Approximate Multimatroid Intersection via Iterative Refinement We introduce a new iterative rounding technique to round a point in a matroid polytope subject to further matroid constraints. This technique returns an independent set in one matroid with limited violations of the other ones. On top of the classical steps of iterative relaxation approaches, we iteratively refinesplit involved matroid constraints to obtain a more restrictive constraint system, that is amenable to iterative relaxation techniques. Hence, throughout the iterations, we both tighten constraints and later relax them by dropping constraints under certain conditions. Due to the refinement step, we can deal with considerably more general constraint classes than existing iterative relaxationrounding methods, which typically round on one matroid polytope with additional simple cardinality constraints that do not overlap too much. We show how our rounding method, combined with an application of a matroid intersection algorithm, yields the first 2approximation for finding a maximumweight common independent set in 3 matroids. Moreover, our 2approximation is LPbased, and settles the integrality gap for the natural relaxation of the problem. Prior to our work, no upper bound better than 3 was known for the integrality gap, which followed from the greedy algorithm. We also discuss various other applications of our techniques, including an extension that allows us to handle a mixture of matroid and knapsack constraints.

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.