Sublinear Algorithms for TSP via Path Covers

TL;DR

This work presents an $\widetilde{O}(n)$ time algorithm that estimates the cost of graphic TSP within a factor of $1.83$ and shows that the approximation can be further improved to $1.66$ using $n^{2-\Omega(1)}$ time.

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 $ε> 0$, there is an algorithm that $(1/2 - ε)$-approximates the maximum path cover size of an $n$-vertex graph in $\widetilde{O}(n)$ time. This improves upon a $(3/8-ε)$-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 $(1.5+ε)$ which is an improvement over a folklore $(1.75 + ε)$-approximate $\widetilde{O}(n)$-time algorithm, as well as a $(1.625+ε)$-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 $n^{2-Ω(1)}$ 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.

Figures (3)

• Figure 1: Examples of why the output of \ref{['alg:path-cover11']} will not have cycles.
• Figure 2: Illustration of proof of \ref{['clm:new-branch']}. The highlighted blue trails show query-trails $\vec{P}$ and $\vec{P}'$. Query-trail $\vec{P}$ is not valid since $\textup{EO}(e_i, \cdot, \pi)$ terminates upon calling $\textup{EO}(f, \cdot, \pi)$.
• Figure 3: Illustration of graph $G' = (V', U', E')$. Each $G_i$ is shown by a rectangle and each $H_i$ is shown by a parallelogram. Top and bottom horizontal lines illustrate $V_i$ and $U_i$. Blue highlighted parts represent the vertex cover of the graph.

Theorems & Definitions (103)

• Proposition 3.1: König’s Theorem
• Proposition 3.2: Chernoff Bound
• Proposition 3.3: Hoeffding’s Inequality
• Claim 4.1
• proof
• Claim 4.2
• proof
• proof
• proof
• Lemma 4.5