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.

翻译：我们证明，图同构问题的Lasserre半定规划层级中第$t$层的可行性可表示为同态不可区分关系。换言之，我们定义了一类图$\mathcal{L}_t$，使得图$G$和$H$无法被Lasserre层级第$t$层区分当且仅当它们从$\mathcal{L}_t$中任意图接收到的同态数量相同。通过分析$\mathcal{L}_t$中图的树宽，我们证明Sherali–Adams线性规划层级的第$3t$层与Lasserre第$t$层具有同等表达能力。此外，我们证明这一结果是最优的，即对于任意$t$，$3t$不能降低至$3t-1$。同样的结论也适用于带有非负性约束的Lasserre层级，我们进一步通过图族$\mathcal{L}_t^+$上的同态不可区分性对其进行了刻画。同时，我们给出了带非负性约束的Lasserre第$t$层在逻辑等价及类Weisfeiler–Leman图着色算法框架下的表征。这为判定两个给定图是否被带非负性约束的Lasserre层级第$t$层所区分提供了一种多项式时间算法。