TY - JOUR
AU - BĂ¶ckenhauer, Hans-Joachim
AU - Forlizzi, Luca
AU - Hromkovic, Juraj
AU - Kneis, Joachim
AU - Kupke, Joachim
AU - Proietti, Guido
AU - Widmayer, Peter
PY - 2007/09/02
Y2 - 2023/06/09
TI - On the Approximability of TSP on Local Modifications of Optimally Solved Instances
JF - Algorithmic Operations Research
JA - AOR
VL - 2
IS - 2
SE - Articles
DO -
UR - https://journals.lib.unb.ca/index.php/AOR/article/view/2803
SP - 83
AB - Given an instance of TSP together with an optimal solution, we consider the scenario in which this instance is modified locally, where a local modification consists in the alteration of the weight of a single edge. More generally, for a problem <em>U</em>, let LM-<em>U</em> (local-modification-<em>U</em>) denote the same problem as <em>U</em>, but in LM-<em>U</em>, we are also given an optimal solution to an instance from which the input instance can be derived by a local modification. The question is how to exploit this additional knowledge, i.e., how to devise better algorithms for LM-<em>U</em> than for <em>U</em>. Note that this need not be possible in all cases: The general problem of LM-TSP is as hard as TSP itself, i.e., unless P=NP, 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-Δ<sub>β</sub>-TSP) remains NP-hard for all β>½. However, for LM-Δ-TSP (i.e., metric LM-TSP), we will present an efficient 1.4-approximation algorithm. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. Similarly, for all 1<β<3.34899, we achieve a better approximation ratio for LM-Δ-TSP than for Δ<sub>β</sub>-TSP. For ½≤β<1, we show how to obtain an approximation ratio arbitrarily close to 1, for sufficiently large input graphs.
ER -