# Reducing Path TSP to TSP

## TL;DR

This paper introduces a reduction from Path TSP to TSP, showing their approximability is essentially equivalent, and applies this to improve approximation algorithms for Graph Path TSP.

## Contribution

It provides a black-box reduction from Path TSP to TSP, establishing their approximability equivalence and improving approximation ratios for Graph Path TSP.

## Key findings

- Reduction from Path TSP to TSP with arbitrarily small error
- Improved approximation algorithm for Graph Path TSP to 1.4+ε
- New techniques including a recursive dynamic program for generalized TSP

## Abstract

We present a black-box reduction from the path version of the Traveling Salesman Problem (Path TSP) to the classical tour version (TSP). More precisely, we show that given an $\alpha$-approximation algorithm for TSP, then, for any $\epsilon >0$, there is an $(\alpha+\epsilon)$-approximation algorithm for the more general Path TSP. This reduction implies that the approximability of Path TSP is the same as for TSP, up to an arbitrarily small error. This avoids future discrepancies between the best known approximation factors achievable for these two problems, as they have existed until very recently. A well-studied special case of TSP, Graph TSP, asks for tours in unit-weight graphs. Our reduction shows that any $\alpha$-approximation algorithm for Graph TSP implies an $(\alpha+\epsilon)$-approximation algorithm for its path version. By applying our reduction to the $1.4$-approximation algorithm for Graph TSP by Seb\H{o} and Vygen, we obtain a polynomial-time $(1.4+\epsilon)$-approximation algorithm for Graph Path TSP, improving on a recent $1.497$-approximation algorithm of Traub and Vygen. We obtain our results through a variety of new techniques, including a novel way to set up a recursive dynamic program to guess significant parts of an optimal solution. At the core of our dynamic program we deal with instances of a new generalization of (Path) TSP which combines parity constraints with certain connectivity requirements. This problem, which we call $\Phi$-TSP, has a constant-factor approximation algorithm and can be reduced to TSP in certain cases when the dynamic program would not make sufficient progress.