Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth

Bodlaender, Hans L.; Cygan, Marek; Kratsch, Stefan; Nederlof, Jesper

doi:10.1016/j.ic.2014.12.008

Cited by 189 publications

(304 citation statements)

References 32 publications

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“…Each of the vertices t 5 and t 6 must be one of types f3, u3, f4, u4, f5, and a u5-vertex, and each of their weights decreases by at least m 3 = min{w 3 , Δ 3 , w 4 , Δ 4 , w 5 , Δ 5 }. Thus, the total weight decrease in the branch of delete(vt 1 ) is at least w 5 − w 4 + w 3 + 2m 3 .…”

Section: Branching On Edges Around F5-vertices (Branchingmentioning

confidence: 99%

“…Hence, the weight of vertex v decreases by Δ 5,3 , and both weights of vertices t 1 and t 2 each decrease by w 3 . Thus, the total weight decrease in the branch of delete(vt 1 ) is at least w 5 − w 3 + 2w 3 .…”

Section: Fig 17mentioning

confidence: 99%

“…Let μ(I) denote the measure of the TSP in the degree-4 instance I, whereμ(I) =ŵ 3 n 3 +ŵ 3 n 3 +ŵ 4 n 4 +ŵ 4 n 4 , and the weightŵ 3 for an f3-vertex is 0.21968, the weightŵ 3 for an u3-vertex is 0.45540, the weightŵ 4 for an f4-vertex is 0.59804 and the weightŵ 4 for an u4-vertex is 1 [16]. Hence, the running time bound of the TSP in the degree-4 instance I with measureμ(I) is given by 1.69193μ (I) .…”

Section: Switching To Tsp In Degree-4 Graphsmentioning

confidence: 99%

“…Ever since, this running time has only been improved for special types of graphs. Primarily, investigation efforts have been focused on graphs in for degree-3 graphs has been presented by Bodlaender et al [3], where the authors make use of a general approach for speeding up straightforward dynamic programming algorithms. For the TSP in degree-4 graphs, Gebauer [8] has shown a time bound of O * (1.733 n ), by using a dynamic programming approach.…”

Section: Introductionmentioning

confidence: 99%

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An Improved-time Polynomial-space Exact Algorithm for TSP in Degree-5 Graphs

Yunos

Shurbevski

Nagamochi

2017

Journal of Information Processing

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Abstract:The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard optimization problems. Following a recent trend of research which focuses on developing algorithms for special types of TSP instances, namely graphs of limited degree, in an attempt to reduce a part of the time and space complexity, we present a polynomialspace branching algorithm for the TSP in an n-vertex graph with degree at most 5, and show that it has a running time of O * (2.3500 n ), which improves the previous best known time bound of O * (2.4723 n ) given by the authors (the 12 th International Symposium on Operations Research and Its Application (ISORA 2015(ISORA ), pp.45-58, 2015. While the base of the exponent in the running time bound of our algorithm is greater than 2, it still outperforms Gurevich and Shelah's O * (4 n n log n ) polynomial-space exact algorithm for the TSP in general graphs (SIAM Journal of Computation, Vol.16, No.3, pp.486-502, 1987). In the analysis of the running time, we use the measure-and-conquer method, and we develop a set of branching rules which foster the analysis of the running time.

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Section: Branching On Edges Around F5-vertices (Branchingmentioning

confidence: 99%

Section: Fig 17mentioning

confidence: 99%

Section: Switching To Tsp In Degree-4 Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Improved-time Polynomial-space Exact Algorithm for TSP in Degree-5 Graphs

Yunos

Shurbevski

Nagamochi

2017

Journal of Information Processing

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“…Saether and Telle address this problem in their paper [13] by introducing a new parameter, split-matching-width, which lies between treewidth and clique-width in terms of generality. They show that even though graphs of restricted split-matching-width might be dense, solving problems such as Hamiltonian Cycle can be done in FPT time.Recently, it was shown that Hamiltonian Cycle parameterized by treewidth is in EPT [1,6], meaning it can be solved in n O(1) 2 O(k) -time. In this paper, using tools from [6], we show that also parameterized by split-matching-width Hamiltonian Cycle is EPT.…”

