Exact and Approximate High-Multiplicity Scheduling on Identical Machines

TL;DR

This paper improves the framework for solving high-multiplicity scheduling problems on identical machines by introducing new preprocessing and encoding techniques, leading to faster algorithms and new complexity bounds.

Contribution

It presents three tools to enhance the existing framework, including problem-specific preprocessing, LP relaxation with Frank-Tardos encoding, and a new bound on the integer hull vertices, improving algorithm efficiency.

Findings

Improved running time from $( ext{log}(C_{ ext{max}}))^{2^{O(d)}}enc(I)^{O(1)}$ to $( ext{log}(p_{ ext{max}}))^{2^{O(d)}}enc(I)^{O(1)}$.

Established new parameterized lower bounds for $P||C_{ ext{max}}$.

Connected the fixed-parameter tractability of $P||C_{ ext{max}}$ with that of $Q||C_{ ext{max}}$ based on job and machine types.

Abstract

Goemans and Rothvoss (SODA'14) gave a framework for solving problems which can be described as finding a point in int$.$cone$(P\cap\mathbb{Z}^N)\cap Q$, where $P,Q\subset\mathbb{R}^N$ are (bounded) polyhedra. The running time for solving such a problem is $enc(P)^{2^{O(N)}}enc(Q)^{O(1)}$. This framework can be used to solve various scheduling problems, but the encoding length $enc(P)$ usually involves large parameters like the makespan. We describe three tools to improve the framework: - Problem-specific preprocessing can be used to greatly reduce $enc(P)$. - By solving a certain LP relaxation and then using the classical result by Frank and Tardos (J. Comb. '87), we get a more compact encoding of $P$ in general. - A result by Jansen and Klein (SODA'17) makes the running time depend on the number of vertices of the integer hull of $P$. We provide a new bound for this number that is similar to the one by Berndt et al. (SOSA'21) but better for our setting. For example, applied to the scheduling problem $P||C_{\max}$, these tools improve the running time from $(\log(C_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$ to the possibly much better $(\log(p_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$. Here, $p_{\max}$ is the largest processing time, $d$ is the number of different processing times, $C_{\max}$ is the makespan and $enc(I)$ is the encoding length of the instance. On the complexity side, we use reductions from the literature to provide new parameterized lower bounds for $P||C_{\max}$. Finally, we show that the big open question asked by Mnich and van Bevern (Comput. Oper. Res. '18) whether $P||C_{\max}$ is FPT w.r.t. the number of job types $d$ has the same answer as the question whether $Q||C_{\max}$ is FPT w.r.t. the number of job and machine types $d+\tau$ (all in high-multiplicity encoding). The same holds for objective $C_{\min}$.