20.500.12556/RUP-22213 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^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 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. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant k ≥ 4 it is NP-complete to decide whether a given graph has the tree-independence number at most k. tree-independence number approximation parameterized algorithm drevesno neodvisnostno število aproksimacija parameteriziran algoritem true false true Angleški jezik Slovenski jezik Članek v reviji 2025-12-16 17:51:13 2025-12-16 17:51:14 2025-12-20 03:03:24 0000-00-00 00:00:00 2026 0 0 str. 1-26 no. 1, [article no.] 12 Vol. 22 Jan. 2026 0000-00-00 Zaloznikova NiDoloceno NiDoloceno 0000-00-00 0000-00-00 2025-11-10 51 1549-6325 10.1145/3767730 261744131 RAZ_Dallard_Clement_Jean_2026.pdf RAZ_Dallard_Clement_Jean_2026.pdf 1 429042850FF9E8EA628A32E483E28311 8c84a858865966d6d5f6b2c88d48a397e1fd5458c21051fea0608104b12dc5c6 4c6ba72a-daa0-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=32250 https://dl.acm.org/doi/10.1145/3767730 1 bcb1aa09-da9f-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=32248 Fakulteta za matematiko, naravoslovje in informacijske tehnologije Inštitut Andrej Marušič 0 0 0