(Nearly) Efficient Algorithms for the Graph Matching Problem on Correlated Random Graphs

We give a quasipolynomial time algorithm for the graph matching problem (also known as noisy or robust graph isomorphism) on correlated random graphs. Specifically, for every γ>0, we give a nO(log n) time algorithm that given a pair of γ-correlated G(n,p) graphs G0,G1 with average degree between nvarepsilon and n1/153 for varepsilon = o(1), recovers the "ground truth" permutation π∈ Sn that matches the vertices of G0 to the vertices of Gn in the way that minimizes the number of mismatched edges. We also give a recovery algorithm for a denser regime, and a polynomial-time algorithm for distinguishing between correlated and uncorrelated graphs. Prior work showed that recovery is information-theoretically possible in this model as long the average degree was at least log n, but sub-exponential time algorithms were only known in the dense case (i.e., for p > n-o(1)). Moreover, "Percolation Graph Matching", which is the most common heuristic for this problem, has been shown to require knowledge of nOmega(1) "seeds" (i.e., input/output pairs of the permutation π) to succeed in this regime. In contrast our algorithms require no seed and succeed for p which is as low as no(1)-1.