Seeded Graph Matching via Large Neighborhood Statistics

We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated ErdHos-R'enyi random graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. For seeded problems, our result provides a significant improvement over previously known results. We show that it is possible to achieve the information-theoretic limit of graph sparsity in time polynomial in the number of vertices n. Moreover, we show the number of seeds needed for exact recovery in polynomial-time can be as low as n3ε in the sparse graph regime (with the average degree smaller than nε) and Omega(log n) in the dense graph regime. Our results also shed light on the unseeded problem. In particular, we give sub-exponential time algorithms for sparse models and an nO(log n) algorithm for dense models for some parameters, including some that are not covered by recent results of Barak et al.