On the Complexity of Computing Treelength

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 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 $\frac{3}{2}$. 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.