Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

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 Lₜ⁺ 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.