Finding the Submodularity Hidden in Symmetric Difference

A fundamental property of convex functions in continuous space is that the convexity is preserved under affine transformations. 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 M S) for X ⊆ V , where X M S = (X \S)∪ (S \X) is the symmetric difference between X and S. Despite that submodular functions and SD-transformations are regarded as counterparts of convex functions and affine transformations in finite discrete space, not all SD-transformations do not preserve the submodularity. Starting with a characterization of SD-stransformations that preserve the submodularity, this paper investigates 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 an 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 solvable by using O(|V |) oracle calls when f is strictly submodular, although it requires exponentially many oracle calls in general.