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

TL;DR

This paper explores the limitations of the exact subgraph hierarchy (ESH) for Paley graphs and introduces a vertex-transitive ESH that provides tighter bounds on the stable set problem for such graphs.

Contribution

The paper introduces a vertex-transitive ESH that improves bounds on the stability number for vertex-transitive graphs like Paley graphs, surpassing the standard ESH.

Findings

Standard ESH bounds do not improve beyond a certain level for Paley graphs.

Vertex-transitive ESH provides tighter upper bounds on the stability number.

Computational results show vertex-transitive ESH outperforms standard ESH for Paley graphs.

Abstract

The stability number of a graph, defined as the cardinality of the largest set of pairwise non-adjacent vertices, is NP-hard to compute. The exact subgraph hierarchy (ESH) provides a sequence of increasingly tighter upper bounds on the stability number, starting with the Lov\'asz theta function at the first level and including all exact subgraph constraints of subgraphs of order $k$ into the semidefinite program to compute the Lov\'asz theta function at level $k$. In this paper, we investigate the ESH for Paley graphs, a class of strongly regular, vertex-transitive graphs. We show that for Paley graphs, the bounds obtained from the ESH remain the Lov\'asz theta function up to a certain threshold level, i.e., the bounds of the ESH do not improve up to a certain level. To overcome this limitation, we introduce the vertex-transitive ESH for the stable set problem for vertex-transitive graphs such as Paley graphs. We prove that this new hierarchy provides upper bounds on the stability number of vertex-transitive graphs that are at least as tight as those obtained from the ESH. Additionally, our computational experiments reveal that the vertex-transitive ESH produces superior bounds compared to the ESH for Paley graphs.