A fix-and-optimize heuristic for the minmax regret shortest path arborescence problem under interval uncertainty

The minmax regret shortest path arborescence (M-SPA) problem under interval uncertainty is a robust counterpart of the shortest path (SP) arborescence problem, where arc costs are modeled as intervals of possible values. This problem finds applications in the design of topologies for low-power wireless personal area networks. The previous work presented a mixed-integer linear programming formulation and heuristics for this problem. In this paper, we propose a new heuristic for M-SPA, called the fix-and-optimize through heuristic decomposition (FO-HD). It consists of two steps. The first step solves multiple instances of the minmax regret SP problem under interval uncertainty, while the second step combines their solutions into an arborescence. As far as we know, this is the first work in the literature to apply the fix-and-optimize metaheuristic to minmax regret problems. We also show that a lower bound for the optimal solution of M-SPA can be computed from the first step of FO-HD. This lower bound allows us to show that the maximum average optimality gap of this heuristic was 0.2% among all classical and novel instances used in the computational experiments.