On the complexity of computing treelength |
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Authors: | Daniel Lokshtanov |
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Institution: | Department of Informatics, University of Bergen, N-5020 Bergen, Norway |
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Abstract: | 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 . Additionally, we show that treelength can be computed in time O∗(1.7549n) 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. |
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Keywords: | Graph treelength NP-complete Approximation Inapproximability Exact exponential algorithm Chordal sandwich problem |
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