Worst Case Analysis of Max-Regret, Greedy and Other Heuristics for Multidimensional Assignment and Traveling Salesman Problems

Optimization heuristics are often compared with each other to determine which one performs best by means of worst-case performance ratio reflecting the quality of returned solution in the worst case. The domination number is a complement parameter indicating the quality of the heuristic in hand by determining how many feasible solutions are dominated by the heuristic solution. We prove that the MaxRegret heuristic introduced by Balas and Saltzman (1991) finds the unique worst possible solution for some instances of the s-dimensional (s ≥ 3) assignment and asymmetric traveling salesman problems of each possible size. We show that the Triple Interchange heuristic (for s = 3) also introduced by Balas and Saltzman and two new heuristics (Part and Recursive Opt Matching) have factorial domination numbers for the s-dimensional (s ≥ 3) assignment problem.