We present a simple neighborhood search that aims at efficiently approximating the Pareto set of particularly difficult multiobjective combinatorial optimization problems. Unlike (elitist) local search which exclusively accepts improving neighbors, the proposed walk explores a subset of λ neighbors only, among which the best one is accepted, whether it improves the current solution or not. This resembles the comma-selection from evolutionary computation, which is to contrast with the plus-selection. This principle was shown to better escape basins of attraction in the context of single-objective optimization. Non-elitism has also been recently revised in the context of multiobjective optimization. However, we emphasize that we do not generate random offspring here. Instead, we create a whole set of λ neighbors among which one is carefully selected, thus providing better control of the selection pressure. In order to extend the partial neighborhood walk principles to multiobjective search, we rely on decomposition: multiple scalarizing subproblems are uniformly defined and optimized (independently or cooperatively) in order to form a whole approximation set. Based on a benchmark of difficult NK-landscapes with two and three objectives, we show that even independent sub-problem solving results in a clear improvement over more advanced multiobjective decomposition approaches such as MOEA/D. We further report an in-depth analysis of the neighborhood sample size, the number of sub-problems, and the cooperation among them.