Symmetric Linear Programming Formulations for Minimum Cut with Applications to TSP

TL;DR

This paper develops symmetric linear programming relaxations for minimum cut problems, which either provide optimal solutions or improve approximations for TSP, leading to the smallest known LP formulations with strong theoretical guarantees.

Contribution

Introduces new symmetric LP relaxations for min-cut and TSP, achieving optimal or near-optimal solutions and surpassing previous LP formulations in size and approximation quality.

Findings

LP relaxations yield optimal or near-optimal solutions for Hamiltonian cycles.

New LP formulations outperform existing relaxations for TSP.

Achieves the smallest known LP size with a 9/8-approximation for min-cut.

Abstract

We introduce multiple symmetric LP relaxations for minimum cut problems. The relaxations give optimal and approximate solutions when the input is a Hamiltonian cycle. We show that this leads to one of two interesting results. In one case, these LPs always give optimal and near optimal solutions, and then they would be the smallest known symmetric LPs for the problems considered. Otherwise, these LP formulations give strictly better LP relaxations for the traveling salesperson problem than the subtour relaxation. We have the smallest known LP formulation that is a 9/8-approximation or better for min-cut. In addition, the LP relaxation of min-cut investigated in this paper has interesting constraints; the LP contains only a single typical min-cut constraint and all other constraints are typically only used for max-cut relaxations.