Submodular Function Maximization on the Bounded Integer Lattice

• Published in: Approximation and Online Algorithms Vol. 9499; pp. 133 - 144
• Main Authors: Gottschalk, Corinna, Peis, Britta
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 01.01.2015; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Algorithms & data structures; Discrete Convex Analysis; Discrete mathematics; Infinite series; Integer Lattice; Submodular Function; Submodular Maximization; Tight Example
• ISBN: 3319286838; 9783319286839
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-28684-6_12

We consider the problem of maximizing a submodular function on the bounded integer lattice. As a direct generalization of submodular set functions, f:0,…,Cn→R+ $$f: \{0, \ldots , C\}^n \rightarrow \mathbb {R}_+$$ is submodular, if f(x)+f(y)≥f(x∧y)+f(x∨y) $$f(x) + f(y) \ge f(x \wedge y) + f(x \vee y)$$ for all x,y∈0,…,Cn $$x,y \in \{0, \ldots , C\}^n$$ where ∧ $$\wedge $$ and ∨ $$\vee $$ denote element-wise minimum and maximum. The objective is finding a vector x maximizing f(x). In this paper, we present a deterministic 13 $$\frac{1}{3}$$ -approximation using a framework inspired by [2]. We also provide an example that shows the analysis is tight and yields additional insight into the possibilities of modifying the algorithm. Moreover, we examine some structural differences to maximization of submodular set functions which make our problem harder to solve.