Sublinear Algorithms for TSP via Path Covers

TL;DR

This paper develops sublinear time algorithms for TSP and related problems, achieving improved approximation ratios and establishing fundamental limits, all within near-optimal time complexity.

Contribution

It introduces a sublinear time algorithm for maximum path cover with better approximation, leading to improved TSP cost estimation algorithms, and proves approximation barriers linked to maximum matching.

Findings

Achieves a (1/2 - ε)-approximate maximum path cover in near-linear time.

Provides improved approximation algorithms for (1,2)-TSP and graphic TSP in sublinear time.

Shows that better approximations require advances in maximum matching algorithms.

Abstract

We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related {\em maximum path cover} problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges. We show that for any fixed $\epsilon > 0$, there is an algorithm that $(1/2 - \epsilon)$-approximates the maximum path cover size of an $n$-vertex graph in $\widetilde{O}(n)$ time. This improves upon a $(3/8-\epsilon)$-approximate $\widetilde{O}(n \sqrt{n})$-time algorithm of Chen, Kannan, and Khanna [ICALP'20]. Equipped with our path cover algorithm, we give an $\widetilde{O}(n)$ time algorithm that estimates the cost of $(1,2)$-TSP within a factor of $(1.5+\epsilon)$ which is an improvement over a folklore $(1.75 + \epsilon)$-approximate $\widetilde{O}(n)$-time algorithm, as well as a $(1.625+\epsilon)$-approximate $\widetilde{O}(n\sqrt{n})$-time algorithm of [CHK ICALP'20]. For graphic TSP, we present an $\widetilde{O}(n)$ algorithm that estimates the cost of graphic TSP within a factor of $1.83$ which is an improvement over a $1.92$-approximate $\widetilde{O}(n)$ time algorithm due to [CHK ICALP'20, Behnezhad FOCS'21]. We show that the approximation can be further improved to $1.66$ using $n^{2-\Omega(1)}$ time. All of our $\widetilde{O}(n)$ time algorithms are information-theoretically time-optimal up to poly log n factors. Additionally, we show that our approximation guarantees for path cover and $(1,2)$-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.