  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. For
dense graphs and sparse graphs, this can be improved to $\delta = O(
\log^{-2/3}(n) )$ and $\delta = O( \log^{-2}(d) )$ respectively, both in
polynomial time. The methodology is based on appropriately chosen distance
statistics of the degree profiles (empirical distribution of the degrees of
neighbors). Before this work, the best known result achieves $\delta=O(1)$ and
$n^{o(1)} \leq d \leq n^c$ for some constant $c$ with an $n^{O(\log n)}$-time
algorithm \cite{barak2018nearly} and $\delta=\tilde O((d/n)^4)$ and $d =
\tilde{\Omega}(n^{4/5})$ with a polynomial-time algorithm
\cite{dai2018performance}.
