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

We consider the graph matching/similarity problem of determining how similar two given graphs G(0),G(1) are and recovering the permutation pi on the vertices of G(1) that minimizes the symmetric difference between the edges of G(0) and pi(G(1)). Graph matching/similarity has applications for pattern matching, computer vision, social network anonymization, malware analysis, and more. We give the first efficient algorithms proven to succeed in the correlated Erdos-Renyi model (Pedarsani and Grossglauser, 2011). Specifically, we give a polynomial time algorithm for the graph similarity/hypothesis testing task which works for every constant level of correlation between the two graphs that can be arbitrarily close to zero. We also give a quasi-polynomial (n(O(log n)) time) algorithm for the graph matching task of recovering the permutation minimizing the symmetric difference in this model. This is the first algorithm to do so without requiring as additional input a "seed" of the values of the ground truth permutation on at least n(Omega(1)) vertices. Our algorithms follow a general framework of counting the occurrences of subgraphs from a particular family of graphs allowing for tradeoffs between efficiency and accuracy.