Improved Multi-Pass Streaming Algorithms for Submodular Maximization with Matroid Constraints
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We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general p-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of p arbitrary matroid constraints and p-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of p+1+ε using O(p/ε)-passes and requires storing only O(k) elements, where k is the maximum size of feasible solution. This immediately gives an O(1/ε)-pass (2+ε)-approximation algorithms for monotone submodular maximization in a matroid and (3+ε)-approximation for monotone submodular matching. Our algorithm is oblivious to the choice ε and can be stopped after any number of passes, delivering the appropriate guarantee. We extend our techniques to obtain the first multi-pass streaming algorithm for general, non-negative submodular functions subject to a p-matchoid constraint with a number of passes independent of the size of the ground set and k. We show that a randomized O(p/ε)-pass algorithm storing O(p^3klog(k)/ε^3) elements gives a (p+1+γ̅+O(ε))-approximation, where g̅a̅m̅m̅a̅ is the guarantee of the best-known offline algorithm for the same problem.
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