Maximizing Non-Monotone DR-Submodular Functions with Cardinality Constraints

We consider the problem of maximizing a nonmonotone DR-submodular function subject to a cardinality constraint. Diminishing returns (DR) submodularity is a generalization of the diminishing returns property for functions defined over the integer lattice. This generalization can be used to solve many machine learning or combinatorial optimization problems such as optimal budget allocation, revenue maximization, etc. In this work we propose the first polynomial-time approximation algorithms for non-monotone constrained maximization. Our first algorithm achieves an approximation ratio of (1 + 12 ‖B‖1 √ k(‖B‖1−k) )−1, where B ∈ R represents the limit for each coordinate of the function and k is the parameter of cardinality constraint. This ratio is maximized when k = ‖B‖1 /2 and evaluates to a (1/2)-approximation. Our second algorithm is a random greedy algorithm that gives a (1/e)-approximation ratio in expectation. We implement our algorithms for a revenue maximization problem with a real-world dataset to check their efficiency and performance.