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：我们证明了图同构的Lasserre半定规划层级第$t$阶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$H$被Lasserre层级第$t$阶区分当且仅当它们从$\mathcal{L}_t$中任意图接受相同数量的同态。通过分析$\mathcal{L}_t$中图的树宽，我们证明Sherali-Adams线性规划层级第$3t$阶与Lasserre第$t$阶具有相同强度。此外，我们表明此结果在以下意义上是最优的：对于任意$t$，$3t$无法降低至$3t-1$。该结论同样适用于带非负约束的Lasserre层级，我们通过图族$\mathcal{L}_t^+$上的同态不可区分性对其进行了类似刻画。进一步地，我们基于逻辑等价性以及一种类似于Weisfeiler-Leman算法的图着色算法，给出了带非负约束的Lasserre第$t$阶的特征描述。这为判断两个给定图是否被带非负约束的Lasserre层级第$t$阶区分提供了多项式时间算法。