Maximizing Submodular or Monotone Functions under Partition Matroid Constraints by Multi-objective Evolutionary Algorithms

TL;DR

This paper extends the theoretical analysis of the GSEMO evolutionary algorithm to partition matroid constraints, demonstrating its efficiency and effectiveness in maximizing submodular functions, with experimental validation on graph cut problems.

Contribution

It generalizes existing runtime analysis of GSEMO from cardinality to partition matroid constraints and empirically compares its performance against GREEDY algorithms.

Findings

GSEMO guarantees good approximation within polynomial runtime.

GSEMO outperforms GREEDY in quadratic runtime on graph cut problems.

Theoretical extension to partition matroid constraints enhances applicability.

Abstract

Many important problems can be regarded as maximizing submodular functions under some constraints. A simple multi-objective evolutionary algorithm called GSEMO has been shown to achieve good approximation for submodular functions efficiently. While there have been many studies on the subject, most of existing run-time analyses for GSEMO assume a single cardinality constraint. In this work, we extend the theoretical results to partition matroid constraints which generalize cardinality constraints, and show that GSEMO can generally guarantee good approximation performance within polynomial expected run time. Furthermore, we conducted experimental comparison against a baseline GREEDY algorithm in maximizing undirected graph cuts on random graphs, under various partition matroid constraints. The results show GSEMO tends to outperform GREEDY in quadratic run time.