Constrained Submodular Maximization via Greedy Local Search

Abstract

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 show that the algorithm can be extended to yield a ratio of $\frac{1-e^{-(k+1)}}{k+1}$ for the problem with a single knapsack and the intersection of $k$ matroid constraints, for any fixed $k>1$. Our algorithms, which combine the greedy algorithm of [Khuller, Moss and Naor, 1999] and [Sviridenko, 2004] with local search, show the power of this natural framework in submodular maximization with combined constraints.

Title changed from "Interleaved Algorithms for Constrained Submodular Function Maximization"