An Improvement of Reed's Treewidth Approximation

10/07/2020
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by   Mahdi Belbasi, et al.
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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^Ξ©(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 π’ͺ(f(k)nlogn) too, but with a much smaller dependence on k. In our case, f(k) = 2^π’ͺ(k). This algorithm is simple and fast, especially for small values of k. We should mention that Bodlaender et al. [2016] have an asymptotically faster algorithm running in time 2^π’ͺ(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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