Optimizing Monotone Chance-Constrained Submodular Functions Using Evolutionary Multi-Objective Algorithms

TL;DR

This paper presents a runtime analysis of evolutionary multi-objective algorithms for chance-constrained submodular functions, demonstrating their effectiveness and comparing them to greedy algorithms in uncertain optimization scenarios.

Contribution

It provides the first theoretical runtime analysis of Pareto-based evolutionary algorithms for chance-constrained submodular optimization, including performance guarantees and practical insights.

Findings

GSEMO achieves similar worst-case guarantees as greedy algorithms for certain weight distributions.

Tail bounds can hinder GSEMO's performance on non-monotone submodular functions.

Evolutionary algorithms outperform greedy methods in network submodular optimization problems.

Abstract

Many real-world optimization problems can be stated in terms of submodular functions. Furthermore, these real-world problems often involve uncertainties which may lead to the violation of given constraints. A lot of evolutionary multi-objective algorithms following the Pareto optimization approach have recently been analyzed and applied to submodular problems with different types of constraints. We present a first runtime analysis of evolutionary multi-objective algorithms based on Pareto optimization for chance-constrained submodular functions. Here the constraint involves stochastic components and the constraint can only be violated with a small probability of alpha. We investigate the classical GSEMO algorithm for two different bi-objective formulations using tail bounds to determine the feasibility of solutions. We show that the algorithm GSEMO obtains the same worst case performance guarantees for monotone submodular functions as recently analyzed greedy algorithms for the case of uniform IID weights and uniformly distributed weights with the same dispersion when using the appropriate bi-objective formulation. As part of our investigations, we also point out situations where the use of tail bounds in the first bi-objective formulation can prevent GSEMO from obtaining good solutions in the case of uniformly distributed weights with the same dispersion if the objective function is submodular but non-monotone due to a single element impacting monotonicity. Furthermore, we investigate the behavior of the evolutionary multi-objective algorithms GSEMO, NSGA-II and SPEA2 on different submodular chance-constrained network problems. Our experimental results show that the use of evolutionary multi-objective algorithms leads to significant performance improvements compared to state-of-the-art greedy algorithms for submodular optimization.