solver’s solution pool,

• •

the branching statistics can initialize a hybrid MIP branching rule, see [4], and

• •

if the CP solver completely solves the problem, the MIP solver can abort.

All five types of information may be beneficial for a MIP solver, and are potentially generated by our algorithm which we now describe more formally.

The Rapid Learning algorithm is outlined in Figure 1. Here, $l(P)$ and $u(P)$ are lower and upper bound vectors, respectively, of the problem at hand, $P$. For the moment we assume $P$ is the root problem, in the next section we will examine the use of Rapid Learning at subproblem nodes. The symbol $C$ refers to a single globally valid conflict constraint explaining the infeasibility of the current subproblem. Rapid Learning is an incomplete CP search: a branch-and-bound algorithm which traverses the search space in a depth-first manner (Line 1), using propagation (Line 1) and conflict analysis (Line 1), but no LP relaxation. Instead, the pseudo-solution [2], i.e., an optimal solution of a relaxation consisting only of the variable bounds (Line 1), is used for the bounding step.

Propagation of linear constraints is conducted by the bound strengthening technique of Brearley et al. [18] which uses the residual activity of linear constraints within the local bounds. For special cases of linear constraints, SCIP implements special, more efficient propagators. Knapsack constraints use efficient integer arithmetic instead of floating point arithmetic, and sort by coefficient values to propagate each variable only once. SCIP also features methods to extract clique information about the binary variables of a problem. A clique is a set of binary variables of which at most one variable can take the value 1 in a feasible solution. Clique information can be used to strengthen the propagation of knapsack constraints. Set cover constraints are propagated by the highly efficient two-watched literal scheme [33], which is based on the fact that the only domain reduction to be inferred from a set cover constraint is to fix a variable to 1 if all other variables have already been fixed to 0.

Variable and value selection takes place in Line 1; inference branching [2] is used as branching rule. Inference branching maintains statistics about how often the fixing of a variable led to fixings of other variables, i.e., it is a history rule, its essentially a MIP equivalent of impact-based search [29, 36]. Since history rules are often weak in the beginning of the search, we seed the CP solver with statistics that the MIP solver has collected in probing [39] during MIP presolving.

We assume that the propagation routines in Line 1 may also deduce global bound changes and modify the global bound vectors $l(P)$ and $u(P)$. Single-clause conflicts are automatically upgraded to global bound changes in Line 1. Note that it suffices to check constraint feasibility in Line 1, since the pseudo-solution $\bar{x}$ (see Line 1) will always take the value of one of the (integral) bounds for each variable.

Our implementation of the Rapid Learning heuristic uses a secondary SCIP instance to perform the CP search. Only a few parameters need to be altered from their default values to turn SCIP into a CP solver, an overview is given in Table 1. Most importantly, we disabled the LP relaxation and use a pure depth-first search with inference branching (but without any additional tie breakers). Further, we switch from All-UIP to 1-UIP in order to generate only one conflict per infeasibility. This is a typical behavior of CP solvers, but not for MIP solvers. Expensive feasibility checks and propagation of the objective function as a constraint are also avoided.

In order to avoid spending too much time in Rapid Learning, the number of nodes explored during the CP search is limited to at most $5000$. The actual number of allowed nodes is determined by the number of simplex iterations $\mathrm{iter}_{\mathrm{LP}}$ performed so far in the main SCIP but at least $500$, i.e.,

[TABLE]

The idea is to restrict Rapid Learning more rigorously for problems where processing of a single MIP node is cheap already. The number of simplex iterations is a deterministic estimate for node processing cost.

We aim to generate short conflict constraints, since these are most likely to frequently trigger propagations in the upcoming MIP search. Thus, we only collect conflicts that contain at most $5\text{\,}\mathrm{\char 37\relax}$ of the problem variables. Finally, we adapt the collection of branching statistics such that history information on general integer variables are collected per value in the domain rather than having one counter for down- and one for up-branches regardless of the value on which was branched. This can be essential for performing an efficient CP search on general integer variables, and was a building block that enabled us to use Rapid Learning on IPs rather than solely on BPs, as in [13].

