On the approximation ability of evolutionary optimization with application to minimum set cover

TL;DR

This paper analyzes the approximation capabilities of a specific evolutionary algorithm, SEIP, demonstrating its effectiveness in solving NP-hard set cover problems with near-optimal approximation ratios.

Contribution

The paper provides the first analysis of the approximation performance of SEIP, showing it achieves optimal and best-known approximation ratios for set cover problems.

Findings

SEIP achieves an $H_n$-approximation ratio for unbounded set cover.

SEIP attains an $(H_k-rac{k-1}{8k^9})$-approximation for k-set cover.

SEIP can overcome greedy algorithm limitations using mutation strategies.

Abstract

Evolutionary algorithms (EAs) are heuristic algorithms inspired by natural evolution. They are often used to obtain satisficing solutions in practice. In this paper, we investigate a largely underexplored issue: the approximation performance of EAs in terms of how close the solution obtained is to an optimal solution. We study an EA framework named simple EA with isolated population (SEIP) that can be implemented as a single- or multi-objective EA. We analyze the approximation performance of SEIP using the partial ratio, which characterizes the approximation ratio that can be guaranteed. Specifically, we analyze SEIP using a set cover problem that is NP-hard. We find that in a simple configuration, SEIP efficiently achieves an $H_n$-approximation ratio, the asymptotic lower bound, for the unbounded set cover problem. We also find that SEIP efficiently achieves an $(H_k-\frac{k-1}/{8k^9})$-approximation ratio, the currently best-achievable result, for the k-set cover problem. Moreover, for an instance class of the k-set cover problem, we disclose how SEIP, using either one-bit or bit-wise mutation, can overcome the difficulty that limits the greedy algorithm.