A new bound for the midpoint solution in minmax regret optimization with an application to the robust shortest path problem

Minmax regret optimization aims at finding robust solutions that perform best in the worst-case, compared to the respective optimum objective value in each scenario. Even for simple uncertainty sets like boxes, most polynomially solvable optimization problems have strongly NP-hard minmax regret counterparts. Thus, heuristics with performance guarantees can potentially be of great value, but only few such guarantees exist. A very easy but effective approximation technique is to compute the midpoint solution of the original optimization problem, which aims at optimizing the average regret, and also the average nominal objective. It is a well-known result that the regret of the midpoint solution is at most 2 times the optimal regret. Besides some academic instances showing that this bound is tight, most instances reveal a way better approximation ratio. We introduce a new lower bound for the optimal value of the minmax regret problem. Using this lower bound we state an algorithm that gives an instance dependent performance guarantee of the midpoint solution for combinatorial problems that is at most 2. The computational complexity of the algorithm depends on the minmax regret problem under consideration; we show that the sharpened guarantee can be computed in strongly polynomial time for several classes of combinatorial optimization problems. To illustrate the quality of the proposed bound, we use it within a branch and bound framework for the robust shortest path problem. In an experimental study comparing this approach with a bound from the literature, we find a considerable improvement in computation times. ∗Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-13-1-3066. The U.S Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon. †Corresponding author. Email: [email protected]