TY - JOUR

T1 - Graph isomorphism

T2 - physical resources, optimization models, and algebraic characterizations

AU - Mančinska, Laura

AU - Roberson, David E.

AU - Varvitsiotis, Antonios

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

U2 - 10.1007/s10107-023-01989-7

DO - 10.1007/s10107-023-01989-7

M3 - Journal article

AN - SCOPUS:85164818975

SN - 0025-5610

VL - 2023

JO - Mathematical Programming

JF - Mathematical Programming

ER -