Computing Tree Decompositions with Small Independence Number
Abstract
The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The treeindependence number of a graph is the minimum independence number of a tree decomposition of it. Several NPhard graph problems, like maximum weight independent set, can be solved in time $n^{O(k)}$ if the input graph is given with a tree decomposition of independence number at most $k$. However, it was an open problem if treeindependence number could be computed or approximated in $n^{f(k)}$ time, for some function $f$, and in particular it was not known if maximum weight independent set could be solved in polynomial time on graphs of bounded treeindependence number. In this paper, we resolve the main open problems about the computation of treeindependence number. First, we give an algorithm that given an $n$vertex graph $G$ and an integer $k$, in time $2^{O(k^2)} n^{O(k)}$ either outputs a tree decomposition of $G$ with independence number at most $8k$, or determines that the treeindependence number of $G$ is larger than $k$. This implies $2^{O(k^2)} n^{O(k)}$ time algorithms for various problems, like maximum weight independent set, parameterized by treeindependence number $k$ without needing the decomposition as an input. Then, we show that the exact computing of treeindependence number is paraNPhard, in particular, that for every constant $k \ge 4$ it is NPhard to decide if a given graph has treeindependence number at most $k$.
 Publication:

arXiv eprints
 Pub Date:
 July 2022
 arXiv:
 arXiv:2207.09993
 Bibcode:
 2022arXiv220709993D
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics