Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

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.