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

TL;DR

This paper develops semi-streaming algorithms for maximizing submodular functions under b-matching, matroid, and matchoid constraints, providing new approximation ratios for various function types and constraints.

Contribution

It introduces novel semi-streaming algorithms with improved approximation ratios for submodular maximization under complex combinatorial constraints.

Findings

Achieves approximation ratios of 2+ε, 3+2√2, and 4+2√3 for linear, monotone, and non-monotone functions.

Provides approximation bounds for generalized problems with hypergraphs and additional matroid or matchoid constraints.

Extends results to complex constraints, including k-matchoids, with explicit approximation ratios.

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.