Subexponential time algorithms for finding small tree and path decompositions

The Minimum Size Tree Decomposition (MSTD) and Minimum Size Path Decomposition (MSPD) problems ask for a given nvertex graph G and integer k, what is the minimum number of bags of a tree decomposition (respectively, path decomposition) of width at most k. The problems are known to be NP-complete for each fixed k ≥ 4. In this paper we present algorithms that solve both problems for fixed k in 2O(n/ log n) time and show that they cannot be solved in 2o(n/ log n) time, assuming the Exponential Time Hypothesis