On the complexity of computing treelength

We resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourisboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of a bounded treelength Dourisboure and Gavoille (2007)  [6]. Whil...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 158; no. 7; pp. 820 - 827
Main Author: Lokshtanov, Daniel
Format: Journal Article
Language:English
Published: Elsevier B.V 06.04.2010
Subjects:
Approximation
Chordal sandwich problem
Exact exponential algorithm
Graph treelength
Inapproximability
NP-complete
Chordal sandwich problem
Exact exponential algorithm
NP-complete
Approximation
Graph treelength
Inapproximability
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary:We resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourisboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of a bounded treelength Dourisboure and Gavoille (2007)  [6]. While recognizing graphs with treelength 1 is easily seen as equivalent to recognizing chordal graphs, which can be done in linear time, the computational complexity of recognizing graphs with treelength 2 was unknown until this result. We show that the problem of determining whether a given graph has a treelength at most k is NP-complete for every fixed k ≥ 2 , and use this result to show that treelength in weighted graphs is hard to approximate within a factor smaller than 3 2 . Additionally, we show that treelength can be computed in time O ∗ ( 1.754 9 n ) by giving an exact exponential time algorithm for the Chordal Sandwich problem and showing how this algorithm can be used to compute the treelength of a graph.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2009.10.007