Approximation Complexity of min-max (Regret) Versions of Shortest Path, Spanning Tree, and Knapsack

This paper investigates, for the first time in the literature, the approximation of min-max (regret) versions of classical problems like shortest path, minimum spanning tree, and knapsack. For a bounded number of scenarios, we establish fully polynomial-time approximation schemes for the min-max versions of these problems, using relationships between multi-objective and min-max optimization. Using dynamic programming and classical trimming techniques, we construct a fully polynomial-time approximation scheme for min-max regret shortest path. We also establish a fully polynomial-time approximation scheme for minmax regret spanning tree and prove that min-max regret knapsack is not at all approximable. We also investigate the case of an unbounded number of scenarios, for which min-max and min-max regret versions of polynomial-time solvable problems usually become strongly NP -hard. In this setting, non-approximability results are provided for min-max (regret) versions of shortest path and spanning tree.