TY - JOUR
AU - Hans-Joachim BĂ¶ckenhauer
AU - Luca Forlizzi
AU - Juraj Hromkovic
AU - Joachim Kneis
AU - Joachim Kupke
AU - Guido Proietti
AU - Peter Widmayer
PY - 2007/09/02
Y2 - 2019/10/14
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
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 U, let LM-U (local-modification-U) denote the same problem as U, but in LM-U, 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-U than for U. 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-Δβ-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 Δβ-TSP. For ½≤β<1, we show how to obtain an approximation ratio arbitrarily close to 1, for sufficiently large input graphs.
ER -