Approximations for Monotone and Nonmonotone Submodular Maximization with Knapsack Constraints

Ariel Kulik, Hadas Shachnai, Tami Tamir
2013 Mathematics of Operations Research  
Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints, where d is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through extension by expectation of the
more » ... modular function. Formally, we show that, for any non-negative submodular function, an α-approximation algorithm for the continuous relaxation implies a randomized (α − ε)-approximation algorithm for the discrete problem. We use this relation to improve the best known approximation ratio for the problem to 1/4 − ε, for any ε > 0, and to obtain a nearly optimal (1−e −1 −ε)−approximation ratio for the monotone case, for any ε > 0. We further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial size domain, given an oracle for the extension by expectation. This leads to a deterministic version of our technique. Our approach has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and Maximum Coverage with additional knapsack constraints.
doi:10.1287/moor.2013.0592 fatcat:calt4f635zaehn37pduhxsl4ce