(Nearly) Efficient Algorithms for the Graph Matching Problem on Correlated Random Graphs
This addresses a fundamental algorithmic challenge in network alignment and graph isomorphism for sparse graphs, offering efficient solutions where previous heuristics failed without seed information.
The paper tackles the graph matching problem on correlated random graphs by providing a quasipolynomial time algorithm that recovers the ground truth permutation without seeds, achieving recovery for average degrees as low as n^{o(1)-1}, where prior methods required seeds or were limited to denser regimes.
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 $n^{O(\log n)}$ time algorithm that given a pair of $γ$-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 $π\in S_n$ 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 $n^{Ω(1)}$ "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 $n^{o(1)-1}$.