TY - JOUR

T1 - Finding submodularity hidden in symmetric difference

AU - Nakashima, Junpei

AU - Yamauchi, Yukiko

AU - Kijima, Shuji

AU - Yamashita, Masafumi

N1 - Funding Information:
\ast Received by the editors February 7, 2019; accepted for publication (in revised form) November 27, 2019; published electronically March 3, 2020. https://doi.org/10.1137/19M1243361 Funding: This work is partly supported by JST PRESTO grant JPMJPR16E4, Japan. \dagger Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka, 819-0395, Japan (junpei.nakashima@inf.kyushu-u.ac.jp, yamauchi@inf.kyushu-u.ac.jp, kijima@inf.kyushu-u.ac.jp, mak@inf.kyushu-u.ac.jp). 1Clearly,thecondition\Phi f(X,Y)\geq 0isequivalenttof(X)+f(Y)\geq f(X\cup Y)+f(X\cap Y), which is often used.
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

PY - 2020

Y1 - 2020

N2 - 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 Δ S) for X ⊆ V, where X Δ 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-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 ∪ 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.

AB - 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 Δ S) for X ⊆ V, where X Δ 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-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 ∪ 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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U2 - 10.1137/19M1243361

DO - 10.1137/19M1243361

M3 - Article

AN - SCOPUS:85091338603

SN - 0895-4801

VL - 34

SP - 571

EP - 585

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -