An O(C(K)N) 5-Approximation Algorithm For Treewidth

We give an algorithm that for an input n-vertex graph G and integer k > 0, in time O(c(k)n) either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single-exponential in k 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.