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Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability
Authors
David E. Roberson 
,
Tim Seppelt
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50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Colloquium on Automata, Languages, and Programming (ICALP) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2023-07-05
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Author Details
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Copenhagen, Denmark
- QMATH, Department of Mathematical Sciences, University of Copenhagen, Denmark
Acknowledgements
Initial discussions for this work took place at Dagstuhl Seminar 22051 "Finite and Algorithmic Model Theory".
Cite AsGet BibTex
David E. Roberson and Tim Seppelt. Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 101:1-101:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.101
https://doi.org/10.4230/LIPIcs.ICALP.2023.101
Abstract
We show that feasibility of the t^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 ℒ_t of graphs such that graphs G and H are not distinguished by the t^th level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in ℒ_t. By analysing the treewidth of graphs in ℒ_t we prove that the 3t^th level of Sherali-Adams linear programming hierarchy is as strong as the t^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 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^th level of the Lasserre hierarchy with non-negativity constraints.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Combinatorics
- Mathematics of computing → Graph theory
Keywords
- Lasserre hierarchy
- homomorphism indistinguishability
- Sherali-Adams hierarchy
- treewidth
- semidefinite programming
- linear programming
- graph isomorphism
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References
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