Improved FPT Approximation for Non-metric TSP

TL;DR

This paper presents an improved fixed-parameter tractable (FPT) approximation algorithm for the non-metric TSP, reducing the approximation ratio from 3 to 2.5, based on a parameter measuring deviation from the metric case.

Contribution

It introduces a 2.5-approximation FPT algorithm for non-metric TSP, improving upon previous 3-approximation results.

Findings

Achieved a 2.5-approximation ratio in FPT time.

Enhanced the approximation factor from 3 to 2.5.

Built upon the parameter measuring triangle inequality violations.

Abstract

In the Traveling Salesperson Problem (TSP) we are given a list of locations and the distances between each pair of them. The goal is to find the shortest possible tour that visits each location exactly once and returns to the starting location. Inspired by the fact that general TSP cannot be approximated in polynomial time within any constant factor, while metric TSP admits a (slightly better than) $1.5$-approximation in polynomial time, Zhou, Li and Guo [Zhou et al., ISAAC '22] introduced a parameter that measures the distance of a given TSP instance from the metric case. They gave an FPT $3$-approximation algorithm parameterized by $k$, where $k$ is the number of triangles in which the edge costs violate the triangle inequality. In this paper, we design a $2.5$-approximation algorithm that runs in FPT time, improving the result of [Zhou et al., ISAAC '22].