On the kernelization of ranking r-CSPs: Linear vertex-kernels for generalizations of Feedback Arc Set and Betweenness in tournaments

An instance of a rankingr-constraint satisfaction problem (ranking r-CSP for short) consists of a ground set of vertices V, an arity r⩾2, a parameter k∈N and a constraint systemc, where c is a function which maps rankings of r-sized sets S⊆V to {0,1}. The objective is to decide if there exists a ran...

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Vydáno v:Discrete Applied Mathematics Ročník 186; s. 214 - 225
Hlavní autor: Perez, Anthony
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 11.05.2015
Elsevier
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ISSN:0166-218X, 1872-6771
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Abstract An instance of a rankingr-constraint satisfaction problem (ranking r-CSP for short) consists of a ground set of vertices V, an arity r⩾2, a parameter k∈N and a constraint systemc, where c is a function which maps rankings of r-sized sets S⊆V to {0,1}. The objective is to decide if there exists a ranking σ of the vertices satisfying all but at most k constraints (i.e ∑S⊆V,|S|=rc(σ(S))⩽k). We mainly focus on dense instances, that is instances where the constraint system is defined for everyr-sized subset S⊆V. Famous dense ranking r-CSPs include Feedback Arc Set and Betweenness in tournaments, two well-studied problems (Alon et al., 2009; Bessy et al., 2011; Karpinski and Schudy, 2010, 2011; Paul et al., 2011). We consider such problems from the kernelization viewpoint (Niedermeier, 2006). We prove that so-called pr-simply characterized ranking r-CSPs admit linear vertex-kernels whenever they admit constant-factor approximation algorithms. This implies that r-Dense Betweenness and r-Dense Transitive Feedback Arc Set, two natural generalizations of the previously mentioned problems (Karpinski and Schudy, 2010, 2011), admit linear vertex-kernels. Moreover, we introduce another generalization of Feedback Arc Set in Tournaments, which does not fit the aforementioned framework. Based on techniques from Coppersmith (2006) we obtain a 5-approximation, and then provide a linear vertex-kernel for this problem as well.
AbstractList An instance of a rankingr-constraint satisfaction problem (ranking r-CSP for short) consists of a ground set of vertices V, an arity r⩾2, a parameter k∈N and a constraint systemc, where c is a function which maps rankings of r-sized sets S⊆V to {0,1}. The objective is to decide if there exists a ranking σ of the vertices satisfying all but at most k constraints (i.e ∑S⊆V,|S|=rc(σ(S))⩽k). We mainly focus on dense instances, that is instances where the constraint system is defined for everyr-sized subset S⊆V. Famous dense ranking r-CSPs include Feedback Arc Set and Betweenness in tournaments, two well-studied problems (Alon et al., 2009; Bessy et al., 2011; Karpinski and Schudy, 2010, 2011; Paul et al., 2011). We consider such problems from the kernelization viewpoint (Niedermeier, 2006). We prove that so-called pr-simply characterized ranking r-CSPs admit linear vertex-kernels whenever they admit constant-factor approximation algorithms. This implies that r-Dense Betweenness and r-Dense Transitive Feedback Arc Set, two natural generalizations of the previously mentioned problems (Karpinski and Schudy, 2010, 2011), admit linear vertex-kernels. Moreover, we introduce another generalization of Feedback Arc Set in Tournaments, which does not fit the aforementioned framework. Based on techniques from Coppersmith (2006) we obtain a 5-approximation, and then provide a linear vertex-kernel for this problem as well.
Author Perez, Anthony
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  email: anthony.perez@univ-orleans.fr
  organization: Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France
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Cites_doi 10.1007/s00453-011-9492-7
10.1016/S0020-0190(00)00004-1
10.1137/120880240
10.1145/1721837.1721848
10.1017/S0963548306007887
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10.1016/j.ic.2007.02.006
10.1016/j.disopt.2013.02.001
10.1016/j.jcss.2009.09.002
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Keywords Dense instances
Kernelization algorithms
Parameterized complexity
Ranking CSP
Language English
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Snippet An instance of a rankingr-constraint satisfaction problem (ranking r-CSP for short) consists of a ground set of vertices V, an arity r⩾2, a parameter k∈N and a...
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SubjectTerms Computer Science
Data Structures and Algorithms
Dense instances
Kernelization algorithms
Parameterized complexity
Ranking CSP
Title On the kernelization of ranking r-CSPs: Linear vertex-kernels for generalizations of Feedback Arc Set and Betweenness in tournaments
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