Submodular Maximization Subject to Matroid Intersection on the Fly

Abstract

Despite a surge of interest in submodular maximization in the data stream model, there remain significant gaps in our knowledge about what can be achieved in this setting, especially when dealing with multiple constraints. In this work, we nearly close several basic gaps in submodular maximization subject to k matroid constraints in the data stream model. We present a new hardness result showing that super polynomial memory in k is needed to obtain an o( k /log k)-approximation. This implies near optimality of prior algorithms. For the same setting, we show that one can nevertheless obtain a constant-factor approximation by maintaining a set of elements whose size is independent of the stream size. Finally, for bipartite matching constraints, a well-known special case of matroid intersection, we present a new technique to obtain hardness bounds that are significantly stronger than those obtained with prior approaches. Prior results left it open whether a 2-approximation may exist in this setting, and only a complexity-theoretic hardness of 1.91 was known. We prove an unconditional hardness of 2.69.

Ranking with submodular functions on the fly

“…In terms of lower bounds, the analysis in Badanidiyuru [3] is shown to be optimal for any online algorithm (i.e., a streaming algorithm that maintains only a feasible solution at any moment). Besides, a 2.692approximation is impossible even subject to a bipartite matching constraint, and there exists evidence that the lower bound can be as high as 3-approximation [9]. For p-matroid constraint, a lower bound of p has been proven for any streaming algorithm with sub-linear memory; moreover, any logarithmic improvement over the best-known 4p-approximation requires memory super polynomial in p [9].…”