Greedy Algorithm for Maximization of Non-submodular Functions Subject to Knapsack Constraint

Abstract

Although submodular maximization generalizes many fundamental problems in discrete optimization, lots of real-world problems are non-submodular. In this paper, we consider the maximization problem of non-submodular function with a knapsack constraint, and explore the performance of the greedy algorithm. Our guarantee is characterized by the submodularity ratio β and curvature α. In particular, we prove that the greedy algorithm enjoys a tight approximation guarantee of (Formula Presented) for the above problem. To our knowledge, it is the first tight constant factor for this problem. In addition, we experimentally validate our algorithm by an important application, the Bayesian A-optimality.