Iterative Graph Alignment via Supermodular Approximation

Graph matching, the problem of aligning a pair of graphs so as to minimize their edge disagreements, has received widespread attention owing to its broad spectrum of applications in data science. As the problem is NP-hard in the worst-case, a variety of approximation algorithms have been proposed for obtaining high quality, suboptimal solutions. In this paper, we approach the task of designing an efficient polynomial-time approximation algorithm for graph matching from a previously unconsidered perspective. Our key result is that graph matching can be formulated as maximizing a monotone, supermodular set function subject to matroid intersection constraints. We leverage this fact to apply a discrete optimization variant of the minorization-maximization algorithm which exploits supermodularity of the objective function to iteratively construct and maximize a sequence of global lower bounds on the objective. At each step, we solve a maximum weight matching problem in a bipartite graph. Differing from prior approaches, the algorithm exploits the combinatorial structure inherent in the problem to generate a sequence of iterates featuring monotonically non-decreasing objective value while always adhering to the combinatorial matching constraints. Experiments on real-world data demonstrate the empirical effectiveness of the algorithm relative to the prevailing state-of-the-art.