In addition to the particular parameters listed in Table 1, we set the emphasis333In SCIP, emphasis settings correspond to a group of individual parameters being changed. for presolving to “fast”. Emphasis settings for cutting are not necessary, since no LP relaxation is solved, from the armada of primal heuristics only a few are applied that do not require an LP relaxation, see [5]. Note that since Rapid Learning will be called at the end of the MIP root node, or even locally, see next Section, the problem that the CP solver considers has already been presolved, might contain cutting planes as additional linear constraints and have an objective cutoff constraint if a primal solution has been found by a primal heuristic during root node processing.

4 Local Rapid Learning

The original Rapid Learning algorithm [13] was used as part of a root preprocessing, i.e., for every instance it was run exactly once at the end of the root node. But only running Rapid Learning at the root limits its effectiveness. We now discuss the factors that arise when we allow Rapid Learning to be run at local nodes inside the search tree

When running in the root only all information returned by the CP solver is globally valid, and the overhead to maintain the information gathered by Rapid Learning is negligible [13]. In contrast, when applying Rapid Learning at a local node within the tree conflicts and bound changes will only be locally valid in general. Since Rapid Learning uses a secondary SCIP instance to perform the CP search, all local information of the current node becomes part of the initial problem formulation for the CP search. Thus, conflicts gathered by Rapid Learning do not include bound changes made along the path from the root to the current node, they are simply considered as valid for this local node. As a consequence, these conflicts will only be locally valid and hence only applied to the current node of the MIP search. Using an assumption interface [34], local conflicts could be lifted to be globally valid. However, this is subject to future investigation and not considered in the current implementation of Rapid Learning.

In practice, all local information needs to be maintained when switching from one node of the tree to another. In CP solvers, switching nodes is typically very cheap, because depth-first search is used. However, a MIP solver frequently “jumps” within the tree. Therefore, two consecutively processed nodes can be quite different. In what follows, we will refer to the time spent for moving from one node to another node as switching time. The switching time can be used as an indicator to quantify the overhead introduced by all locally added information found by Rapid Learning.

To ensure that the amount of locally added information does not increase the switching time too much, we apply Rapid Learning very rarely by using a exponentially decreasing frequency of execution. Rapid Learning is executed at every node of depth $d$ with

[TABLE]

where $\beta$ and $f$ are two parameters to control the speed of decrease. For example, if $\beta=1$ Rapid Learning is executed at every depth $d=i\cdot f$ with $i\in\mathbb{Z}_{+}$.

Unfortunately, the amount of locally valid information produced by Rapid Learning still leads to an increase of switching time by $21\text{\,}\%$. Consequently, the overall performance decreased by $20\text{\,}\%$ in our first experiments. At the same time the number of explored branch-and-bound nodes decreased by $16\text{\,}\%$. This indicates the potential gains possible using local Rapid Learning.

To control at which subproblem Rapid Learning is applied we propose six criteria to forecast the potential of Rapid Learning. These criteria aim at identifying one of two situations. The first is to estimate whether the (sub)problem is infeasible or a pure feasibility problem. In these cases propagating conflicts is expected to be particularly beneficial. The second is to estimate the dual degeneracy of a problem. In this case, VSIDS branching statistics are expected to be particularly beneficial. The VSIDS [31] (variable state independent decaying sum) statistics takes the contribution of every variable (and its negated complement) in conflict constraints found so far into account. For every variable, the number of clauses (in MIP speaking: conflict constraints) the variable is part of is counted. In the remainder of the search the VSIDS are periodically scaled by a predefined constant. By this, the weight of older clauses is reduced over time and more recent observations have a bigger impact.

A basic solution of an LP is called dual degenerate, when it has nonbasic variables with zero reduced costs. One can define the dual degeneracy of a MIP as the average number of nonbasic variables with zero reduced costs appearing in a basic solution of its LP relaxation. The higher the dual degeneracy, the higher the chance that the LP objective will not change by branching and hence many of the costs involved in the pseudo-cost computation are zero. Therefore, for highly dual degenerate problems, using other branching criteria, such as VSIDS or inference scores, is crucial for solving the problem.

