Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

Abstract

We show that feasibility of the $t^\text{th}$ level of the Lasserresemidefinite programming hierarchy for graph isomorphism can be expressed as ahomomorphism indistinguishability relation. In other words, we define a class$\mathcal{L}_t$ of graphs such that graphs $G$ and $H$ are not distinguished bythe $t^\text{th}$ level of the Lasserre hierarchy if and only if they admit thesame number of homomorphisms from any graph in $\mathcal{L}_t$. By analysingthe 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 possiblein the sense that $3t$ cannot be lowered to $3t-1$ for any $t$. The same resultholds for the Lasserre hierarchy with non-negativity constraints, which wesimilarly characterise in terms of homomorphism indistinguishability over afamily $\mathcal{L}_t^+$ of graphs. Additionally, we give characterisations oflevel-$t$ Lasserre with non-negativity constraints in terms of logicalequivalence and via a graph colouring algorithm akin to the Weisfeiler--Lemanalgorithm. This provides a polynomial time algorithm for determining if twogiven graphs are distinguished by the $t^\text{th}$ level of the Lasserrehierarchy with non-negativity constraints.Comment: Full version. 36 pages, 6 figure