Worst case analysis of Max-Regret, Greedy and other heuristics for Multidimensional Assignment and Traveling Salesman Problems

Abstract

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 Max-Regret heuristic introduced by Balas and Saltzman (Oper. Res. 39:150-161, 1991) finds the unique worst possible solution for some instances of the sdimensional (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 Match-G. Gutin ( )

Greedy Like Algorithms for the Traveling Salesman and Multidimensional Assignment Problems

“…However, after conducting wider computational experiments, Robertson [25] came to a different conclusion: the greedy algorithm and max-regret are of similar quality for 3-AP. Gutin, Goldengorin and Huang [11] share the conclusion of Robertson: both greedy algorithm and max-regret are of domination number 1 for s-AP for each s ≥ 3. Moreover, there exists a common family of s-AP instances for which both heuristics find the unique worst assignment [11] (for each s ≥ 3).…”

“…Gutin, Goldengorin and Huang [11] share the conclusion of Robertson: both greedy algorithm and max-regret are of domination number 1 for s-AP for each s ≥ 3. Moreover, there exists a common family of s-AP instances for which both heuristics find the unique worst assignment [11] (for each s ≥ 3). Similarly to TSP, we may obtain MAP heuristics of factorial domination number if we consider not-worth-than-average heuristics.…”

“…Extensive computational experiments in [8] demonstrated a clear superiority of max-regret-fc over the greedy algorithm and several other construction heuristics from [9]. Therefore, the result of Theorem 6 obtained by Gutin, Goldengorin and Huang [11] was somewhat unexpected. Let Q be a collection of disjoint directed paths in and let V = V ( ) ={1, 2, … , n}.…”

“…) s-1 . Using Theorem 10, it is proved in [11] We assign the weights as follows: w(e i ) = 1 for every i = 1, 2, … , n, w(e (i,j,F ) ) = 2 for each of the 2 s-1 -1 sets F and 1 ≤ i ≠ j ≤ n and w(e) = 0 for every vector e ∈ X that has at least three coordinates of different value. Let F 0 be the first set F chosen by MDV.…”

The Pollyanna Problem

“…Approximations to the multi-dimensional assignment problem were generated using the least squares method and results improved the previously known bound (Kuroki and Matsui, 2007). A worst case analysis of collected heuristics for the multidimensional assignment problem was completed and a domination number established for each heuristic (Gutin et al, 2008). The asymptotic behavior of large scale multidimensional assignment problems (Krokhmal et al, 2007).…”