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 $2+\epsilon$, $3+2\sqrt{2}\approx 5.828$, 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 $k^{\prime }$-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 $k+1+\epsilon$, $k+2\sqrt{k+1}+2$, and $k+2\sqrt{k+2}+3$ for linear, monotone and non-monotone submodular functions, respectively. When the extra constraint is a $k^{\prime }$-matchoid, we attain the approximation ratio $\frac{8}{3}k+\frac{64}{9}{k}^{\prime }+O(1)$ for general submodular functions.