Subexponential time algorithms for finding small tree and path decompositions

TL;DR

This paper introduces subexponential algorithms for the Minimum Size Tree and Path Decomposition problems, solving them efficiently for fixed parameters and establishing tight complexity bounds under the Exponential Time Hypothesis.

Contribution

It provides the first subexponential algorithms for these problems and proves their optimality assuming ETH.

Findings

Algorithms run in 2^{O(n/ log n)} time for fixed k

Proves no algorithms can do better than 2^{o(n/ log n)} assuming ETH

Establishes tight complexity bounds for the problems

Abstract

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