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confidence: 99%

Solving Hamiltonian Cycle by an EPT algorithm for a non-sparse parameter

Sæther

2017

Discrete Applied Mathematics

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Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter cliquewidth, Hamiltonian Cycle becomes W[1]-hard, as shown by Fomin et al. [5]. Saether and Telle address this problem in their paper [13] by introducing a new parameter, split-matching-width, which lies between treewidth and clique-width in terms of generality. They show that even though graphs of restricted split-matching-width might be dense, solving problems such as Hamiltonian Cycle can be done in FPT time.Recently, it was shown that Hamiltonian Cycle parameterized by treewidth is in EPT [1,6], meaning it can be solved in n O(1) 2 O(k) -time. In this paper, using tools from [6], we show that also parameterized by split-matching-width Hamiltonian Cycle is EPT. To the best of our knowledge, this is the first EPT algorithm for any "globally constrained" graph problem parameterized by a non-trivial and non-sparse structural parameter. To accomplish this, we also give an algorithm constructing a branch decomposition approximating the minimum split-matching-width to within a constant factor. Combined, these results show that the algorithms in [13] for Edge Dominating Set, Chromatic Number and Max Cut all can be improved. We also show that for Hamiltonian Cycle and Max Cut the resulting algorithms are asymptotically optimal under the Exponential Time Hypothesis.

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Parameterized algorithms and data reduction for the short secluded s‐t‐path problem

2019

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Given a graph G = (V, E), two vertices s, t ∈ V, and two integers k, , the SHORT SECLUDED PATH problem is to find a simple s-t-path with at most k vertices and neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge number. In particular, we completely settle the question of the existence of problem kernels with size polynomial in these parameters and their combinations with k and . We also obtain a 2 O(tw) ⋅ 2 ⋅ n-time algorithm for n-vertex graphs of treewidth tw, which yields subexponential-time algorithms in several graph classes. ]. This version contains full proof details, new kernelization results with respect to the feedback vertex number as parameter, and the algorithm for graphs of bounded treewidth has been generalized to a more general problem variant and accelerated. Networks. 2020;75:34-63. wileyonlinelibrary.com/journal/net © 2019 Wiley Periodicals, Inc. 34 VAN BEVERN ET AL. 35 TABLE 1 Overview of our results par. Positive results Negative results vc Size vc O(r) -kernel in K r,r -subgraph-free graphs (Theorem 3.8) No polynomial kernel and WK[1]-hard w.r.t. vc (Theorem 3.1) fes Size poly(fes)-kernel (Theorem 5.15) fvs O(fvs ⋅ (k + ) 2 )-vertex kernel (Theorem 5.4) No kernel with size poly(fvs + ) (Theorem 5.20) tw 2 O(tw) ⋅ 2 ⋅ n-time algorithm (Theorem 4.2) No kernel with size poly(tw + k + ) even in planar graphs with const. Δ (Theorem 4.14)Herein, n, tw, vc, fes, fvs, and Δ denote the number of vertices, treewidth, vertex cover number, feedback edge number, feedback vertex number, and maximum degree of the input graph, respectively. FIGURE 1Overview on the existence of polynomial kernelization. Gray: no polynomial-size kernel unless coNP ⊆ NP/poly. White: polynomial-size kernel exists. An arrow from parameter p to p ′ means that the value of p can be upper-bounded by a polynomial in p ′ [25]. Thus, hardness results for p ′ also hold for p and polynomial-size kernels for p also hold for p ′ call a problem fixed-parameter tractable if it can be solved in f (k) ⋅ n O(1) time on inputs of length n and some function f depending only on some parameter k. In contrast to an algorithm that merely runs in polynomial time for fixed k, say, in O(n k ) time, which is intractable even for small values of k, fixed-parameter algorithms can solve NP-hard problems quickly if k is small.Provably effective polynomial-time data reduction. Parameterized complexity theory also provides a framework for data reduction with performance guarantees-problem kernelization [16,21,27,47].Kernelization allows for provably effective polynomial-time data reduction. Note that a result of the form "our polynomial-time data reduction algorithm reduces the input size by at least one bit, preserving optimality of solutions" is impossible for NP-hard problems unless P = NP. In contrast, a kernelization algorithm reduces a problem instance into an equivalent one (the problem kernel) whose size depends o...

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Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth

Cited by 189 publications

References 32 publications

An Improved-time Polynomial-space Exact Algorithm for TSP in Degree-5 Graphs

An Improved-time Polynomial-space Exact Algorithm for TSP in Degree-5 Graphs

Solving Hamiltonian Cycle by an EPT algorithm for a non-sparse parameter

Parameterized algorithms and data reduction for the short secluded s‐t‐path problem

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