We now describe the six criteria we use to identify infeasible or dual degenerate problems, already using the criteria abbreviations from the tables in Section 5:

Criterion I: Dual Bound Improvement.

During the tree search a valid lower bound for each individual subproblem is given by the respective LP solution. A globally valid lower bound is given by the minimum over all individual lower bounds. This global bound is called the dual bound. If the dual bound has not changed after processing a certain number of nodes, i.e., the dual bound is equal to the lower bound of the root node, it might be the case that the MIP lies inside a level plane of the objective, i.e., all feasible LP (and MIP) solutions will have the same objective. In other words, the instance might be a feasibility instance for which Rapid Learning was already shown to be very successful [13]. Feasibility instances are typically highly dual degenerate. The dualbound criterion means to call local Rapid Learning if the dual bound never changed during the MIP search.

Criterion II: Leaves Pruned by Infeasibility or Exceeding the Cutoff bound.

During the tree search every leaf node either provides a new incumbent solution (the rare case), is proven to be infeasible or to exceed the current cutoff bound which is given by the incumbent solution. The ratio of the latter two cases is used in SCIP’s default branching rule. Hybrid branching [4] combines pseudo-costs, inference scores, and conflict information into one single branching score. The current implementation in SCIP puts a higher weight on conflict information, e.g., VSIDS [31], and a lower weight on pseudo-costs when the ratio of infeasible and cutoff nodes is larger than a predefined threshold. The leaves criterion means to call local Rapid Learning if the ratio of infeasible leaves over those exceeding the cutoff bound is larger than $10$. The rationale is that we expect (local) conflicts to be most beneficial, when infeasibility detection appears to be the main driver for pruning the tree.

Criterion III: LP Degeneracy.

As mentioned above, the more nonbasic variables are dual degenerate, the less information can be gained during strong branching or pseudo-cost computation. As a consequence, Berthold et al. [14] introduced a modification to strong branching that considers the dual degeneracy of the LP solution. In rough terms, if either the share of dual degenerate nonbasic variables or the variable-constraint ratio of the optimal face exceed certain thresholds, strong branching will be deactivated. We adapt this idea of using the dual degeneracy of the current LP solution. The degeneracy criterion means to call local Rapid Learning if more than $80\text{\,}\%$ of the nonbasic variables are degenerate or the variable-constraint ratio of the optimal face is larger than $2$, as proposed in [14]. In both cases we expect that “strong conflict generation” will be useful.

Criterion IV: (Local) Objective Function.

If all variables with non-zero objective coefficients are fixed at the local subproblem, i.e., the objective is constant, Criteria I and II will apply: every LP solution is fully dual degenerate and the only possibility to prune a leaf node is by infeasibility. If there are only very few unfixed variables with nonzero objective are left, the criteria might not apply. However, it is likely that the targeted situations occur frequently in the tree rooted at the current subproblem, at the latest, when all the variables occurring in the objective are fixed. The obj criterion means to call local Rapid Learning once the objective support is small enough, in anticipation of the current subproblem turning into a feasibility problem. In our implementation we apply this criterion very conservatively, and call Rapid Learning only if the local objective is zero.

Criterion V: Number of Solutions.

The most obvious evidence, and indeed a necessary one, that a MIP instance is infeasible, is that no feasible solution has been found during the course of the MIP search. Note that for most (feasible) MIP instances, primal heuristics find a feasible solution at the root node [11] or at the latest during the first dive in the branch-and-bound. The nsols criterion means to call local Rapid Learning if no feasible solution has been found so far.

Criterion VI: Strong Branching Improvements.

In the beginning of the tree search it is very unlikely that enough leaf nodes are explored to reliably guess whether the actual MIP is a feasibility instance. Therefore, we consider the subproblems evaluated during strong branching, which are concentrated at the top of the search tree. Similarly to Criterion II, we compute the ratio between the number of strong branching problems that gave no improvement in the objective or went infeasible to the number of strong branching problems where we observed an objective change. The sblps criterion means to call local Rapid Learning if this ratio exceeds a threshold of $10$, hence strong branching does not appear to be efficient for generating pseudo-cost information.

In addition to the exponentially decreasing frequency and the six criteria above, we applied the following three changes to the original implementation of Rapid Learning used in [13].

