Computing Tree Decompositions with Small Independence Number

The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP -hard graph problems, like maximum-weight independent set, can be solved in time \(n^{\mathcal{O}(k)}\) if the input \( n \) -vertex graph is given together with a tree decomposition of independence number \( k \) . Yolov, in SODA 2018, gave an algorithm that, given an \( n \) -vertex graph \( G \) and an integer \( k \) , in time \(n^{\mathcal{O}(k^{3})}\) either constructs a tree decomposition of \( G \) whose independence number is \(\mathcal{O}(k^{3})\) or correctly reports that the tree-independence number of \( G \) is larger than \( k \) . In this article, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time \(2^{\mathcal{O}(k^{2})}n^{\mathcal{O}(k)}\) and either outputs a tree decomposition of \( G \) with independence number at most \(8k\) or determines that the tree-independence number of \( G \) is larger than \( k \) . This implies \(2^{\mathcal{O}(k^{2})}n^{\mathcal{O}(k)}\) -time algorithms for various problems, like maximum-weight independent set, parameterized by the tree-independence number \( k \) without needing the decomposition as an input. Assuming Gap-ETH, an \(n^{\Omega(k)}\) factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP -hard: We show that for every constant \(k\geq 4\) it is NP -complete to decide whether a given graph has the tree-independence number at most \( k \) .