A Reduction for Optimizing Lattice Submodular Functions with Diminishing Returns

06/27/2016

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by   Alina Ene, et al.

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A function f: Z_+^E →R_+ is DR-submodular if it satisfies f( + χ_i) -f () > f( + χ_i) - f() for all <, i∈ E. Recently, the problem of maximizing a DR-submodular function f: Z_+^E →R_+ subject to a budget constraint _1 ≤ B as well as additional constraints has received significant attention SKIK14,SY15,MYK15,SY16. In this note, we give a generic reduction from the DR-submodular setting to the submodular setting. The running time of the reduction and the size of the resulting submodular instance depends only logarithmically on B. Using this reduction, one can translate the results for unconstrained and constrained submodular maximization to the DR-submodular setting for many types of constraints in a unified manner.