  We present a new approximation algorithm for the treewidth problem which
finds an upper bound on the treewidth and 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 of 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 $\mathcal{O}(f(k)
g(n))$ where $f$ is a computable function, and $g(n)$ is polynomial in $n$,
where $n$ is the number of vertices. An analysis of Reed's algorithm shows
$f(k) = 2^{\mathcal{O}(k \log k)}$ and $g(n) = n \log n$ for a 5-approximation.
Reed simply claims time $\mathcal{O}(n \log n)$ for bounded $k$ for his
constant factor approximation algorithm, but the bound of $2^{\Omega(k \log k)}
n \log n$ is well known. From a practical point of view, we notice that the
time of Reed's algorithm also contains a term of $\mathcal{O}(k^2 2^{24k} n
\log n)$, which for small $k$ is much worse than the asymptotically leading
term of $2^{\mathcal{O}(k \log k)} n \log n$. 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 $\mathcal{O}(f(k)n\log{n})$ too, but with a much
smaller dependence on $k$. In our case, $f(k) = 2^{\mathcal{O}(k)}$. This
algorithm is simple and fast, especially for small values of $k$. We should
mention that Bodlaender et al. [2016] have an algorithm with a linear
dependence on $n$, and Korhonen [2021] obtains the much better approximation
ratio of 2, while the current paper achieves a better dependence on $k$.