• •

We limited the number of conflict constraints transferred from Rapid Learning back to the original search tree to ten. This corresponds to the SCIP parameter conflict/maxconss for the maximal allowed number of added conflicts per call of conflict analysis. We greedily use the shortest conflicts.

• •

We prefer conflict constraints that have a linear representation over bound disjunction constraints (see Definition 2).

• •

To exploit performance variability [19, 30] every CP search is initialized with a different pseudo-random seed.

5 Computational results

To evaluate how local Rapid Learning impacts IP solving performance we used the academic constraint integer programming solver SCIP 6.0 [24] (with SoPlex 4.0 as LP solver) and extended the existing code of Rapid Learning. The original implementation of Rapid Learning was already shown to significantly improve the performance of SCIP 1.2.0.5 on pure binary instances [13]. In this setting, Rapid Learning was applied exactly once at the root node. However, during the last eight years SCIP has changed in many places. In SCIP 6.0, Rapid Learning is deactivated by default, since it led to a big performance variability.

Therefore, we use SCIP without Rapid Learning (as it is the current default) as a baseline. We will refer to this setting as default. In our computational experiments we evaluate the impact of local Rapid Learning if one or more of the criteria described in Section 4 are fulfilled. In the following, we will refer to the criteria I–VI as dualbound, leaves, degeneracy, obj, nsols, and sblps, respectively. Within the tree, Rapid Learning is applied with an exponentially decreasing frequency (see Section 4). In our experiments, we used $f=5$ and $\beta=4$, i.e., Rapid Learning is called at depths $d$ with $log_{4}(d/5)\in\mathbb{Z}$, i.e., $d=0,5,20,80,320\ldots$, if one of the six criteria is fulfilled.

As a test set we used all pure integer problems of Miplib 3 [16], Miplib 2003 [8] and the Miplib 2010 [28] benchmark set. This test set consists of 71 publicly available instances, which we will refer to as MMM-IP. To account for the effect of performance variability [19, 30] all experiments were performed with five different global random seeds. Every pair of instance and seed was treated as an individual observation, effectively resulting in a test set of 355 instances. We will use the term “instance” when actually referring to an instance-seed-combination. The experiments were run on a cluster of identical machines, each with an Intel Xeon E5-2690 with 2.6 GHz and 128 GB of RAM; a time limit of $3600$ seconds was set.

In a first experiment we evaluated the efficacy of each individual criterion and global Rapid Learning as published in [13]. Aggregated results are shown in Table 2, section Exp.1. For detailed results see Table LABEL:tab:MMM-IP_exp1_detailed in the appendix. For every setting, the table shows the number of solved instances out of $71$ (solved), shifted geometric means [3] of the absolute solving time in seconds (time, shift = $1$) and number of explored nodes (nodes, shift = $100$), as well as the relative solving time (timeQ) and number of nodes (nodesQ) w.r.t. to default as a baseline. Local Rapid Learning without any of the presented criteria (nochecks) leads to a performance decrease of $20\text{\,}\%$ on the complete test set MMM-IP compared to default. Always applying Rapid Learning only at the root (onlyroot), which corresponds to Rapid Learning as published in [13], leads to slowdown of $7.8\text{\,}\%$ but solves three instance more. For this settings, we could observe a performance decrease of $38.1\text{\,}\%$ on the group of instances that are not affected444An instance is called affected when the solving path changes. by Rapid Learning. To avoid a computational overhead and performance variability on instances where Rapid Learning is not expected to be beneficial, we apply the criteria degeneracy, obj, and nsols at the root node, too. Afterwards, the performance decrease of global Rapid Learning reduced to $0.9\text{\,}\%$. The computational results indicate that almost all individual criteria are useful on their own. The solving time and generated nodes can be reduced by up to $6\text{\,}\%$ and $15\text{\,}\%$, respectively, on the complete test set of $71$. The exception is the obj criterion, which leads to a marginal slowdown of $0.6\text{\,}\%$, but solves three more instance than default. On the group of affected instances the solving time and number of generated nodes can be reduced up to $8\text{\,}\%$ and $19\text{\,}\%$, respectively, both by using degeneracy.

