Semi-streaming Algorithms for Submodular Function Maximization Under b-Matching, Matroid, and Matchoid Constraints

Abstract

We consider the problem of maximizing a non-negative submodular function under the b-matching constraint, in the semi-streaming model. When the function is linear, monotone, and non-monotone, we obtain the approximation ratios of \(2+\varepsilon \), \(3 + 2 \sqrt{2} \approx 5.828\), and \(4 + 2 \sqrt{3} \approx 7.464\), respectively. We also consider a generalized problem, where a k-uniform hypergraph is given, along with an extra matroid or a \(k'\)-matchoid constraint imposed on the edges, with the same goal of finding a b-matching that maximizes a submodular function. When the extra constraint is a matroid, we obtain the approximation ratios of \(k + 1 + \varepsilon \), \(k + 2\sqrt{k+1} + 2\), and \(k + 2\sqrt{k + 2} + 3\) for linear, monotone and non-monotone submodular functions, respectively. When the extra constraint is a \(k'\)-matchoid, we attain the approximation ratio \(\frac{8}{3}k+ \frac{64}{9}k' + O(1)\) for general submodular functions.