FINDING SUBMODULARITY HIDDEN IN SYMMETRIC DIFFERENCE
SIAM JOURNAL ON DISCRETE MATHEMATICS(2020)
摘要
A set function f on a finite set V is submodular if f (X) + (Y) >= f (X boolean OR Y) + f (X boolean AND Y) for any pair X , Y subset of V. The symmetric difference transformation (SD-transformation) of f by a canonical set S subset of V is a set function g given by g(X) = f(X Delta S) for X subset of V, where X Delta S = (X \ S) boolean OR (S \ X) denotes the symmetric difference between X and S. Submodularity and SD-transformations are regarded as the counterparts of convexity and affine transformations in a discrete space, respectively. However, submodularity is not preserved under SD-transformations, in contrast to the fact that convexity is invariant under affine transformations. This paper presents a characterization of SD-transformations preserving submodularity. Then, we are concerned with the problem of discovering a canonical set S, given the SD-transformation g of a submodular function f by S, provided that g(X) is given by a function value oracle. A submodular function f on V is said to be strict if f(X) + f(Y) > f(X boolean OR Y) + f(X boolean AND Y) holds whenever both X \ Y and Y \ X are nonempty. We show that the problem is solved by using O(vertical bar V vertical bar) oracle calls when f is strictly submodular, although it requires exponentially many oracle calls in general.
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关键词
submodular functions,symmetric difference
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