Abstract
We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes o1,…,ok. The TSP solution must have that oi+1 is visited at some point after oi for each 1≤i<k. This is the special case of Precedence-Constrained TSP (PTSP) in which the precedence constraints are given by a single chain on a subset of nodes. In k-Person TSP Path (k-TSPP), we are given pairs of nodes (s1,t1),…,(sk,tk). The goal is to find an si-ti path with minimum total cost such that every node is visited by at least one path.
We obtain a 3/2+e−1<1.878 approximation for OTSP, the first improvement over a trivial α+1 approximation where α is the current best TSP approximation. We also obtain a 1+2⋅e−1/2<2.214 approximation for k-TSPP, the first improvement over a trivial 3-approximation.
These algorithms both use an adaptation of the Bridge Lemma that was initially used to obtain improved Steiner Tree approximations [Byrka et al., 2013]. Roughly speaking, our variant states that the cost of a cheapest forest rooted at a given set of terminal nodes will decrease by a substantial amount if we randomly sample a set of non-terminal nodes to also become terminals such provided each non-terminal has a constant probability of being sampled. We believe this view of the Bridge Lemma will find further use for improved vehicle routing approximations beyond this paper.
Joint work with Zachary Friggstad, Tobias Mömke and Joachim Spoerhase.
ArXiv link: https://arxiv.org/abs/2405.12876
