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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-012-9671-1