TY - GEN

T1 - Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

AU - Roberson, David E.

AU - Seppelt, Tim

N1 - Publisher Copyright: © David E. Roberson and Tim Seppelt.

PY - 2023

Y1 - 2023

N2 - We show that feasibility of the tth 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 Lt of graphs such that graphs G and H are not distinguished by the tth level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in Lt. By analysing the treewidth of graphs in Lt we prove that the 3tth level of Sherali–Adams linear programming hierarchy is as strong as the tth 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 tth level of the Lasserre hierarchy with non-negativity constraints.

AB - We show that feasibility of the tth 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 Lt of graphs such that graphs G and H are not distinguished by the tth level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in Lt. By analysing the treewidth of graphs in Lt we prove that the 3tth level of Sherali–Adams linear programming hierarchy is as strong as the tth 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 tth level of the Lasserre hierarchy with non-negativity constraints.

KW - graph isomorphism

KW - homomorphism indistinguishability

KW - Lasserre hierarchy

KW - linear programming

KW - semidefinite programming

KW - Sherali–Adams hierarchy

KW - treewidth

U2 - 10.4230/LIPIcs.ICALP.2023.101

DO - 10.4230/LIPIcs.ICALP.2023.101

M3 - Article in proceedings

AN - SCOPUS:85167368828

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023

A2 - Etessami, Kousha

A2 - Feige, Uriel

A2 - Puppis, Gabriele

PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik

T2 - 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023

Y2 - 10 July 2023 through 14 July 2023

ER -