On the Approximability of TSP on Local Modifications of Optimally Solved Instances

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 $\textsc{lm-}U$ (local-modification-$U$) denote the same problem as~$U$, but in $\textsc{lm-}U$, 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, \ie how to devise better algorithms for $\textsc{lm-}U$ than for~$U$. Note that this need not be possible in all cases: The general problem of \tweakTSP is as hard as \TSP itself, \ie unless $P=NP$, there is no polynomial-time $p(n)$-approximation algorithm for \tweakTSP for any polynomial~$p$. Moreover, \tweakTSP where inputs must satisfy the $\beta$-triangle inequality (\tweakTSPDB) remains NP-hard for all $\beta>\frac{1}{2}$. However, for \tweakTSPD (\ie metric \tweakTSP), we will present an efficient $1.4$-approx\-i\-ma\-tion 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 $1<\beta<3.34899$, we achieve a better approximation ratio for \tweakTSPDB than for \TSPDB. For $\frac{1}{2}\leq\beta<1$, we show how to obtain an approximation ratio arbitrarily close to~$1$, for sufficiently large input graphs.