Efficient random graph matching via degree profiles

Understand

Random graph matching refers to recovering the underlying vertex correspondence between two random graphs with correlated edges; a prominent example is when the two random graphs are given by Erd\H{o}s-R\'{e}nyi graphs $G(n,\frac{d}{n})$.

• This can be viewed as an average-case and noisy version of the graph isomorphism problem.
• Under this model, the maximum likelihood estimator is equivalent to solving the intractable quadratic assignment problem.
• This work develops an $\tilde{O}(n d^2+n^2)$-time algorithm which perfectly recovers the true vertex correspondence with high probability, provided that the average degree is at least $d = \Omega(\log^2 n)$ and the two graphs differ by at most $\delta = O( \log^{-2}(n) )$ fraction of edges.