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• W2170610459 abstract "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 $f:mathbb{Z}_+^n to mathbb{R}_+$ is given via an evaluation oracle. Furthermore, we assume that $f$ satisfies the diminishing return property, which is not an immediate consequence of the submodularity when the domain is the integer lattice. Then, we show (i) a $(1-1/e-epsilon)$-approximation algorithm for a cardinality constraint with $widetilde{O}(frac{n}{epsilon}log frac{r}{epsilon})$ queries, where $r$ is the maximum cardinality of feasible solutions, (ii) a $(1-1/e-epsilon)$-approximation algorithm for a polymatroid constraint with $widetilde{O}(frac{nr}{epsilon^4}+n^6)$ queries, where $r$ is the rank of the polymatroid, and (iii) a $(1-1/e-epsilon)$-approximation algorithm for a knapsack constraint with $widetilde{O}(frac{n^2}{epsilon^{18}}log frac{1}{w})(frac{1}{epsilon})^{O(1/epsilon^8)}$ queries, where $w$ is the minumum weight of elements. Our algorithms for polymatroid constraints and knapsack constraints first extend the domain of the objective function to the Euclidean space and then run the continuous greedy algorithm. We give two different kinds of continuous extensions, one is for knapsack constraints and the other is for polymatroid constraints, which might be of independent interest." @default.
• W2170610459 created "2016-06-24" @default.
• W2170610459 creator A5019397097 @default.
• W2170610459 creator A5038701345 @default.
• W2170610459 date "2015-03-04" @default.
• W2170610459 modified "2023-09-27" @default.
• W2170610459 title "Maximizing Submodular Functions with the Diminishing Return Property over the Integer Lattice." @default.
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