Towards Nearlylinear Time Algorithms for Submodular Maximization with a Matroid Constraint. We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a emphgeneral matroid constraint with a 1  1e  epsilon approximation that achieves a fast running time provided we have a fast data structure for maintaining a maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the emphfirst algorithms for these classes of matroids that achieve a nearlyoptimal, 1  1e  epsilon approximation, using a nearlylinear number of function evaluations and arithmetic operations.

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 NPxe2x80x93hard in the worstcase, 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 polynomialtime 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 minorizationmaximization 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 nondecreasing objective value while always adhering to the combinatorial matching constraints. Experiments on realworld data demonstrate the empirical effectiveness of the algorithm relative to the prevailing stateoftheart.