A c^κn 5-Approximation algorithm for treewidth

We give an algorithm that for an input n-vertex graph G and integer κ > 0, in time 2^O(κ)n, either outputs that the treewidth of G is larger than κ, or gives a tree decomposition of G of width at most 5κ + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single exponential in κ and linear in n. Treewidth-based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single exponential in the treewidth and linear in the input size.