Finding Submodularity Hidden in Symmetric Difference
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A set function f on a finite set V is submodular if f(X) + f(Y) ≥ f(X ∪ Y) + f(X ∩ Y) for any pair X, Y ⊆ V. The symmetric difference transformation ( SD-transformation) of f by a canonical set S ⊆ V is a set function g given by g(X) = f(X S) for X ⊆ V,where X S = (X ∖ S) ∪ (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-stransformations 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 ∪ Y) + f(X ∩ Y) holds whenever both X ∖ Y and Y ∖ X are nonempty. We show that the problem is solved by using O(|V|) oracle calls when f is strictly submodular, although it requires exponentially many oracle calls in general.
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