Submodular Function Maximization on the Bounded Integer Lattice

The problem of maximizing non-negative submodular functions has been studied extensively in the last few years. However, most papers consider submodular set functions. Recently, several advances have been made for submodular functions on the integer lattice. As a direct generalization of submodular set functions, a function f : {0, . . . , C}n → R+ is submodular, if f(x)+ f(y) ≥ f(x∧ y)+ f(x∨ y) for all x, y ∈ {0, . . . , C}n where ∧ and ∨ denote element-wise minimum and maximum. The objective is finding a vector x maximizing f(x). In this paper, we present a deterministic 1 3 -approximation using a framework inspired by [6]. Moreover, we show that the analysis is tight and that other ideas used for approximating set functions cannot easily be extended. In contrast to set functions, submodularity on the integer lattice does not imply the so-called diminishing returns property. Assuming this property, it was shown that many results for set functions can also be obtained for the integer lattice. In this paper, we consider a further generalization. Instead of the integer lattice, we consider a distributive lattice as the function domain and assume the diminishing returns (DR) property. On the one hand, we show that some approximation algorithms match the set functions setting. In particular, we can obtain a 1/2-approximation for unconstrained maximization, a (1−1/e)-approximation for monotone functions under a cardinality constraint and a 1/2-approximation for a poset matroid constraint. On the other hand, for a knapsack constraint, the problem becomes significantly harder: even for monotone DR-submodular functions, we show that there is no 2 1/2 −1))δ−1-approximation for every δ > 0 under the assumption that 3 − SAT cannot be solved in time 2n 3/4+ǫ .