🤖 AI Summary
This paper studies the *tree-independence number* τ(G)—the maximum independence number among the subgraphs induced by the bags of an optimal tree decomposition of G. This parameter enables efficient algorithms for NP-hard problems such as Maximum Weight Independent Set (MWIS). We present the first fixed-parameter approximation algorithm for τ(G): it computes an 8k-approximation in time $2^{O(k^2)} n^{O(k)}$. We prove that this runtime is conditionally tight under Gap-ETH, and show that exact computation of τ(G) is para-NP-hard for k ≥ 4. This yields the first complete parameterized complexity characterization of τ(G), resolving its approximability and hardness without assuming a precomputed tree decomposition. Moreover, our result implies a $2^{O(k^2)} n^{O(k)}$-time algorithm for MWIS on graphs with τ(G) ≤ k—bypassing the traditional requirement of being given a tree decomposition with bounded 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 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^{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^{O(k^3)} either constructs a tree decomposition of G whose independence number is O(k^3) or correctly reports that the tree-independence number of G is larger than k. In this paper, 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^{O(k^2)} n^{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^{O(k^2)} n^{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 ge 4 it is NP-hard to decide if a given graph has the tree-independence number at most k.
