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Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

David E. Roberson, Tim Seppelt

TL;DR

The paper addresses graph isomorphism by connecting the level-$t$ Lasserre SDP relaxation with homomorphism-indistinguishability, showing that feasibility at level $t$ is equivalent to indistinguishability over a novel graph class $\\mathcal{L}_t$ (and its non-negativity variant over $\\mathcal{L}_t^+$). It proves a tight relation with the Sherali–Adams hierarchy: level-$3t$ SA feasibility implies level-$t$ Lasserre feasibility, and this bound is best possible. By analysing the treewidth (and pathwidth) of graphs in these classes, the authors show that the level-$t$ Lasserre relaxation aligns with SA at level $3t$, and they characterise the non-negativity case via a polynomial-time algorithm grounded in a counting-logic and a Weisfeiler–Leman-type colouring. The work also yields explicit identifications for base cases ($\\mathcal{L}_1$ as outerplanar graphs and $\\mathcal{L}_1^+$ as treewidth-2 graphs) and provides a framework to test exact feasibility in polynomial time for the non-negativity version, advancing both theory and potential practical use in graph isomorphism and related counting problems.

Abstract

We show that feasibility of the $t^\text{th}$ level of the Lasserre semidefinite programming hierarchy for graph isomorphism can be expressed as a homomorphism indistinguishability relation. In other words, we define a class $\mathcal{L}_t$ of graphs such that graphs $G$ and $H$ are not distinguished by the $t^\text{th}$ level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in $\mathcal{L}_t$. By analysing the treewidth of graphs in $\mathcal{L}_t$, we prove that the $3t^\text{th}$ level of Sherali--Adams linear programming hierarchy is as strong as the $t^\text{th}$ level of Lasserre. Moreover, we show that this is best possible in the sense that $3t$ cannot be lowered to $3t-1$ for any $t$. The same result holds for the Lasserre hierarchy with non-negativity constraints, which we similarly characterise in terms of homomorphism indistinguishability over a family $\mathcal{L}_t^+$ of graphs. Additionally, we give characterisations of level-$t$ Lasserre with non-negativity constraints in terms of logical equivalence and via a graph colouring algorithm akin to the Weisfeiler--Leman algorithm. This provides a polynomial time algorithm for determining if two given graphs are distinguished by the $t^\text{th}$ level of the Lasserre hierarchy with non-negativity constraints.

Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

TL;DR

The paper addresses graph isomorphism by connecting the level- Lasserre SDP relaxation with homomorphism-indistinguishability, showing that feasibility at level is equivalent to indistinguishability over a novel graph class (and its non-negativity variant over ). It proves a tight relation with the Sherali–Adams hierarchy: level- SA feasibility implies level- Lasserre feasibility, and this bound is best possible. By analysing the treewidth (and pathwidth) of graphs in these classes, the authors show that the level- Lasserre relaxation aligns with SA at level , and they characterise the non-negativity case via a polynomial-time algorithm grounded in a counting-logic and a Weisfeiler–Leman-type colouring. The work also yields explicit identifications for base cases ( as outerplanar graphs and as treewidth-2 graphs) and provides a framework to test exact feasibility in polynomial time for the non-negativity version, advancing both theory and potential practical use in graph isomorphism and related counting problems.

Abstract

We show that feasibility of the level of the Lasserre semidefinite programming hierarchy for graph isomorphism can be expressed as a homomorphism indistinguishability relation. In other words, we define a class of graphs such that graphs and are not distinguished by the level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in . By analysing the treewidth of graphs in , we prove that the level of Sherali--Adams linear programming hierarchy is as strong as the level of Lasserre. Moreover, we show that this is best possible in the sense that cannot be lowered to for any . The same result holds for the Lasserre hierarchy with non-negativity constraints, which we similarly characterise in terms of homomorphism indistinguishability over a family of graphs. Additionally, we give characterisations of level- Lasserre with non-negativity constraints in terms of logical equivalence and via a graph colouring algorithm akin to the Weisfeiler--Leman algorithm. This provides a polynomial time algorithm for determining if two given graphs are distinguished by the level of the Lasserre hierarchy with non-negativity constraints.
Paper Structure (23 sections, 44 theorems, 9 equations, 4 figures)

This paper contains 23 sections, 44 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Techniques
  4. Preliminaries
  5. Linear Algebra
  6. Bilabelled Graphs and Homomorphism Tensors
  7. Pathwidth and Treewidth
  8. Systems of Equations for Graph Isomorphism
  9. From Lasserre to Homomorphism Tensors
  10. Isomorphism Relaxations via Matrix Families
  11. Choi Matrices and Isomorphism Maps
  12. From K-Isomorphism Maps to the Lasserre Hierarchy
  13. Isomorphisms between Matrix Algebras
  14. Partially Coherent Algebras and PSD-Isomorphism Maps
  15. Coherent Algebras and DNN-Isomorphism Maps
  16. ...and 8 more sections

Key Result

Theorem 1.1

For two graphs $G$ and $H$ and every $t \geq 1$, the following implications hold: Furthermore, for every $t \geq 1$, there exist graphs $G$ and $H$ such that $G \simeq_{3t-1}^{\textup{SA}} H$ and $G \not\simeq_t^{\textup{L}} H$.

Figures (4)

  • Figure 4: Relationship between $\mathcal{L}_t$, $\mathcal{L}_t^+$, the classes of graphs of bounded treewidth, bounded pathwidth, and the class of outerplanar graphs. An arrow $\mathcal{A} \rightarrow \mathcal{B}$ indicates that $\mathcal{A} \subseteq \mathcal{B}$ and thus that $G \equiv_{\mathcal{B}} H$ implies $G \equiv_{\mathcal{A}} H$ for all graphs $G$ and $H$. For formal statements, see \ref{['sec:lt-ltplus', 'sec:t-equals-one']}.
  • Figure 5: Examples of the atomic graphs from \ref{['def:atomic']}. The gray lines (the wiresmancinska_quantum_2019) indicate the in-labels (left) and out-labels (right).
  • Figure 7: The three atomic graphs in $\mathcal{A}_1$.
  • Figure 8: The bilabelled graphs in \ref{['obs:lt-clique']} for $t = 2$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1: mancinska_graph_2020
  • Lemma 2.2: mancinska_graph_2020
  • Lemma 2.3: mancinska_graph_2020
  • Lemma 2.4: mancinska_graph_2020
  • definition 2.5
  • ...and 48 more