The impact of the individual criteria on the solving time is illustrated in Figure 2. For each criterion, the box plot [32] shows the median (dashed line), and the 1st and 3rd quartile (shaded box) of all observations. The plot shows that degeneracy performs best since it leads by far to the best improvement on the 1st quartile. In contrast to that, the 3rd quartile of nsols indicates this criterion leads to a deterioration of more than $10\text{\,}\mathrm{\char 37\relax}$ on $25\text{\,}\mathrm{\char 37\relax}$ of the instances.

Grouping all instances of MMM-IP based on the degeneracy at the end of the root node shows the importance of this criterion. On the group of instances where at least $1\text{\,}\%$ of the variables is dual degenerate at the end of the root node Rapid Learning leads to a performance improvement of $9.1\text{\,}\%$. On all instances where at least $80\text{\,}\%$ of the variable are dual degenerate at the root node, we could observe a reduction of solving time by $28.8\text{\,}\%$. Note that this was one of the two thresholds for the degeneracy criterion.

In a second experiment (Table 2, section Exp.2; Table LABEL:tab:MMM-IP_exp2_detailed in the appendix) we combined all individual criteria. Combining two or more criteria leads to more aggressive version of Rapid Learning since it runs if at least one of the chosen criteria is satisfied. The two (out of fifteen) best pairwise combinations as well as the (most aggressive) combination of all six criteria are shown in Table 2. Interestingly, no combined setting is superior to degeneracy. The combination of degeneracy and leaves, which were the two outstanding criteria in the individual test, performs almost the same as the degeneracy criterion alone.

For a final experiment we choose degeneracy as the best criterion, since it was one of two criteria that solved an additional instance, clearly showed the best search reduction, and was a close second to leaves with respect to running time. Our final experiment evaluates the impact of the individual information gained from local Rapid Learning. To this end, we individually deactivated transferring variable bounds, conflict constraints, inference information, and primal feasible solutions (see Table 3; Table LABEL:tab:MMM-IP_exp3_detailed in the appendix). This experiment indicates that primal solutions are the most important information for the remainder of the MIP search. When ignoring solutions found during the CP search, the overall solving time increased by $9.9\text{\,}\%$ (primsols). When ignoring conflict constraints, the original motivation of Rapid Learning, solving time increased by $4.4\text{\,}\%$ (conflicts). Both transferring variable bounds and inference information proved beneficial, with a $1.4\text{\,}\%$ (variablebounds) and $0.6\text{\,}\%$ (infervals) impact on performance, respectively. It is not surprising that finding primal solutions has the largest effect. Firstly, they are applied globally, in contrast to bound changes and conflicts. Secondly, highly dual degenerate problems are known to be cumbersome not only for MIP branching but also for primal heuristics [11], which means that solution-generating procedures that do not rely on solving LPs are particularly promising for such problems.

6 Conclusion

In this paper, we extended the idea of Rapid Learning [13]. Firstly, we generalized Rapid Learning to integer programs and described the details that were necessary for doing so: value-based inference branching, additional propagators and generalized conflict constraints, most of which were already available in SCIP. Secondly, we applied Rapid Learning repeatedly during the search. This generates a true hybrid CP/MIP approach, with two markedly different search strategies communicating information forth and back. To this end, we introduced six heuristic criteria to decide when to start local Rapid Learning. Those criteria are based on degeneracy information, branch-and-bound statistics, and the local structure of the problem. Our computational experiments showed a speed-up of up to $7\text{\,}\%$ when applying local Rapid Learning in SCIP. Calling local Rapid Learning depending on the local degree of dual degeneracy is the best strategy found in our experiments.

Interesting future work in this direction includes: extending the CP search to generate global conflicts at local nodes using an assumption interface, running the CP search in a parallel thread where whenever the MIP solver moves to a new node the CP search restarts from that node, and extending the method to handle problems that include continuous variables.

Acknowledgments

The work for this article has been partly conducted within the Research Campus MODAL funded by the German Federal Ministry of Education and Research (BMBF grant number 05M14ZAM). We thank the anonymous reviewers for their valuable suggestions and helpful comments.