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Stable Set Polytopes with Rank $|V(G)|/3$ for the Lovász--Schrijver SDP Operator

Yu Hin Au, Levent Tunçel

TL;DR

This work analyzes the Lovász–Schrijver LS_+ lift-and-project hierarchy for the stable set problem. It proves a tight bound n_+(\ell)=3\ell by constructing \ell-minimal graphs and, separately, vertex-transitive graphs on 4\ell+12 vertices achieving LS_+-rank at least \ell, resolving longstanding conjectures. The authors develop a robust framework based on stretched-clique decompositions and a general LS_+-relaxation toolkit, enabling rank lower bounds without explicit certificates via perturbations and a constructive lifting mechanism. They also show that stretched cliques can exhibit unbounded CG-rank, via families such as H_k' and A_{k,S}, illustrating hardness beyond LS_+ and highlighting rich combinatorial structure in worst-case instances. Overall, the paper advances understanding of LS_+-rank behavior, provides large families of minimal and near-minimal graphs, and opens multiple directions for characterizing minimal obstructions and CG-rank phenomena in stable set relaxations.

Abstract

We study the lift-and-project rank of the stable set polytope of graphs with respect to the Lovász--Schrijver SDP operator $\text{LS}_+$ applied to the fractional stable set polytope. In particular, we show that for every positive integer $\ell$, the smallest possible graph with $\text{LS}_+$-rank $\ell$ contains $3\ell$ vertices. This result is sharp and settles a conjecture posed by Lipták and the second author in 2003, as well as answers a generalization of a problem posed by Knuth in 1994. We also show that for every positive integer $\ell$ there exists a vertex-transitive graph on $4\ell+12$ vertices with $\text{LS}_+$-rank at least $\ell$.

Stable Set Polytopes with Rank $|V(G)|/3$ for the Lovász--Schrijver SDP Operator

TL;DR

This work analyzes the Lovász–Schrijver LS_+ lift-and-project hierarchy for the stable set problem. It proves a tight bound n_+(\ell)=3\ell by constructing \ell-minimal graphs and, separately, vertex-transitive graphs on 4\ell+12 vertices achieving LS_+-rank at least \ell, resolving longstanding conjectures. The authors develop a robust framework based on stretched-clique decompositions and a general LS_+-relaxation toolkit, enabling rank lower bounds without explicit certificates via perturbations and a constructive lifting mechanism. They also show that stretched cliques can exhibit unbounded CG-rank, via families such as H_k' and A_{k,S}, illustrating hardness beyond LS_+ and highlighting rich combinatorial structure in worst-case instances. Overall, the paper advances understanding of LS_+-rank behavior, provides large families of minimal and near-minimal graphs, and opens multiple directions for characterizing minimal obstructions and CG-rank phenomena in stable set relaxations.

Abstract

We study the lift-and-project rank of the stable set polytope of graphs with respect to the Lovász--Schrijver SDP operator applied to the fractional stable set polytope. In particular, we show that for every positive integer , the smallest possible graph with -rank contains vertices. This result is sharp and settles a conjecture posed by Lipták and the second author in 2003, as well as answers a generalization of a problem posed by Knuth in 1994. We also show that for every positive integer there exists a vertex-transitive graph on vertices with -rank at least .
Paper Structure (6 sections, 25 theorems, 40 equations, 11 figures)

This paper contains 6 sections, 25 theorems, 40 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Stretched cliques and their properties
  3. Structural results for $\mathop{\mathrm{LS}}\nolimits_+$-relaxations
  4. $\ell$-minimal graphs and implications
  5. $\mathop{\mathrm{CG}}\nolimits$-rank of stretched cliques
  6. Future research directions

Key Result

Theorem 1

For every graph $G$, $r_+(G) \leq \frac{|V(G)|}{3}$.

Figures (11)

  • Figure 1: Several known $\ell$-minimal graphs due to LiptakT03EscalanteMN06AuT24b
  • Figure 2: Illustrating the vertex-stretching operation
  • Figure 3: Illustrating the definition of the set of graphs $\hat{\mathcal{K}}_{n,d}$ (see Example \ref{['eg405']})
  • Figure 4: Several members of the family of graphs $\mathcal{A}_{k}$
  • Figure 5: The $13$ graphs $G \in \hat{\mathcal{K}}_{5,2}$ with $\omega(G) \leq 3$, and their corresponding optimal values of $\max\left\{\bar{e}^{\top}x : x \in \mathop{\mathrm{LS}}\nolimits_+^2(G)\right\}$ according to CVX
  • ...and 6 more figures

Theorems & Definitions (48)

  • Theorem 1
  • Lemma 2
  • proof
  • Example 3
  • Lemma 4
  • proof
  • Example 5
  • Lemma 6
  • Lemma 7
  • proof
  • ...and 38 more