    An Improvement of Reed's Treewidth Approximation

We present a new approximation algorithm for the treewidth problem which constructs a corresponding tree decomposition as well. Our algorithm is a faster variation of Reed's classical algorithm. For the benefit of the reader, and to be able to compare these two algorithms, we start with a detailed time analysis for Reed's algorithm. We fill in many details that have been omitted in Reed's paper. Computing tree decompositions parameterized by the treewidth k is fixed parameter tractable (FPT), meaning that there are algorithms running in time O(f(k) g(n)) where f is a computable function, g is a polynomial function, and n is the number of vertices. An analysis of Reed's algorithm shows f(k) = 2^O(k log k) and g(n) = n log n for a 5-approximation. Reed simply claims time O(n log n) for bounded k for his constant factor approximation algorithm, but the bound of 2^Ω(k log k) n log n is well known. From a practical point of view, we notice that the time of Reed's algorithm also contains a term of O(k^2 2^24k n log n), which for small k is much worse than the asymptotically leading term of 2^O(k log k) n log n. We analyze f(k) more precisely, because the purpose of this paper is to improve the running times for all reasonably small values of k. Our algorithm runs in 𝒪(f(k)nlogn) too, but with a much smaller dependence on k. In our case, f(k) = 2^𝒪(k). This algorithm is simple and fast, especially for small values of k. We should mention that Bodlaender et al.  have an asymptotically faster algorithm running in time 2^𝒪(k) n. It relies on a very sophisticated data structure and does not claim to be useful for small values of k.

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