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

Abstract In the ( G , H )-isomorphism game, a verifier interacts with two non-communicating players (called provers), by privately sending each of them random vertex from either or . The goal is to convince that graphs and are isomorphic. recent work along Atserias et al. (J Comb Theory Ser B 136:89–328, 2019) we showed can be convinced non-isomorphic isomorphic, if provers allowed share quantum resources. this paper model classical graph isomorphism linear constraints over certain complicated convex cones, which then relax pair tractable models (semidefinite programs). Our main result complete algebraic characterization corresponding equivalence relations on in terms appropriate matrix algebras. techniques an interesting mix algebra, combinatorics, optimization, information.