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An SDP-based approach for computing the stability number of a graph.

Elisabeth Gaar¹, Melanie Siebenhofer², Angelika Wiegele²

¹Institute of Production and Logistics Management, Johannes Kepler University Linz, Altenberger Straß 69, 4040 Linz, Austria.

Mathematical Methods of Operations Research (Heidelberg, Germany)

|April 11, 2022

Summary

This study introduces faster graph stability number algorithms using new semidefinite programming relaxations. These methods improve branch and bound efficiency by reducing computation time and nodes explored.

Keywords:

Branch and bound Combinatorial optimization Lovász theta function Semidefinite programming Stable set

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Area of Science:

Combinatorial Optimization
Graph Theory
Computer Science

Background:

The stability number problem is NP-hard with real-world applications.
Existing semidefinite programming methods (Lovász theta function) have limitations.
Exact subgraph constraints (ESC) enhance Lovász theta bounds but increase computational cost.

Purpose of the Study:

Develop faster and effective algorithms for finding the graph stability number.
Introduce novel relaxations to improve semidefinite programming bounds.
Enhance the efficiency of branch and bound algorithms for this NP-hard problem.

Main Methods:

Introduced two new semidefinite programming relaxations for graph stability number bounds.
Relaxations involve violated facets of the ESC polytope and separating hyperplanes.
Implemented a branch and bound (B&B) algorithm incorporating these new relaxations.

Main Results:

New relaxations provide bounds comparable in quality to existing methods but faster to compute.
The B&B algorithm with new relaxations significantly reduces overall runtime.
The number of nodes explored in the B&B tree remains small, improving efficiency.

Conclusions:

The proposed relaxations offer a practical improvement for solving the graph stability number problem.
Faster bounding routines lead to more efficient branch and bound algorithms.
This work advances the state-of-the-art in combinatorial optimization for graph problems.