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Minimal Conditions for Beneficial Neighbourhood Search and Local Descent

Mark G. Wallace

TL;DR

It is shown 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.

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.

Minimal Conditions for Beneficial Neighbourhood Search and Local Descent

TL;DR

It is shown 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.

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.

Paper Structure

This paper contains 45 sections, 12 theorems, 92 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Neighbourhood search and local descent
  3. Structure of the paper
  4. Related Work
  5. Cost Function
  6. Locality
  7. Local descent
  8. Properties needed for neighbourhood search to be beneficial
  9. Definitions
  10. Neighbourhood search symbols
  11. Neighbourhood Weight
  12. Starting cost
  13. Neighbours with no cost difference
  14. Probability of Improvement
  15. Definitions and Lemmas
  16. ...and 30 more sections

Key Result

Lemma 1

Assuming: it follows that

Figures (5)

  • Figure 1: MAX-2-SAT probability at each cost level
  • Figure 2: TSP 10 satisfies conditions for theorem \ref{['thm:avrgt1']}
  • Figure 3: TSP 80 cost probabilities and NWeights
  • Figure 4: TSP 10 instance, rates of improvement
  • Figure 5: neighbourhood weights

Theorems & Definitions (35)

  • Definition 1: NWeight
  • Definition 2
  • Definition 3
  • proof
  • Definition 4
  • Definition 5
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • ...and 25 more