On the Effect of Connectedness for Biobjective Multiple and Long Path Problems

TL;DR

This paper investigates how the property of connectedness affects local search algorithms in multiobjective combinatorial optimization, revealing that connectedness alone does not determine the effectiveness of local search versus evolutionary algorithms.

Contribution

The study introduces biobjective multiple and long path problems to experimentally analyze the impact of connectedness on local search performance in MOCO.

Findings

Connectedness does not guarantee local search superiority.

Local search can outperform evolutionary algorithms on connected efficient sets.

Disconnected efficient sets can be effectively approximated by local search.

Abstract

Recently, the property of connectedness has been claimed to give a strong motivation on the design of local search techniques for multiobjective combinatorial optimization (MOCO). Indeed, when connectedness holds, a basic Pareto local search, initialized with at least one non-dominated solution, allows to identify the efficient set exhaustively. However, this becomes quickly infeasible in practice as the number of efficient solutions typically grows exponentially with the instance size. As a consequence, we generally have to deal with a limited-size approximation, where a good sample set has to be found. In this paper, we propose the biobjective multiple and long path problems to show experimentally that, on the first problems, even if the efficient set is connected, a local search may be outperformed by a simple evolutionary algorithm in the sampling of the efficient set. At the opposite, on the second problems, a local search algorithm may successfully approximate a disconnected efficient set. Then, we argue that connectedness is not the single property to study for the design of local search heuristics for MOCO. This work opens new discussions on a proper definition of the multiobjective fitness landscape.