Fixed-Parameter Complexity of Minimum Profile Problems
The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem....
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| Vydané v: | Algorithmica Ročník 52; číslo 2; s. 133 - 152 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York
Springer-Verlag
01.10.2008
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order
n
is at most
n
−1+
k
, considering
k
as the parameter; this is a parameterization above guaranteed value, since
n
−1 is a tight lower bound for the profile. We present two fixed-parameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest. |
|---|---|
| AbstractList | The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order
n
is at most
n
−1+
k
, considering
k
as the parameter; this is a parameterization above guaranteed value, since
n
−1 is a tight lower bound for the profile. We present two fixed-parameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest. |
| Author | Gutin, Gregory Szeider, Stefan Yeo, Anders |
| Author_xml | – sequence: 1 givenname: Gregory surname: Gutin fullname: Gutin, Gregory email: gutin@cs.rhul.ac.uk organization: Department of Computer Science, Royal Holloway University of London, Department of Computer Science, University of Haifa – sequence: 2 givenname: Stefan surname: Szeider fullname: Szeider, Stefan organization: Department of Computer Science, Durham University – sequence: 3 givenname: Anders surname: Yeo fullname: Yeo, Anders organization: Department of Computer Science, Royal Holloway University of London |
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| Cites_doi | 10.1145/1233481.1233493 10.1016/0020-0190(96)00050-6 10.1007/s00224-007-1330-6 10.1145/1250790.1250847 10.1007/BFb0028251 10.1016/S0022-0000(76)80045-1 10.1007/BF02006264 10.1137/S0895480199350477 10.1051/ro/1986200302451 10.1007/978-1-4612-0515-9 10.2140/pjm.1969.28.565 10.1145/167088.167170 10.1093/acprof:oso/9780198566076.001.0001 10.4064/fm-51-1-45-64 10.1089/cmb.1995.2.139 |
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| Keywords | Graph profile Above guaranteed value Kernel Fixed parameter tractability |
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| References | Gutin, Rafiey, Szeider, Yeo (CR13) 2007; 41 Niedermeier (CR19) 2006 Kendall (CR16) 1969; 28 Guo, Niedermeier (CR12) 2007; 38 Billionnet (CR4) 1986; 20 Booth, Lueker (CR1) 1976; 13 Fomin, Golovach (CR9) 2000; 13 Guo, Niedermeier (CR2) 2007; 38 CR15 Bodlaender, Downey, Fellows, Hallett, Wareham (CR5) 1995; 11 Diaz, Gibbons, Paterson, Toran (CR7) 1991; 519 Garey, Johnson (CR10) 1979 West (CR21) 2001 Lin, Yuan (CR18) 1994; 10 Cai (CR6) 1996; 58 Goldberg, Golumbic, Kaplan, Shamir (CR11) 1995; 2 Downey, Fellows (CR3) 1999 Lekkerkerker, Boland (CR17) 1962; 51 Heggernes, Paul, Telle, Villanger (CR14) 2007 Serna, Thilikos (CR20) 2005; 86 Flum, Grohe (CR8) 2006 K.S. Booth (9144_CR1) 1976; 13 G. Gutin (9144_CR13) 2007; 41 J. Flum (9144_CR8) 2006 D.G. Kendall (9144_CR16) 1969; 28 9144_CR15 C.G. Lekkerkerker (9144_CR17) 1962; 51 M.R. Garey (9144_CR10) 1979 L. Cai (9144_CR6) 1996; 58 J. Guo (9144_CR12) 2007; 38 M. Serna (9144_CR20) 2005; 86 P. Heggernes (9144_CR14) 2007 J. Guo (9144_CR2) 2007; 38 R.G. Downey (9144_CR3) 1999 P.W. Goldberg (9144_CR11) 1995; 2 H.L. Bodlaender (9144_CR5) 1995; 11 R. Niedermeier (9144_CR19) 2006 D.B. West (9144_CR21) 2001 A. Billionnet (9144_CR4) 1986; 20 J. Diaz (9144_CR7) 1991; 519 F.V. Fomin (9144_CR9) 2000; 13 Y. Lin (9144_CR18) 1994; 10 |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation |
| Title | Fixed-Parameter Complexity of Minimum Profile Problems |
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