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

Recently, the property of connectedness has been claimed to give a strong motivation on the design of local search techniques for multiobjective combinatorial optimization. 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, ideally a representative sample of efficient solutions. In this paper, we propose the biobjective long and multiple path problems. We show experimentally that, on the first problem, 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 problem, 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 multiobjective local search algorithms. This work opens new discussions on a proper definition of multiobjective fitness landscapes.