Approximating Traveling Salesman Problems Using a Bridge Lemma

TL;DR

This paper presents improved approximation algorithms for two variants of the metric Traveling Salesman Problem, utilizing a novel adaptation of the Bridge Lemma to achieve better bounds than previous trivial approximations.

Contribution

It introduces new approximation algorithms for Ordered TSP and k-Person TSP Path that surpass trivial bounds by adapting the Bridge Lemma in innovative ways.

Findings

Achieved a 1.878-approximation for Ordered TSP, improving over the trivial bound.

Achieved a 2.214-approximation for k-Person TSP Path, surpassing the previous 3-approximation.

Demonstrated the utility of a modified Bridge Lemma in vehicle routing problems.

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 $o_1, \ldots, o_k$. The TSP solution must have that $o_{i+1}$ is visited at some point after $o_i$ for each $1 \leq i < k$. 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 $(s_1,t_1), \ldots, (s_k,t_k)$. The goal is to find an $s_i$-$t_i$ 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 $\alpha+1$ approximation where $\alpha$ is the current best TSP approximation. We also obtain a $1 + 2 \cdot e^{-1/2} < 2.214$ 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.