Approximating covering problems by randomized search heuristics using multi-objective models

Abstract

The main aim of randomized search heuristics is to produce good approximations of optimal solutions within a small amount of time. In contrast to numerous experimental results, there are only a few theoretical ones on this subject. We consider the approximation ability of randomized search heuristics for the class of covering problems and compare single-objective and multi-objective models for such problems. For the VertexCover problem, we point out situations where the multi-objective model leads to a fast construction of optimal solutions while in the single-objective case even no good approximation can be achieved within expected polynomial time. Examining the more general SetCover problem we show that optimal solutions can be approximated within a factor of log n, where n is the problem dimension, using the multi-objective approach while the approximation quality obtainable by the single-objective approach in expected polynomial time may be arbitrarily bad.

Homogeneous and Heterogeneous Island Models for the Set Cover Problem

“…This behaviour can be mimicked by SEMO [14] and the homogeneous island model. Friedrich et al [14] define the potential of the population of the archipelago as the largest k such that there is an individual in the population that covers k elements and costs at most (H m − H m−k ) · OPT.…”

“…Theorem 1 (Friedrich et al [14]). For any initialisation and every SetCover instance, SEMO and global SEMO find an H m -approximate solution in O(m 2 n + mn log(nc max )) expected generations.…”

“…In their analysis of SEMO, Friedrich et al [14] consider the time until SEMO finds an empty selection, and how long it takes to get a H m -approximate solution from there. Their results are as follows.…”

“…This classic NP-hard problem is one of the most fundamental problems in computer science. Friedrich et al [14] studied a SEMO algorithm on a biobjective formulation of the problem and showed that SEMO efficiently computes an H m -approximation, where…”

“…Both island models are initialised with empty selections. This is a sensible strategy for SetCover and theoretical results [14] as well as preliminary experiments have shown that this only speeds up computation.…”