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On Rank-Monotone Graph Operations and Minimal Obstruction Graphs for the Lovász--Schrijver SDP Hierarchy

Yu Hin Au, Levent Tunçel

TL;DR

This work advances the understanding of the Lovász–Schrijver $\mathrm{LS}_+$ SDP hierarchy for the stable set problem by developing graph-operations that preserve or increase $\mathrm{LS}_+$-rank. Central contributions include a generalized vertex-stretching operation that yields many $\mathrm{LS}_+$-minimal graphs, the construction and certification of a 12-vertex $4$-minimal graph $G_{4,1}$, and the demonstration that families like $H_k$ can realize high rank while remaining sparse. The paper also links $\ell$-minimal graphs to $2$-stretching of cliques, analyzes facets with full support, and demonstrates how stretching can generate an abundance of high-rank graphs, including very sparse cubic graphs with $\mathrm{LS}_+$-rank $\Omega(\sqrt{|V|})$. Together, these results illuminate fundamental obstructions to SDP relaxations of the stable set polytope and offer tools likely applicable to other lift-and-project hierarchies.

Abstract

We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász--Schrijver SDP operator $\text{LS}_+$, with a particular focus on finding and characterizing the smallest graphs with a given $\text{LS}_+$-rank (the needed number of iterations of the $\text{LS}_+$ operator on the fractional stable set polytope to compute the stable set polytope). We introduce a generalized vertex-stretching operation that appears to be promising in generating $\text{LS}_+$-minimal graphs and study its properties. We also provide several new $\text{LS}_+$-minimal graphs, most notably the first known instances of $12$-vertex graphs with $\text{LS}_+$-rank $4$, which provides the first advance in this direction since Escalante, Montelar, and Nasini's discovery of a $9$-vertex graph with $\text{LS}_+$-rank $3$ in 2006.

On Rank-Monotone Graph Operations and Minimal Obstruction Graphs for the Lovász--Schrijver SDP Hierarchy

TL;DR

This work advances the understanding of the Lovász–Schrijver SDP hierarchy for the stable set problem by developing graph-operations that preserve or increase -rank. Central contributions include a generalized vertex-stretching operation that yields many -minimal graphs, the construction and certification of a 12-vertex -minimal graph , and the demonstration that families like can realize high rank while remaining sparse. The paper also links -minimal graphs to -stretching of cliques, analyzes facets with full support, and demonstrates how stretching can generate an abundance of high-rank graphs, including very sparse cubic graphs with -rank . Together, these results illuminate fundamental obstructions to SDP relaxations of the stable set polytope and offer tools likely applicable to other lift-and-project hierarchies.

Abstract

We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász--Schrijver SDP operator , with a particular focus on finding and characterizing the smallest graphs with a given -rank (the needed number of iterations of the operator on the fractional stable set polytope to compute the stable set polytope). We introduce a generalized vertex-stretching operation that appears to be promising in generating -minimal graphs and study its properties. We also provide several new -minimal graphs, most notably the first known instances of -vertex graphs with -rank , which provides the first advance in this direction since Escalante, Montelar, and Nasini's discovery of a -vertex graph with -rank in 2006.
Paper Structure (11 sections, 31 theorems, 70 equations, 16 figures, 1 table)

This paper contains 11 sections, 31 theorems, 70 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $\mathop{\mathrm{LS}}\nolimits_+$-operator
  4. Analyzing the $\mathop{\mathrm{LS}}\nolimits_+$-rank of a graph
  5. Star-homomorphic graphs
  6. The (generalized) vertex-stretching operation
  7. $\mathop{\mathrm{LS}}\nolimits_+$-minimal graphs via $2$-stretching cliques
  8. Existence of $4$-minimal graphs
  9. Revisiting $H_k$ and constructing sparse graphs with high $\mathop{\mathrm{LS}}\nolimits_+$-rank
  10. Some Future Research Directions
  11. Proofs of Lemmas \ref{['lemG412']}, \ref{['lemG411']}, and \ref{['lemG413']}

Key Result

Theorem 1

For every graph $G$, the $\mathop{\mathrm{LS}}\nolimits_+$-rank of $G$ is at most $\left\lfloor \frac{|V(G)|}{3} \right\rfloor$.

Figures (16)

  • Figure 1: Known $2$- and $3$-minimal graphs due to LiptakT03 and EscalanteMN06
  • Figure 2: Several graphs in the family $H_k$
  • Figure 3: $G_{4,1}$, a $12$-vertex graph with $\mathop{\mathrm{LS}}\nolimits_+$-rank $4$
  • Figure 4: Constructing $G^{\dagger}$ from $G$
  • Figure 5: Two graphs $H_1,H_2$ that are star-homomorphic to $G$
  • ...and 11 more figures

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 5
  • Proposition 6
  • Lemma 7
  • Example 8
  • Example 9
  • ...and 50 more