2 research outputs found

    Posimodular Function Optimization

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    Given a posimodular function f:2Vβ†’Rf: 2^V \to \mathbb{R} on a finite set VV, we consider the problem of finding a nonempty subset XX of VV that minimizes f(X)f(X). Posimodular functions often arise in combinatorial optimization such as undirected cut functions. In this paper, we show that any algorithm for the problem requires Ξ©(2n7.54)\Omega(2^{\frac{n}{7.54}}) oracle calls to ff, where n=∣V∣n=|V|. It contrasts to the fact that the submodular function minimization, which is another generalization of cut functions, is polynomially solvable. When the range of a given posimodular function is restricted to be D={0,1,...,d}D=\{0,1,...,d\} for some nonnegative integer dd, we show that Ξ©(2d15.08)\Omega(2^{\frac{d}{15.08}}) oracle calls are necessary, while we propose an O(ndTf+n2d+1)O(n^dT_f+n^{2d+1})-time algorithm for the problem. Here, TfT_f denotes the time needed to evaluate the function value f(X)f(X) for a given XβŠ†VX \subseteq V. We also consider the problem of maximizing a given posimodular function. We show that Ξ©(2nβˆ’1)\Omega(2^{n-1}) oracle calls are necessary for solving the problem, and that the problem has time complexity Θ(ndβˆ’1Tf)\Theta(n^{d-1}T_f) when D={0,1,...,d}D=\{0,1,..., d\} is the range of ff for some constant dd.Comment: 18 page
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