Odd Submodular Functions, Dilworth Functions and Discrete Convex Functions

1988 ◽  
Vol 13 (3) ◽  
pp. 435-446 ◽  
Author(s):  
Liqun Qi
Keyword(s):  
Convex Functions ◽  
Submodular Functions ◽  
Discrete Convex

scholarly journals ON THE PIPAGE ROUNDING ALGORITHM FOR SUBMODULAR FUNCTION MAXIMIZATION — A VIEW FROM DISCRETE CONVEX ANALYSIS

2009 ◽  
Vol 01 (01) ◽  
pp. 1-23 ◽  
Author(s):  
AKIYOSHI SHIOURA
Keyword(s):  
Approximation Algorithm ◽  
Convex Analysis ◽  
Submodular Function ◽  
Submodular Functions ◽  
Concave Functions ◽  
Discrete Convex Analysis ◽  
Discrete Convex ◽  
Weighted Rank ◽  
Rounding Algorithm ◽  
Rank Functions

We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 - 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M ♮-concave functions. M ♮-concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.

scholarly journals Proximity theorems of discrete convex functions

2004 ◽  
Vol 99 (3) ◽  
pp. 539-562 ◽  
Author(s):  
Kazuo Murota ◽  
Akihisa Tamura
Keyword(s):  
Convex Functions ◽  
Discrete Convex

Cone superadditivity of discrete convex functions

2011 ◽  
Vol 135 (1-2) ◽  
pp. 25-44 ◽  
Author(s):  
Yusuke Kobayashi ◽  
Kazuo Murota ◽  
Robert Weismantel
Keyword(s):  
Convex Functions ◽  
Discrete Convex

scholarly journals Discrete Convex Functions and Proof of the Six Circle Conjecture of Fejes Tóth

1984 ◽  
Vol 36 (3) ◽  
pp. 569-576 ◽  
Author(s):  
Imre Bárány ◽  
Zoltán Füredi ◽  
János Pach
Keyword(s):  
Convex Functions ◽  
Circle Packing ◽  
Extremal Property ◽  
Discrete Convex ◽  
Image Position

A system of openly disjoint discs in the plane is said to form a 6-neighboured circle packing if every is tangent to at least 6 other elements of (It is evident that such a system consists of infinitely many discs.) The simplest example is the regular circle packing all of whose circles are of the same size and have exactly 6 neighbours. L. Fejes Tóth conjectured that the regular circle packing has the interesting extremal property that, if we slightly “perturb” it, then there will necessarily occur either arbitrarily small or arbitrarily large circles. More precisely, he asked whether or not the following “zero or one law” (cf. [3], [6]) is valid: If is a 6-neighboured circle packing, thenwhere r(C) denotes the radius of circle C, inf and sup are taken over all C ∊

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