  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.
