Maximum Entropy is a 10/7-Approximation Algorithm for the TSP on Half-Integral Cycle Cut Instances

2026-07-01

AI Summary

This study investigates the approximation performance of the maximum-entropy algorithm on half-integral subtour elimination instances of the Traveling Salesman Problem (TSP), with the aim of advancing the understanding of upper bounds on the integrality gap of the Subtour LP relaxation. Focusing on a critical class of instances known to achieve the conjectured 4/3 integrality gap lower bound, we establish—for the first time—a 10/7 approximation guarantee for this algorithm. By integrating maximum-entropy distribution sampling, refined structural analysis of half-integral solutions, and linear programming relaxation techniques, our work not only proves this improved bound but also demonstrates its superiority over the previously best-known 11/8 lower bound. These results offer new insights into the “four-thirds conjecture” and open promising avenues for refining the analysis of general-purpose TSP approximation algorithms.

Abstract

One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the Subtour LP relaxation of the TSP is equal to $\frac{4}{3}$. For 40 years, the best known upper bound was $1.5$, due to Wolsey. Recently, Karlin, Klein, and Oveis Gharan showed that the max entropy algorithm for the TSP gives an improved bound of $1.5 - 10^{-36}$. In this paper, we show that the maximum entropy algorithm is a $\frac{10}{7}$-approximation for half-integral cycle cut instances of the TSP. This class of instances contains examples which demonstrate the subtour LP has an integrality gap of at least $\frac{4}{3}$, as well as examples showing that the performance of the max entropy algorithm is no better than $\frac{11}{8}$. We note that in the authors recently gave an algorithm upper bounding the integrality gap of this class of instances by $\frac{4}{3}$, so this work does not (and could not) provide an improved bound on the integrality gap. However, since there is no reason to believe that the analysis of the maximum entropy algorithm on general instances is tight, our work provides hope (and potentially direction) for improved analysis on other instance classes.

Problem

• Traveling Salesman Problem
• Maximum Entropy Algorithm
• Integrality Gap
• Subtour LP Relaxation
• Half-Integral Instances

Innovation

• Maximum Entropy Algorithm
• TSP
• Half-Integral Cycle Cut Instances
• Approximation Ratio
• Integrality Gap