Computing Tree Decompositions with Small Independence Number
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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 graph is given with a tree decomposition of independence number at most k. However, it was an open problem if tree-independence 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 tree-independence number. In this paper, we resolve the main open problems about the computation of tree-independence 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 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 tree-independence number k without needing the decomposition as an input. Then, we show that the exact computing of tree-independence number is para-NP-hard, in particular, that for every constant k ≥ 4 it is NP-hard to decide if a given graph has tree-independence number at most k.
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