Minimal Conditions for Beneficial Local Search

TL;DR

This paper explores the theoretical foundations and practical benefits of local search in optimization problems, demonstrating conditions under which neighborhood search outperforms blind search.

Contribution

It provides novel proofs showing when neighborhood search is more effective than blind search and investigates how problem properties influence this advantage.

Findings

Neighborhood search is more likely to find improvements in a single step.

Sequences of neighborhood searches tend to yield greater improvements.

Theoretical properties are validated through experiments on combinatorial problems.

Abstract

This paper investigates why it is beneficial, when solving a problem, to search in the neighbourhood of a current solution. The paper identifies properties of problems and neighbourhoods that support two novel proofs that neighbourhood search is beneficial over blind search. These are: firstly a proof that search within the neighbourhood is more likely to find an improving solution in a single search step than blind search; and secondly a proof that a local improvement, using a sequence of neighbourhood search steps, is likely to achieve a greater improvement than a sequence of blind search steps. To explore the practical impact of these properties, a range of problem sets and neighbourhoods are generated, where these properties are satisfied to different degrees. Experiments reveal that the benefits of neighbourhood search vary dramatically in consequence. Random problems of a classical combinatorial optimisation problem are analysed, in order to demonstrate that the underlying theory is reflected in practice.