Minimal Conditions for Beneficial Neighbourhood Search and Local Descent

TL;DR

This paper establishes the first proof that certain neighborhood properties enhance local search effectiveness, demonstrating conditions where local blind descent outperforms blind search in reaching better solutions efficiently.

Contribution

It introduces a novel proof linking neighborhood locality and cost reduction to improved local search success, supported by illustrative examples and experimental validation.

Findings

Neighborhood locality supports beneficial local search.

Local blind descent can outperform blind search in expected steps.

Switching to local descent improves efficiency near the optimum.

Abstract

This paper investigates what properties a neighbourhood requires to support beneficial local search. We show that neighbourhood locality, and a reduction in cost probability towards the optimum, support a proof that search among neighbours is more likely to find an improving solution in a single search step than blind search. This is the first paper to introduce such a proof. The concepts underlying these properties are illustrated on a satisfiability problem class, and on travelling salesman problems. Secondly, for a given cost target t, we investigate a combination of blind search and local descent termed local blind descent, and present various conditions under which the expected number of steps to reach a cost better than t using local blind descent, is proven to be smaller than with blind search. Experiments indicate that local blind descent, given target cost t, should switch to local descent at a starting cost that reduces as t approaches the optimum.