A 1.5-Approximation for Path TSP

TL;DR

This paper introduces a simplified 1.5-approximation algorithm for the Metric Path TSP, leveraging larger cuts and Karger's near-minimum cut results, matching the best known approximation ratio.

Contribution

It presents a novel approach that handles larger cuts without complex recursion, improving simplicity and avoiding additional error terms in the approximation.

Findings

Achieves a 1.5-approximation ratio for Path TSP.

Simplifies previous algorithms by avoiding recursive dynamic programming.

Utilizes Karger's near-minimum cut results to handle larger cuts.

Abstract

We present a $1.5$-approximation for the Metric Path Traveling Salesman Problem (Path TSP). All recent improvements on Path TSP crucially exploit a structural property shown by An, Kleinberg, and Shmoys [Journal of the ACM, 2015], namely that narrow cuts with respect to a Held-Karp solution form a chain. We significantly deviate from these approaches by showing the benefit of dealing with larger $s$-$t$ cuts, even though they are much less structured. More precisely, we show that a variation of the dynamic programming idea recently introduced by Traub and Vygen [SODA, 2018] is versatile enough to deal with larger size cuts, by exploiting a seminal result of Karger on the number of near-minimum cuts. This avoids a recursive application of dynamic programming as used by Traub and Vygen, and leads to a considerably simpler algorithm avoiding an additional error term in the approximation guarantee. We match the still unbeaten $1.5$-approximation guarantee of Christofides' algorithm for TSP. Hence, any further progress on the approximability of Path TSP will also lead to an improvement for TSP.