On different Versions of the Exact Subgraph Hierarchy for the Stable Set Problem

TL;DR

This work analyzes SDP-based relaxations toward the stability number $\alpha(G)$ of a graph, focusing on the Lovász theta function and two SDP formulations with different matrix orders. It introduces two hierarchy variants—Compressed ESH (CESH) starting from the smaller $n\times n$ theta form, and Scaled ESCs (SESC) leading to the Scaled ESH (SESH)—and proves that the original ESH yields bounds at least as strong as CESH, with SESH coinciding with CESH. Theoretical results show that ESH reaches $\alpha(G)$ at level $n$, while CESH/SESH share the same feasible region structure under scaling, and computational experiments indicate that ESH often provides the best trade-off between bound quality and runtime when using standard SDP solvers, though CESH can be advantageous in some instances. The findings guide practical choices for tightening theta-based relaxations via subgraph information and point to avenues for specialized solvers to exploit CESH structure.

Abstract

Let $G$ be a graph with $n$ vertices and $m$ edges. One of several hierarchies towards the stability number of $G$ is the exact subgraph hierarchy (ESH). On the first level it computes the Lovász theta function $\vartheta(G)$ as semidefinite program (SDP) with a matrix variable of order $n+1$ and $n+m+1$ constraints. On the $k$-th level it adds all exact subgraph constraints (ESC) for subgraphs of order $k$ to the SDP. An ESC ensures that the submatrix of the matrix variable corresponding to the subgraph is in the correct polytope. By including only some ESCs into the SDP the ESH can be exploited computationally. In this paper we introduce a variant of the ESH that computes $\vartheta(G)$ through an SDP with a matrix variable of order $n$ and $m+1$ constraints. We show that it makes sense to include the ESCs into this SDP and introduce the compressed ESH (CESH) analogously to the ESH. Computationally the CESH seems favorable as the SDP is smaller. However, we prove that the bounds based on the ESH are always at least as good as those of the CESH. In computational experiments sometimes they are significantly better. We also introduce scaled ESCs (SESCs), which are a more natural way to include exactness constraints into the smaller SDP and we prove that including an SESC is equivalent to including an ESC for every subgraph.

Theorems & Definitions (29)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Definition 3
• proof
• Definition 4
• Definition 5
• Lemma 2
• proof