Reusing Optimal TSP Solutions for Locally Modified Input Instances

Abstract

Abstract. Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e. g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification-U ) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i. e., whether lm-U is computationally more tractable than U . Here, we give non-trivial examples both of problems where this is and problems where this is not the case. Our main results are these:1. The local modification to change the cost of a singular edge turns the traveling salesperson problem (TSP) into a problem lm-TSP which is as hard as TSP itself, i. e., unless P = N P , there is no polynomial-time p(n)-approximation algorithm for lm-TSP for any polynomial p. Moreover, lm-TSP where inputs must satisfy the β-triangle inequality (lm-∆ β -TSP) remains NP-hard for all β > 1 2 . 2. For lm-∆-TSP (i. e., metric lm-TSP), an efficient 1.4-approximation algorithm is presented. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. 3. Similarly, for all 1 < β < 3.34899, we achieve a better approximation ratio for lm-∆ β -TSP than for ∆ β -TSP. 4. Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem.

Knowing All Optimal Solutions Does Not Help for TSP Reoptimization

“…Since then, reoptimization has been applied to various problems like the TSP [4][5][6][7], the Steiner tree problem [8][9][10][11], the shortest common superstring problem [12,13], the knapsack problem [14], and several covering problems [15].…”

“…We denote the resulting reoptimization problem by Inc-Edge-Reopt-∆TSP. This reoptimization problem is N Phard [6,19], but it admits an approximation algorithm which improves over the one from Christofides. The proof of this theorem is based on the following idea which can be used for several other reoptimization problems as well.…”

The Steiner Tree Reoptimization Problem with Sharpened Triangle Inequality

“…This concept of reoptimization was mentioned for the first time in [20] in the context of postoptimality analysis for a scheduling problem. Since then, the concept of reoptimization has been investigated for several different problems like the traveling salesman problem [1,3,8,12], knapsack problems [2], covering problems [7], and the shortest common superstring problem [6]. In these papers, it was shown that, for some problems, the reoptimization variant is exactly as hard as the original problem, whereas reoptimization can help a lot for improving the approximation ratio for other problems, for an overview of some results see also [11].…”

Reoptimization of the Shortest Common Superstring Problem

“…Postoptimality analysis deals with the related question of how much an instance may be altered without changing the set of optimal solutions, see, e.g., [19]. Since then, the concept of reoptimization has been successfully applied to various problems like the traveling salesman problem [1,3,7,8], the Steiner tree problem [4,10,11], the knapsack problem [2], and various covering problems [5]. A survey of reoptimization problems can be found in [9].…”