Non-monotone submodular maximization under matroid and knapsack constraints

Abstract

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard.In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for non-monotone submodular functions. In particular, for any constant k, we present a " 1 k+2+ 1 k + " -approximation for the submodular maximization problem under k matroid constraints, and a`1 5 − ´-approximation algorithm for this problem subject to k knapsack constraints ( > 0 is any constant). We improve the approximation guarantee of our algorithm to 1 k+1+ 1 k−1 + for k ≥ 2 partition matroid constraints. This idea also gives a " 1 k+ "-approximation for maximizing a monotone submodular function subject to k ≥ 2 partition matroids, which improves over the previously best known guarantee of 1 k+1 .

Constrained Submodular Maximization: Beyond 1/e

“…A key difficulty that separates the submodular setting from the classical setting is that even finding a fractional solution may be quite challenging, and in particular it is NP-hard to solve the continuous relaxation for maximizing submodular functions that is based on the multilinear extension. Thus, a line of work has been developed to approximately optimize this relaxation [3,13,8,7] culminating in the work [7], which we extend here.…”

Scheduling to minimize power consumption using submodular functions

“…In Section 2, we introduce a general optimization problem, called submodular maximization with budget constraints. Many interesting optimization problems are special cases of this general problem, for example, Set Cover and Max Cover [7,11] and the submodular maximization problems studied in [9,10]. We obtain bicriteria ((1 − ε), O(log 1/ε))-approximation factor for this general problem.…”

“…The authors of [10] studied the problem of submodular maximization under matroid and knapsack constraints (which can be seen as some kind of budget constraints), and they give the first constant factor approximation when the number of constraints is constant. We try to find solutions with more utility by relaxing the budget constraints.…”

Optimizing Peer Teaching to Enhance Team Performance

“…Unlike minimization, maximizing a submodular function is NP-hard, however. To find the best peer teaching strategy, we may use the algorithm, which we refer to as MAX, proposed by Lee et al in [8]. This is a polynomial time algorithm which achieves a 1/(4 + )-approximation, assuming a value oracle model, i.e.…”

“…At each step, it checks whether the proposed better learning profile is feasible. Lee et al [8] do not explicitly provide an algorithm for checking whether a set is independent. However, in our setting the feasibility of a learning profile is not trivial to verify.…”