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

TL;DR

This paper introduces quasipolynomial time algorithms for graph matching in correlated random graphs, achieving near-efficient recovery without seeds in sparser regimes than previously possible.

Contribution

It presents the first quasipolynomial time algorithms for graph matching on correlated random graphs that work in sparser regimes without seed knowledge.

Findings

Recovery is possible in sparser regimes than prior algorithms.

Algorithms succeed without seed knowledge in regimes where $p$ is as low as $n^{o(1)-1}$.

Distinguishing correlated from uncorrelated graphs is achievable in polynomial time.

Abstract

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 $\gamma>0$, we give a $n^{O(\log n)}$ time algorithm that given a pair of $\gamma$-correlated $G(n,p)$ graphs $G_0,G_1$ with average degree between $n^{\varepsilon}$ and $n^{1/153}$ for $\varepsilon = o(1)$, recovers the "ground truth" permutation $\pi\in S_n$ that matches the vertices of $G_0$ to the vertices of $G_n$ 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 $n^{\Omega(1)}$ "seeds" (i.e., input/output pairs of the permutation $\pi$) to succeed in this regime. In contrast our algorithms require no seed and succeed for $p$ which is as low as $n^{o(1)-1}$.