Maximizing monotone submodular functions over the integer lattice

• Published in: Mathematical programming Vol. 172; no. 1-2; pp. 539 - 563
• Main Authors: Soma, Tasuku, Yoshida, Yuichi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2018; Springer Nature B.V
• Subjects: Algorithms; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Maximization; Numerical Analysis; Theoretical; Integer lattice; DR-submodular functions; Submodular functions; 90C27 Combinatorial optimization
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-018-1324-y

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function f : Z + n → R + is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of submodularity when the domain is the integer lattice. Given this, we design polynomial-time ( 1 - 1 / e - ϵ ) -approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a ( 1 - 1 / e - ϵ ) -approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property.