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 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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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. |
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| 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 |
| Author_xml | – sequence: 1 givenname: Anthony surname: Perez fullname: Perez, Anthony 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 10.1137/050623905 10.1016/j.jcss.2011.07.001 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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| References | Guillemot, Havet, Paul, Perez (br000065) 2012 Bessy, Fomin, Gaspers, Paul, Perez, Saurabh, Thomassé (br000025) 2011; 77 Bodlaender (br000035) 2009 Weller, Komusiewicz, Niedermeier, Uhlmann (br000115) 2012; 78 Bodlaender, Downey, Fellows, Hermelin (br000040) 2009; 75 Bodlaender, Jansen, Kratsch (br000045) 2014; 28 Paul, Perez, Thomassé (br000100) 2011; vol. 6907 Niedermeier, Rossmanith (br000095) 2000; 73 Alon (br000015) 2006; 20 Coppersmith, Fleischer, Rudra (br000055) 2006 Downey, Fellows (br000060) 1999 Ailon, Alon (br000010) 2007; 205 Karpinski, Schudy (br000075) 2011; vol. 6845 Bessy, Perez (br000030) 2011; vol. 6914 Karpinski, Schudy (br000070) 2010; vol. 6506 Abu-Khzam (br000005) 2010; 76 Alon, Lokshtanov, Saurabh (br000020) 2009 Kenyon-Mathieu, Schudy (br000080) 2007 Niedermeier (br000090) 2006 Thomassé (br000105) 2010; 6 Charbit, Thomassé, Yeo (br000050) 2007; 16 Kratsch, Wahlström (br000085) 2013; 10 van Bevern, Moser, Niedermeier (br000110) 2012; 62 Guillemot (10.1016/j.dam.2015.01.032_br000065) 2012 Karpinski (10.1016/j.dam.2015.01.032_br000070) 2010; vol. 6506 Charbit (10.1016/j.dam.2015.01.032_br000050) 2007; 16 Paul (10.1016/j.dam.2015.01.032_br000100) 2011; vol. 6907 Alon (10.1016/j.dam.2015.01.032_br000015) 2006; 20 Bessy (10.1016/j.dam.2015.01.032_br000030) 2011; vol. 6914 Bessy (10.1016/j.dam.2015.01.032_br000025) 2011; 77 Kenyon-Mathieu (10.1016/j.dam.2015.01.032_br000080) 2007 Bodlaender (10.1016/j.dam.2015.01.032_br000045) 2014; 28 Thomassé (10.1016/j.dam.2015.01.032_br000105) 2010; 6 Bodlaender (10.1016/j.dam.2015.01.032_br000040) 2009; 75 Niedermeier (10.1016/j.dam.2015.01.032_br000090) 2006 Kratsch (10.1016/j.dam.2015.01.032_br000085) 2013; 10 Weller (10.1016/j.dam.2015.01.032_br000115) 2012; 78 Karpinski (10.1016/j.dam.2015.01.032_br000075) 2011; vol. 6845 Bodlaender (10.1016/j.dam.2015.01.032_br000035) 2009 van Bevern (10.1016/j.dam.2015.01.032_br000110) 2012; 62 Coppersmith (10.1016/j.dam.2015.01.032_br000055) 2006 Abu-Khzam (10.1016/j.dam.2015.01.032_br000005) 2010; 76 Alon (10.1016/j.dam.2015.01.032_br000020) 2009 Downey (10.1016/j.dam.2015.01.032_br000060) 1999 Ailon (10.1016/j.dam.2015.01.032_br000010) 2007; 205 Niedermeier (10.1016/j.dam.2015.01.032_br000095) 2000; 73 |
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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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