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We give an algorithm that for an input n-vertex graph G and integer k>0, intime 2^[O(k)]n either outputs that the treewidth of G is larger than k, orgives a tree decomposition of G of width at most 5k+4. This is the firstalgorithm providing a constant factor approximation for treewidth which runs intime single-exponential in k and linear in n. Treewidth based computations aresubroutines of numerous algorithms. Our algorithm can be used to speed up manysuch algorithms to work in time which is single-exponential in the treewidthand linear in the input size.

A O(c^k n) 5-Approximation Algorithm for Treewidth