Approximation Algorithms for Intersection Graphs

We study three complexity parameters that, for each vertex v , are an upper bound for the number of cliques that are sufficient to cover a subset S ( v ) of its neighbors. We call a graph k-perfectly groupable if S ( v ) consists of all neighbors, k-simplicial if S ( v ) consists of the neighbors wi...

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Published in:	Algorithmica Vol. 68; no. 2; pp. 312 - 336
Main Authors:	Kammer, Frank, Tholey, Torsten
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.02.2014 Springer
Subjects:	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Computer Science Computer science; control theory; systems Computer Systems Organization and Communication Networks Convex and discrete geometry Data Structures and Information Theory Exact sciences and technology Flows in networks. Combinatorial problems Geometry Information retrieval. Graph Mathematics Mathematics of Computing Operational research and scientific management Operational research. Management science Sciences and techniques of general use Theoretical computing Theory of Computation Parameterized approximation algorithms Graph algorithms NP-hard problems Intersection graphs Intersection Partition Tree(graph) Dominating set Similarity Graph clique Chordal graph Graph theory Approximation algorithm Polynomial time Upper bound NP hard problem Graph colouring Independent set Intersection graph Parameterized complexity
ISSN:	0178-4617, 1432-0541
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Abstract	We study three complexity parameters that, for each vertex v , are an upper bound for the number of cliques that are sufficient to cover a subset S ( v ) of its neighbors. We call a graph k-perfectly groupable if S ( v ) consists of all neighbors, k-simplicial if S ( v ) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S ( v ) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.
AbstractList	We study three complexity parameters that, for each vertex v , are an upper bound for the number of cliques that are sufficient to cover a subset S ( v ) of its neighbors. We call a graph k-perfectly groupable if S ( v ) consists of all neighbors, k-simplicial if S ( v ) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S ( v ) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.
Author	Kammer, Frank Tholey, Torsten
Author_xml	– sequence: 1 givenname: Frank surname: Kammer fullname: Kammer, Frank organization: Institut für Informatik, Augsburg University – sequence: 2 givenname: Torsten surname: Tholey fullname: Tholey, Torsten email: tholey@informatik.uni-augsburg.de organization: Institut für Informatik, Augsburg University
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Cites_doi	10.1093/acprof:oso/9780198566076.001.0001 10.1007/978-3-540-78773-0_64 10.1016/0304-3975(76)90059-1 10.1016/0022-0000(91)90023-X 10.1145/1721837.1721856 10.1007/978-3-642-02927-1_64 10.4086/toc.2007.v003a006 10.1002/net.3230250205 10.1016/j.endm.2004.06.012 10.1007/s00373-011-1026-1 10.1016/0020-0190(81)90111-3 10.1016/0196-6774(83)90012-3 10.1007/BF01864170 10.1145/1383369.1383380 10.1006/jagm.1997.0903 10.1137/S0097539793251219 10.1145/1989493.1989520 10.1016/S0196-6774(02)00294-8 10.1016/S0012-365X(99)00231-9 10.1016/0890-5401(90)90043-H 10.1007/BF01196133 10.1007/978-1-4757-3613-7_23 10.1007/978-3-642-15369-3_20 10.1137/S0097539702402676 10.1007/978-1-4612-0515-9 10.1016/j.ipl.2008.09.021 10.1137/0201013 10.1137/S0097539703437843 10.1137/0601001 10.1016/0012-365X(90)90358-O 10.1287/opre.34.2.250 10.1145/800116.803775
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Keywords	Parameterized approximation algorithms Graph algorithms NP-hard problems Intersection graphs Intersection Partition Tree(graph) Dominating set Similarity Graph clique Chordal graph Graph theory Approximation algorithm Polynomial time Upper bound NP hard problem Graph colouring Independent set Intersection graph Parameterized complexity
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Snippet	We study three complexity parameters that, for each vertex v , are an upper bound for the number of cliques that are sufficient to cover a subset S ( v ) of...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Computer Science Computer science; control theory; systems Computer Systems Organization and Communication Networks Convex and discrete geometry Data Structures and Information Theory Exact sciences and technology Flows in networks. Combinatorial problems Geometry Information retrieval. Graph Mathematics Mathematics of Computing Operational research and scientific management Operational research. Management science Sciences and techniques of general use Theoretical computing Theory of Computation
Title	Approximation Algorithms for Intersection Graphs
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