图同构：物理资源、最优化模型和代数特征
Graph isomorphism: Physical resources, optimization models, and algebraic characterizations
摘要：
在$(G,H)$ -同构博弈中，验证者与两个不通信的参与者（称为证明者）进行交互，通过私下向每个参与者发送来自$G$或$H$的随机顶点，其目的是说服验证者两个图$G$和$H$是同构的。在最近与Atserias，Šámal和Severini的工作中[组合理论杂志，系列B，136：89- 328，2019]，我们证明了验证者可以确信两个非同构图是同构的，如果允许证明者共享量子资源。在本文中，我们通过在某些复杂的凸锥上的线性约束来模拟经典图和量子图的同构，然后将其松弛为一对易处理的凸模型（半定规划）。我们的主要结果是图上相应的等价关系在适当的矩阵代数方面的一个完整的代数刻画。我们的技术是代数、组合学、优化和量子信息的有趣组合。
英文摘要：
In the $(G,H)$-isomorphism game, a verifier interacts with two non-communicating players (called provers) by privately sending each of them a random vertex from either $G$ or $H$, whose aim is to convince the verifier that two graphs $G$ and $H$ are isomorphic. In recent work along with Atserias, Šámal and Severini [Journal of Combinatorial Theory, Series B, 136:89--328, 2019] we showed that a verifier can be convinced that two non-isomorphic graphs are isomorphic, if the provers are allowed to share quantum resources. In this paper we model classical and quantum graph isomorphism by linear constraints over certain complicated convex cones, which we then relax to a pair of tractable convex models (semidefinite programs). Our main result is a complete algebraic characterization of the corresponding equivalence relations on graphs in terms of appropriate matrix algebras. Our techniques are an interesting mix of algebra, combinatorics, optimization, and quantum information.
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