We give an algorithm that for an input n-vertex graph G and integer k > 0, in time 2 O(k) n either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single-exponential in k and linear in n.Treewidth based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single-exponential in the treewidth and linear in the input size.
Let F be a family of graphs. In the F -C ompletion problem, we are given an n -vertex graph G and an integer k as input, and asked whether at most k edges can be added to G so that the resulting graph does not contain a graph from F as an induced subgraph. It was shown recently that two special cases of F -C ompletion , namely, (i) the problem of completing into a chordal graph known as M inimum F ill-in (SIAM J. Comput. 2013), which corresponds to the case of F ={ C 4 , C 5 , C 6 , …}, and (ii) the problem of completing into a split graph (Algorithmica 2015), that is, the case of F ={ C 4 , 2 K 2 , C 5 }, are solvable in parameterized subexponential time 2 O (√ k log k ) n O (1) . The exploration of this phenomenon is the main motivation for our research on F -C ompletion . In this article, we prove that completions into several well-studied classes of graphs without long induced cycles and paths also admit parameterized subexponential time algorithms by showing that: —The problem T rivially P erfect C ompletion , which is F - C ompletion for F ={ C 4 , P 4 }, a cycle and a path on four vertices, is solvable in parameterized subexponential time 2 O (√ k log k ) n O (1) . —The problems known in the literature as P seudosplit C ompletion , the case in which F{2 K 2 , C 4 }, and T hreshold C ompletion , in which F =2 K 2 , P 4 , C 4 }, are also solvable in time 2 O (√ k log k ) n O }(1) . We complement our algorithms for F -C ompletion with the following lower bounds: —For F ={2 K 2 }, F = { C 4 }, F ={ P o 4 }, and F ={2 K 2 , P 4 }, F -C ompletion cannot be solved in time 2 o(k) n O (1) unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of F -C ompletion problems for any F ⊆ {2 K 2 , C 4 , P 4 }.
The Weighted Vertex Integrity (wVI) problem takes as input an n-vertex graph G, a weight function w : V (G) → N, and an integer p. The task is to decide if there exists a set X ⊆ V (G) such that the weight of X plus the weight of a heaviest component of G − X is at most p. Among other results, we prove that:(1) wVI is NP-complete on co-comparability graphs, even if each vertex has weight 1;(2) wVI can be solved in O(p p+1 n) time;(3) wVI admits a kernel with at most p 3 vertices.Result (1) 1997), stating that the unweighted version of the problem can be solved in polynomial time on co-comparability graphs of bounded dimension, provided that an intersection model of the input graph is given as part of the input. An instance of the Weighted Component Order Connectivity (wCOC) problem consists of an n-vertex graph G, a weight function w : V (G) → N, and two integers k and ℓ, and the task is to decide if there exists a set X ⊆ V (G) such that the weight of X is at most k and the weight of a heaviest component of G − X is at most ℓ. In some sense, the wCOC problem can be seen as a refined version of the wVI problem. We obtain several classical and parameterized complexity results on the wCOC problem, uncovering interesting similarities and differences between wCOC and wVI. We prove, among other results, that: (4) wCOC can be solved in O(min{k, ℓ} · n 3 ) time on interval graphs, while the unweighted version can be solved in O(n 2 ) time on this graph class;(5) wCOC is W[1]-hard on split graphs when parameterized by k or by ℓ;(6) wCOC can be solved in 2 O(k log ℓ) n time;(7) wCOC admits a kernel with at most kℓ(k + ℓ) + k vertices.We also show that result (6) is essentially tight by proving that wCOC cannot be solved in 2 o(k log ℓ) n O (1) time, even when restricted to split graphs, unless the Exponential Time Hypothesis fails.
The Weighted Vertex Integrity (wVI) problem takes as input an n-vertex graph G, a weight function w : V (G) → N, and an integer p. The task is to decide if there exists a set X ⊆ V (G) such that the weight of X plus the weight of a heaviest component of G − X is at most p. Among other results, we prove that:(1) wVI is NP-complete on co-comparability graphs, even if each vertex has weight 1;(2) wVI can be solved in O(p p+1 n) time;(3) wVI admits a kernel with at most p 3 vertices.Result (1) 1997), stating that the unweighted version of the problem can be solved in polynomial time on co-comparability graphs of bounded dimension, provided that an intersection model of the input graph is given as part of the input. An instance of the Weighted Component Order Connectivity (wCOC) problem consists of an n-vertex graph G, a weight function w : V (G) → N, and two integers k and ℓ, and the task is to decide if there exists a set X ⊆ V (G) such that the weight of X is at most k and the weight of a heaviest component of G − X is at most ℓ. In some sense, the wCOC problem can be seen as a refined version of the wVI problem. We obtain several classical and parameterized complexity results on the wCOC problem, uncovering interesting similarities and differences between wCOC and wVI. We prove, among other results, that: (4) wCOC can be solved in O(min{k, ℓ} · n 3 ) time on interval graphs, while the unweighted version can be solved in O(n 2 ) time on this graph class;(5) wCOC is W[1]-hard on split graphs when parameterized by k or by ℓ;(6) wCOC can be solved in 2 O(k log ℓ) n time;(7) wCOC admits a kernel with at most kℓ(k + ℓ) + k vertices.We also show that result (6) is essentially tight by proving that wCOC cannot be solved in 2 o(k log ℓ) n O (1) time, even when restricted to split graphs, unless the Exponential Time Hypothesis fails.
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