Abstract.
We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2V with ?,V∈? and f(?)=0 and for any vector x∈R V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|2) times, the descent method finds a sequence of subsets Z 1,Z 2,···,Z k of V such that f(Z 1)>f(Z 2)>···>f(Z k )=min{f(Y) | Y∈?}, where k is O(|V|2). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms.
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Received: September 9, 2001 / Accepted: October 15, 2001¶Published online December 6, 2001
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Fujishige, S., Iwata, S. A descent method for submodular function minimization. Math. Program. 92, 387–390 (2002). https://doi.org/10.1007/s101070100273
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DOI: https://doi.org/10.1007/s101070100273