As a fundamental problem in computer vision, graph matching problem can usually be formulated as a Quadratic Programming (QP) problem with doubly stochastic and discrete (integer) constraints. Since it is NP-hard, approximate algorithms are required. In this paper, we present a new algorithm, called Multiplicative Update Graph Matching (MPGM), that develops a multiplicative update technique to solve the QP matching problem. MPGM has three main benefits: (1) theoretically, MPGM solves the general QP problem with doubly stochastic constraint naturally whose convergence and KKT optimality are guaranteed.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Wegivethe first efficient algorithms proven to succeed in the correlated Erdös-Rényi model (Pedarsani and Grossglauser, 2011). Specifically, we give apolynomial time algorithm for thegraphsimilarity/hypothesis testingtaskwhich worksforeveryconstant level of correlation between the two graphs that can be arbitrarily close to zero. We also give a quasi-polynomial (nO(logn) time) algorithm for thegraph matching task of recovering the permutation minimizing the symmetric difference in this model.

We consider the problem of graph matching for a sequence of graphs generated under a time-dependent Markov chain noise model. Our edgelighter error model, a variant of the classical lamplighter random walk, iteratively corrupts the graph $G_0$ with edge-dependent noise, creating a sequence of noisy graph copies $(G_t)$. Much of the graph matching literature is focused on anonymization thresholds in edge-independent noise settings, and we establish novel anonymization thresholds in this edge-dependent noise setting when matching $G_0$ and $G_t$. Moreover, we also compare this anonymization threshold with the mixing properties of the Markov chain noise model. We show that when $G_0$ is drawn from an Erdős-Rényi model, the graph matching anonymization threshold and the mixing time of the edgelighter walk are both of order $Θ(n^2\log n)$. We further demonstrate that for more structured model for $G_0$ (e.g., the Stochastic Block Model), graph matching anonymization can occur in $O(n^α\log n)$ time for some $α<2$, indicating that anonymization can occur before the Markov chain noise model globally mixes. Through extensive simulations, we verify our theoretical bounds in the settings of Erdős-Rényi random graphs and stochastic block model random graphs, and explore our findings on real-world datasets derived from a Facebook friendship network and a European research institution email communication network.