Graph Matching Via the Lens of Supermodularity

• Published in: IEEE transactions on knowledge and data engineering Vol. 34; no. 5; pp. 2200 - 2211
• Main Authors: Konar, Aritra, Sidiropoulos, Nicholas D.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximation; Approximation algorithms; Approximation methods; Bipartite graph; Combinatorial analysis; Data science; discrete optimization; Graph matching; Graph theory; Indexes; Linear programming; Lower bounds; Mathematical analysis; matroid intersection; Maximization; minorization-maximization; Optimization; Polynomials; Proteins; supermodularity; weighted bipartite matching
• ISSN: 1041-4347; 1558-2191
• DOI: 10.1109/TKDE.2020.3008128

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 article, 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.