Multi-Party Multi-Objective Optimization as Consensus Search: Runtime Analysis of Cross-Party Recombination

TL;DR

This paper analyzes the runtime of cross-party recombination in multi-party multi-objective optimization, demonstrating improved efficiency over baseline methods and exploring coverage and approximation bounds in specific problem settings.

Contribution

It provides the first theoretical runtime analysis of cross-party recombination in MPMOPs, introducing novel algorithms and bounds for specific problem instances.

Findings

CPR-NSGA-II discovers Pareto-optimal solutions in O(n log n) evaluations.

Flattening objectives increases coverage burden compared to multi-party formulations.

Derived runtime bounds separate effects of local filling, recombination, and repair ambiguity.

Abstract

Multi-party multi-objective optimization problems (MPMOPs) require consensus among autonomous decision makers and therefore differ from flattened many-objective formulations. Existing runtime theory for multi-objective evolutionary algorithms is largely tailored to single-party Pareto-front approximation and does not directly explain common-solution search in MPMOPs. We investigate cross-party recombination in two representative settings. On MP-JCG, a pseudo-Boolean benchmark with an explicit gap region, we prove that a payoff-guided mutation baseline faces a gap-crossing bottleneck requiring \(\Theta(n^2)\) expected fitness evaluations. In contrast, an analytical CPR-NSGA-II variant discovers both common Pareto-optimal solutions in \(O(n\log n)\) expected evaluations by directly assembling complementary prefix and suffix templates distributed across party populations. Comparing this with the flattened four-objective formulation F-JCG, our full-front coverage analysis illustrates the additional coverage burden introduced by flattening. For BPBOMST, the bi-party, two-objective-per-party specialization of the multi-party multi-objective minimum spanning tree problem, we develop a layered support-cover analysis. For each common Pareto objective vector, the symmetric average projection induces an auxiliary bi-objective MST instance, and suitable support representatives yield a \(2\lambda\)-common approximation cover with \(\lambda\in[1,2]\). We further derive an instance-parameterized expected runtime bound for a representative-pool CPR-NSGA-II variant using edge-union recombination and uniform repair. This bound separates the effects of local auxiliary-front filling, cross-party recombination shortcuts, and edge-union repair ambiguity.