Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

TL;DR

This paper connects the feasibility of the Lasserre hierarchy for graph isomorphism to homomorphism indistinguishability, providing new characterizations and complexity results that relate different hierarchies and logical equivalences.

Contribution

It introduces a homomorphism indistinguishability framework for the Lasserre hierarchy levels and establishes tight bounds relating Lasserre and Sherali--Adams hierarchies, along with logical and algorithmic characterizations.

Findings

Feasibility of Lasserre hierarchy levels can be characterized by homomorphism counts.

The $3t$-th Sherali--Adams level matches the $t$-th Lasserre level in strength.

A polynomial-time algorithm determines graph distinction at Lasserre level $t$.

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.