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A set function $f$ on a finite set $V$ is submodular if $f(X) + f(Y) \geq f(X\cup Y) + f(X \cap Y)$ for any pair $X, Y \subseteq V$. The symmetricdifference transformation (SD-transformation) of $f$ by a canonical set $S\subseteq V$ is a set function $g$ given by $g(X) = f(X \vartriangle S)$ for $X\subseteq V$,where $X \vartriangle S = (X \setminus S) \cup (S \setminus X)$denotes the symmetric difference between $X$ and $S$. Submodularity andSD-transformations are regarded as the counterparts of convexity and affinetransformations in a discrete space, respectively. However, submodularity isnot preserved under SD-transformations, in contrast to the fact that convexityis invariant under affine transformations. This paper presents acharacterization of SD-stransformations preserving submodularity. Then, we areconcerned with the problem of discovering a canonical set $S$, given theSD-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 saidto be strict if $f(X) + f(Y) > f(X \cup Y) + f(X \cap Y)$ holds whenever both$X \setminus Y$ and $Y \setminus X$ are nonempty. We show that the problem issolved by using ${\rm O}(|V|)$ oracle calls when $f$ is strictly submodular,although it requires exponentially many oracle calls in general.

Finding Submodularity Hidden in Symmetric Difference