An Improvement of Reed’s Treewidth Approximation

We present a new approximation algorithm for the treewidth problem which constructs corresponding tree decomposition as well. Our is faster variation of Reed's classical algorithm. For benefit reader, and to be able compare these two algorithms, we start with detailed time analysis fill in many details that have been omitted paper. Computing decompositions parameterized by $k$ fixed parameter tractable (FPT), meaning there are algorithms running $O(f(k) g(n))$ where $f$ computable function, $g$ polynomial $n$ number vertices. An shows $f(k) = 2^{O(k \log k)}$ $g(n) n n$ 5-approximation. Reed simply claims $O(n n)$ bounded his constant factor algorithm, but bound $2^{\Omega(k k)} well known. From practical point view, notice also contains term $O(k^2 2^{24k} n)$, small much worse than asymptotically leading $2^{O(k n$. analyze $f(k)$ more precisely, because purpose this paper improve times all reasonably values $k$. Our runs $\mathcal{O}(f(k)n\log{n})$ too, smaller dependence on In our case, 2^{\mathcal{O}(k)}$. This simple fast, especially should mention Bodlaender et al. [2016] an $2^{\mathcal{O}(k)} It relies very sophisticated data structure does not claim useful