Constrained submodular maximization via greedy local search. (English) Zbl 1476.90289

Summary: We present a simple combinatorial \(\frac{1 - e^{- 2}}{2} \)-approximation algorithm for maximizing a monotone submodular function subject to a knapsack and a matroid constraint. This classic problem is known to be hard to approximate within factor better than \(1 - 1/e\). We extend the algorithm to yield \(\frac{1 - e^{-(k + 1)}}{k + 1}\) approximation for submodular maximization subject to a single knapsack and \(k\) matroid constraints, for any fixed \(k > 1\). Our algorithms, which combine the greedy algorithm of S. Khuller et al. [Inf. Process. Lett. 70, No. 1, 39–45 (1999; Zbl 1002.68203)] and M. Sviridenko [Oper. Res. Lett. 32, No. 1, 41–43 (2004; Zbl 1056.90124)] with local search, show the power of this natural framework in submodular maximization with combined constraints.


90C27 Combinatorial optimization
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