On the Complexity of Computing Treelength 1st Edition by Daniel Lokshtanov ISBN 9783540744566

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ISBN 10: 
ISBN 13: 9783540744566
Author: Daniel Lokshtanov

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 bounded treelength [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 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 <span id="MathJax-Element-1-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="32″>32. Additionally, we show that treelength can be computed in time O ^*(1.8899ⁿ) 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.

 On the Complexity of Computing Treelength 1st Table of contents:

1 Introduction
2 Hardness of treebreadth, pathlength and pathbreadth
3 Graphs with treebreadth one: some polynomial cases

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