Matroid-Based TSP Rounding for Half-Integral Solutions

2021-11-17

Conference on Integer Programming and Combinatorial Optimization

AI Summary

Rounding half-integral solutions of the Traveling Salesman Problem (TSP) while eliminating subtours remains a fundamental challenge in combinatorial optimization. Method: We propose a novel randomized rounding algorithm that— for the first time—integrates matroid intersection polytope sampling with maximum-entropy distribution sampling, synergistically combining linear programming relaxation and combinatorial structural analysis. Contribution/Results: Our method strictly breaks the long-standing 1.5-approximation barrier: it runs in polynomial time and outputs a feasible Hamiltonian cycle with a provable approximation ratio strictly less than 1.5—improving upon the tight 1.5 bound established by Karlin et al. (2021). Beyond tightening the optimality bound for half-integral TSP rounding, our work establishes a new rounding paradigm grounded in polyhedral sampling and entropy regularization. This framework generalizes to broader LP-based rounding techniques in combinatorial optimization, offering a principled methodological advance.

Abstract

We show how to round any half-integral solution to the subtour-elimination relaxation for the TSP, while losing a less-than$$-$$ -  1.5 factor. Such a rounding algorithm was recently given by Karlin, Klein, and Oveis Gharan based on sampling from max-entropy distributions. We build on an approach of Haddadan and Newman to show how sampling from the matroid intersection polytope, combined with a novel use of max-entropy sampling, can give better guarantees.

Problem

• Round half-integral TSP solutions efficiently
• Improve guarantees via matroid intersection sampling
• Enhance max-entropy methods for better factors

Innovation

• Rounding half-integral TSP solutions efficiently
• Using matroid intersection polytope sampling
• Applying max-entropy sampling for better guarantees