We study the problem of maximizing non-monotone diminish return (DR)-submodular function on the bounded integer lattice, which is a generalization of submodular set function. DR-submodular functions consider the case that we can choose multiple copies for each element in the ground set. This generalization has many applications in machine learning. In this paper, we propose a [Formula: see text]-approximation algorithm with a running time of [Formula: see text], where [Formula: see text] is the size of the ground set, [Formula: see text] is the upper bound of integer lattice. Discovering important properties of DR-submodular function, we propose a fast double greedy algorithm which improves the running time.
[Formula: see text]-submodular maximization is a generalization of submodular maximization, which requires us to select [Formula: see text] disjoint subsets instead of one subset. Attracted by practical values and applications, we consider [Formula: see text]-submodular maximization with two kinds of constraints. For total size and individual size difference constraints, we present a [Formula: see text]-approximation algorithm for maximizing a nonnegative k-submodular function, running in time [Formula: see text] at worst. Specially, if [Formula: see text] is multiple of [Formula: see text], the approximation ratio can reduce to [Formula: see text], running in time [Formula: see text] at worst. Besides, this algorithm can be applied to [Formula: see text]-bisubmodular achieving [Formula: see text]-approximation running in time [Formula: see text]. Furthermore, if [Formula: see text] is multiple of 2, the approximation ratio can reduce to [Formula: see text], running in time [Formula: see text] at worst. For individual size constraint, there is a [Formula: see text]-approximation algorithm for maximizing a nonnegative [Formula: see text]-submodular function and an nonnegative [Formula: see text]-bisubmodular function, running in time [Formula: see text] and [Formula: see text] respectively, at worst.
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