Seeded Graph Matching via Large Neighborhood Statistics

TL;DR

This paper introduces a polynomial-time algorithm for seeded graph matching in noisy Erdős-Rényi graphs, achieving the information-theoretic limit with minimal seed sets and providing insights into unseeded cases.

Contribution

It presents a new seeded graph matching algorithm that reaches the theoretical sparsity limit and reduces seed requirements, improving upon prior methods.

Findings

Achieves polynomial-time exact recovery at the information-theoretic sparsity limit.

Requires as few as n^{3ε} seeds in sparse regimes.

Provides sub-exponential algorithms for unseeded graph matching.

Abstract

We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated Erd\H{o}s-R\'{e}nyi random graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. For seeded problems, our result provides a significant improvement over previously known results. We show that it is possible to achieve the information-theoretic limit of graph sparsity in time polynomial in the number of vertices $n$. Moreover, we show the number of seeds needed for exact recovery in polynomial-time can be as low as $n^{3\epsilon}$ in the sparse graph regime (with the average degree smaller than $n^{\epsilon}$) and $\Omega(\log n)$ in the dense graph regime. Our results also shed light on the unseeded problem. In particular, we give sub-exponential time algorithms for sparse models and an $n^{O(\log n)}$ algorithm for dense models for some parameters, including some that are not covered by recent results of Barak et al.