Reusing Optimal TSP Solutions for Locally Modified Input Instances

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=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 β -TSP) remains NP-hard for all β > 1/2. 2. For LM-Δ-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-Δ β -TSP than for Δ’-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. instance. A second construction inflates this advantage. Tours which start at time X, different from those that start between times X+g and X +ςg, may spend some extra time to visit a group of vertices which, unless visited early, will cause belated tours to run k times zigzag across a huge distance γ.