The exact subgraph hierarchy and its vertex-transitive variant for the stable set problem for Paley graphs

2024-12-17

Discrete Applied Mathematics

AI Summary

This paper investigates the maximum stable set (independent set) problem on Paley graphs—a computationally intractable NP-hard problem. To overcome the limitation that the standard Exact Subgraph Hierarchy (ESH) fails to improve upon the Lovász θ-function bound even at low levels for Paley graphs, the authors introduce the vertex-transitive Exact Subgraph Hierarchy (vt-ESH), a novel hierarchy specifically tailored for vertex-transitive graphs. It integrates tools from algebraic graph theory, character sums over finite fields, group actions, and spectral bounding techniques. Key contributions include: determining the exact stability number for all Paley graphs of small order; establishing tight upper bounds for infinitely many Paley graphs; revealing the first deep connection between ESH performance and graph symmetry; and verifying boundary cases of several long-standing conjectures. This work establishes a new paradigm for combinatorial optimization on symmetric graphs.

Technology Category

Constraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionMachine Learning: Graph-based Machine LearningGame Theory and Economic Paradigms: Cooperative Game Theory

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsSecurity and Privacy: Applications of cryptographySemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Problem

• Computing the stability number of graphs is NP-hard
• The exact subgraph hierarchy provides upper bounds for stability numbers
• A new vertex-transitive ESH improves bounds for Paley graphs

Innovation

• Introduces vertex-transitive exact subgraph hierarchy
• Uses semidefinite programming to compute bounds
• Improves stability number bounds for Paley graphs