NDLI: Minimum Transversals in Posimodular Systems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Makino, Kazuhisa Nagamochi, Hiroshi Sakashita, Mariko Fujishige, Satoru
Copyright Year	2009
Abstract	Given a system $(V,f,d)$ on a finite set V consisting of two set functions $f:2^V\to\mathbb{R}$ and $d:2^V\to\mathbb{R}$, we consider the problem of finding a set $R\subseteq V$ of minimum cardinality such that $f(X)\ge d(X)$ for all $X\subseteq V-R$, where the problem can be regarded as a natural generalization of the source location problems and the external network problems in (undirected) graphs and hypergraphs. We give a structural characterization of minimal deficient sets of $(V,f,d)$ under certain conditions. We show that all such sets form a tree hypergraph if f is posimodular and d is modulotone (i.e., each nonempty subset X of V has an element $v\in X$ such that $d(Y)\ge d(X)$ for all subsets Y of X that contain v) and that, conversely, any tree hypergraph can be represented by minimal deficient sets of $(V,f,d)$ for a posimodular function f and a modulotone function d. By using this characterization, we present a polynomial-time algorithm if, in addition, f is submodular and d is given by either $d(X)=\max\{p(v)\mid v\in X\}$ for a function $p:V\to\RR_+$ or $d(X)=\max\{r(v,w)\mid v\in X,w\in V-X\}$ for a function $r:V^2\to\mathbb{R}_+$. Our result provides first polynomial-time algorithms for the source location problem in hypergraphs and the external network problems in graphs and hypergraphs. We also show that the problem is intractable, even if f is submodular and $d\equiv\mathbf{0}$.
Starting Page	858
Ending Page	871
Page Count	14
File Format	PDF
ISSN	08954801
DOI	10.1137/060663970
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	23
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-04-24
Access Restriction	Subscribed
Subject Keyword	deficient set Analysis of algorithms and problem complexity posimodular system Connectivity tree hypergraph source location Hypergraphs Combinatorial optimization transversal problem external network problem
Content Type	Text
Resource Type	Article
Subject	Mathematics

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