Graph isomorphism: physical resources, optimization models, and algebraic characterizations

• 成果类型: Article
• 署名作者: Mancinska, Laura; Roberson, David E.; Varvitsiotis, Antonios
• 署名单位: University of Copenhagen; Technical University of Denmark; Singapore University of Technology & Design
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01989-7
• 发表日期: 2024
• 页码: 617-660
• 关键词: conic formulations

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. The goal of the players is to convince the verifier that the graphs G and H are isomorphic. In recent work along with Atserias et al. (J Comb Theory Ser 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.