Speeding Up Dynamic Programming with Representative Sets: An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions

Fafianie, Stefan; Bodlaender, Hans L.; Nederlof, Jesper

doi:10.1007/s00453-014-9934-0

Cited by 9 publications

(10 citation statements)

References 26 publications

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“…However, in practice it seems reasonable that sometimes it pays off to apply this computationally expensive step less often; that is, allow the set of partial solutions to grow significantly beyond the theoretical bounds, and once in a while trim it at bulk with a single Gaussian elimination step. This intuition has been supported by the results of Fafianie et al [14] for the case of Steiner Tree.…”

Section: Fine-tuning the Frequency Of Gaussian Eliminationsupporting

confidence: 54%

Finding Hamiltonian Cycle in Graphs of Bounded Treewidth

Ziobro

Pilipczuk

2019

ACM J. Exp. Algorithmics

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The notion of treewidth, introduced by Robertson and Seymour in their seminal Graph Minors series, turned out to have tremendous impact on graph algorithmics. Many hard computational problems on graphs turn out to be efficiently solvable in graphs of bounded treewidth: graphs that can be sweeped with separators of bounded size. These efficient algorithms usually follow the dynamic programming paradigm.In the recent years, we have seen a rapid and quite unexpected development of involved techniques for solving various computational problems in graphs of bounded treewidth. One of the most surprising directions is the development of algorithms for connectivity problems that have only single-exponential dependency (i.e., 2 O(t) ) on the treewidth in the running time bound, as opposed to slightly superexponential (i.e., 2 O(t log t) ) stemming from more naive approaches. In this work, we perform a thorough experimental evaluation of these approaches in the context of one of the most classic connectivity problem, namely Hamiltonian Cycle.

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Section: Fine-tuning the Frequency Of Gaussian Eliminationsupporting

confidence: 54%

Finding Hamiltonian Cycle in Graphs of Bounded Treewidth

Ziobro

Pilipczuk

2019

ACM J. Exp. Algorithmics

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“…Then any set of partial colorings S of H can be reduced to a subset S of size 2 k , with the guarantee that if some coloring in S could be extended to a proper coloring of G, then this still holds for S . The reduction can be achieved by an application of Gaussian elimination, which has experimentally been shown to work well for speeding up dynamic programming for other problems [16]. We therefore believe the table-reduction steps presented here may also be useful when solving graph coloring over tree-or path decompositions, and can be applied whenever processing a separator consisting of few edges.…”

Section: Resultsmentioning

confidence: 93%

Computing the chromatic number using graph decompositions via matrix rank

Jansen

Nederlof

2019

Theoretical Computer Science

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Computing the smallest number q such that the vertices of a given graph can be properly qcolored is one of the oldest and most fundamental problems in combinatorial optimization. The q-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. For example, algorithms are known to solve the problem on graphs of treewidth tw in time O * (q tw ), while a running time of O * ((q−ε) tw ) is impossible assuming the Strong Exponential Time Hypothesis (SETH). While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying q-Coloring parameterized by cutwidth, and parameterized by pathwidth in boundeddegree graphs. Our research uncovers interesting new ways to exploit small edge separators.We present two algorithms for q-Coloring parameterized by cutwidth ctw: a deterministic one that runs in time O * (2 ω·ctw ), where ω is the matrix multiplication constant, and a randomized one with runtime O * (2 ctw ). In sharp contrast to earlier work, the running time is independent of q. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in O * ((2−ε) ctw ) time assuming SETH. Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an O * (( d/2 + 1) pw ) time randomized algorithm for q-Coloring on graphs of pathwidth pw and maximum degree d. Such a runtime was first obtained by Björklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no O * (( d/2 +1−ε) pw )-time algorithm exists assuming SETH. ACM Subject ClassificationMathematics of computing → Graph algorithms; Theory of computation → Parameterized complexity and exact algorithms

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“…Indeed, due to the fact that algorithms based on tree decompositions are an area of intensive research, there also arise performance improvements for specialized algorithms, as studied by Fafianie et al (2015) who showed that the efficiency of solving the Steiner Tree problem can be significantly improved by combining tree decompositions with methods from linear algebra. It may be an interesting task in future work to investigate the impact of tree decomposition selection also in this context.…”

Section: Resultsmentioning

confidence: 99%

“…treewidth, the general runtime of such algorithms for an instance of size n is f (k) · n O(1) , where f is an arbitrary function over width k of the tree decomposition used. In fact, this approach has been used for several applications, including inference in probabilistic networks (Lauritzen & Spiegelhalter, 1988), frequency assignment (Koster, van Hoesel, & Kolen, 1999), computational biology (Xu, Jiao, & Berger, 2005), logic programming (Morak, Musliu, Pichler, Rümmele, & Woltran, 2012) and the Steiner Tree problem (Fafianie, Bodlaender, & Nederlof, 2015).…”

Section: Introductionmentioning

confidence: 99%

Improving the Efficiency of Dynamic Programming on Tree Decompositions via Machine Learning

Abseher¹,

Musliu²,

Woltran³

2017

jair

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Dynamic Programming (DP) over tree decompositions is a well-established method to solve problems -that are in general NP-hard -efficiently for instances of small treewidth. Experience shows that (i) heuristically computing a tree decomposition has negligible runtime compared to the DP step; and (ii) DP algorithms exhibit a high variance in runtime when using different tree decompositions; in fact, given an instance of the problem at hand, even decompositions of the same width might yield extremely diverging runtimes. We thus propose here a novel and general method that is based on selection of the best decomposition from an available pool of heuristically generated ones. For this purpose, we require machine learning techniques that provide automated selection based on features of the decomposition rather than on the actual problem instance. Thus, one main contribution of this work is to propose novel features for tree decompositions. Moreover, we report on extensive experiments in different problem domains which show a significant speedup when choosing the tree decomposition according to this concept over simply using an arbitrary one of the same width.

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Speeding Up Dynamic Programming with Representative Sets: An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions

Cited by 9 publications

References 26 publications

Finding Hamiltonian Cycle in Graphs of Bounded Treewidth

Finding Hamiltonian Cycle in Graphs of Bounded Treewidth

Computing the chromatic number using graph decompositions via matrix rank

Improving the Efficiency of Dynamic Programming on Tree Decompositions via Machine Learning

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