Evolution is Still Good: Theoretical Analysis of Evolutionary Algorithms on General Cover Problems

TL;DR

This paper provides a theoretical analysis demonstrating that a generalized multi-objective evolutionary algorithm (GSEMO) can efficiently achieve near-optimal solutions for various complex cover problems, including submodular and non-submodular cases.

Contribution

It introduces a unified analysis framework for GSEMO and proves its effectiveness across a broad class of general cover problems with tight approximation guarantees.

Findings

GSEMO achieves asymptotically tight approximation ratios.

The analysis applies to both submodular and non-submodular cover problems.

Expected polynomial time convergence is established.

Abstract

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently embedded in many evolutionary algorithms. In this paper, we identify such a relation by proposing a unified analysis framework for a generalized simple multi-objective evolutionary algorithm (GSEMO), and apply it on a minimum weight general cover problem. For a wide range of problems (including the the minimum submodular cover problem in which the submodular function is real-valued, and the minimum connected dominating set problem for which the potential function is non-submodular), GSEMO yields asymptotically tight approximation ratios in expected polynomial time.