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

TL;DR

This paper investigates upper bounds on the stability number $\alpha(G)$ for Paley graphs $P_q$ using the exact subgraph hierarchy (ESH) and introduces a vertex-transitive variant (VTESH) to overcome stagnation in the ESH. It analytically derives an explicit optimal SDP solution for the Lovász theta function on Paley graphs, establishing $\vartheta(P_q)=\sqrt{q}$ and showing that ESCs do not improve the bound up to level $\ell(q)=\left\lfloor(\sqrt{q}+3)/2\right\rfloor$ for many $q$. Computational experiments confirm this stagnation and demonstrate that VTESH yields significantly tighter bounds, often matching or exactly determining $\alpha(P_q)$ at second or higher levels, and outperforming the ESH and prior bounds such as Hanson–Mertidis–Magsino-type bounds. The VTESH bound $z'_k(P_q)$ is shown to be at least as strong as the ESH bound $z_k(P_q)$, and in practice substantially improves the quality of upper bounds, indicating strong potential for vertex-transitive graphs beyond Paley graphs. Open questions include whether improvements occur at higher levels for all Paley graphs, extensions to other vertex-transitive families, and possible reductions to linear programming exploiting circulant structure.

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 Lovász theta function at the first level and including all exact subgraph constraints of subgraphs of order $k$ into the semidefinite program to compute the Lovász 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 Lovász 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.

Figures (3)

• Figure 1: The values of $\alpha(P_q)$, $\ell(q)$, and $\tilde{\ell}(q)$ for Paley graphs $P_q$
• Figure 2: The upper bounds on $z_2(P_q)$ and $z^\prime_2(P_q)$ for Paley graphs $P_q$ with $17 \leq q < 200$
• Figure 3: The upper bounds on $z_8(P_q)$ and $z^\prime_8(P_q)$ for Paley graphs $P_q$ with $17 \leq q < 200$

Theorems & Definitions (27)

• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 7
• Lemma 10
• proof
• Lemma 11
• proof
• Lemma 12
• proof