A Computational Study of Exact Subgraph Based SDP Bounds for Max-Cut, Stable Set and Coloring

TL;DR

This paper presents a computational framework for solving hierarchical SDP relaxations of NP-hard graph problems using a partial Lagrangian dual and bundle method, resulting in high-quality bounds.

Contribution

It introduces a novel computational approach that efficiently handles large subgraph constraints in SDP relaxations for Max-Cut, stable set, and coloring problems.

Findings

High-quality bounds achieved for Max-Cut, stable set, and coloring.

Efficient decomposition into independent subproblems enables scalable computation.

The approach outperforms traditional methods in bound tightness.

Abstract

The "exact subgraph" approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into several independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Finally computational experiments on the Max-Cut, stable set and coloring problem show the excellent quality of the bounds obtained with this approach.