# An Improved Approximation Algorithm for TSP in the Half Integral Case

## TL;DR

This paper presents a new approximation algorithm with a ratio of 1.49993 for the metric TSP in cases where the LP relaxation solution is half-integral, addressing a longstanding conjecture about the integrality gap.

## Contribution

It introduces an improved approximation algorithm specifically for half-integral LP solutions in the metric TSP, advancing understanding of the integrality gap.

## Key findings

- Achieves a 1.49993-approximation ratio for the half-integral case.
- Supports the conjecture that the integrality gap is less than 3/2.
- Provides insights into the structure of half-integral solutions.

## Abstract

We design a $1.49993$-approximation algorithm for the metric traveling salesperson problem (TSP) for instances in which an optimal solution to the subtour linear programming relaxation is half-integral. These instances received significant attention over the last decade due to a conjecture of Schalekamp, Williamson and van Zuylen stating that half-integral LP solutions have the largest integrality gap over all fractional solutions. So, if the conjecture of Schalekamp et al. holds true, our result shows that the integrality gap of the subtour polytope is bounded away from $3/2$.