Landscape properties of the very large-scale and the variable neighborhood search metaheuristics for the multidimensional assignment problem

TL;DR

This paper analyzes the landscape properties of large-scale neighborhood search algorithms for the multidimensional assignment problem, revealing structural insights that can guide the development of more effective metaheuristics.

Contribution

It extends the understanding of landscape structures in large-scale neighborhood searches and introduces a generalized variable neighborhood search for the MAP.

Findings

The search graph generalizes a hypercube structure.

The new variable neighborhood search can explore larger neighborhoods.

Landscape analysis reveals correlations between fitness and solution proximity.

Abstract

We study the recent metaheuristic search algorithm for the multidimensional assignment problem (MAP) using fitness landscape theory. The analyzed algorithm performs a very large-scale neighborhood search on a set of feasible solutions to the problem. We derive properties of the landscape graphs that represent these very large-scale search algorithms acting on the solutions of the MAP. In particular, we show that the search graph is generalization of a hypercube. We extend and generalize the original very large-scale neighborhood search to develop the variable neighborhood search. The new search is capable or searching even larger large-scale neighborhoods. We perform numerical analyses of the search graph structures for various problem instances of the MAP and different neighborhood structures of the MAP algorithm based on a very large-scale search. We also investigate the correlation between fitness (i.e., objective values) and distance (i.e., path lengths) of the local minima (i.e., sinks of the landscape). Our results can be used to design improved search-based metaheuristics for the MAP.