Fast exact algorithms for some connectivity problems parameterized by clique-width

Bergougnoux, Benjamin; Kanté, Mamadou Moustapha

doi:10.1016/j.tcs.2019.02.030

Cited by 19 publications

(20 citation statements)

References 19 publications

(37 reference statements)

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“…Pino et al [26] used Cut&Count and rank-based approach to get single-exponential time algorithms for connectivity problems parametrized by branchwidth. Recently, Cut&Count was also applied in the context of cliquewidth [2], and of Qrankwidth, rankwidth, and MIM-width [3]. All these algorithms have exponential space complexity, as they follow the standard dynamic programming approach.…”

Section: Approximation Of Treedepthmentioning

confidence: 99%

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Nederlof¹,

Pilipczuk²,

Swennenhuis³

et al. 2020

Preprint

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For many algorithmic problems on graphs of treewidth t, a standard dynamic programming approach gives an algorithm with time and space complexity 2 O(t) • n O(1) . It turns out that when one considers a more restrictive parameter treedepth, it is often the case that a variation of this technique can be used to reduce the space complexity to polynomial, while retaining time complexity of the form 2 O(d) • n O(1) , where d is the treedepth. This transfer of methodology is, however, far from being automatic. For instance, for problems with connectivity constraints, standard dynamic programming techniques give algorithms with time and space complexity 2 O(t log t) • n O(1) on graphs of treewidth t, but it is not clear how to convert them into time-efficient polynomial space algorithms for graphs of low treedepth.Cygan et al. (FOCS'11) introduced the Cut&Count technique and showed that a certain class of problems with connectivity constraints can be solved in time and space complexity 2 O(t) • n O(1) . Recently, Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the Cut&Count technique can be also applied in the setting of treedepth, and it gives algorithms with running time 2 O(d) • n O(1) and polynomial space usage. However, a number of important problems eluded such a treatment, with the most prominent examples being Hamiltonian Cycle and Longest Path.In this paper we clarify the situation by showing that Hamiltonian Cycle, Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit 5 d •n O(1) -time and polynomial space algorithms on graphs of treedepth d. The algorithms are randomized Monte Carlo with only false negatives.

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Section: Approximation Of Treedepthmentioning

confidence: 99%

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Nederlof¹,

Pilipczuk²,

Swennenhuis³

et al. 2020

Preprint

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“…For other graph problems where boundedness of clique-width is used to classify their computational complexity on hereditary graph classes, see, for example, [28,44]. We refer to [11,41,79,80,89] for parameterized complexity results on clique-width.…”

Section: Algorithmic Consequencesmentioning

confidence: 99%

“…1. Graph Isomorphism is solvable in polynomial time on G if G is equivalent 11 to a class of (H 1 , H 2 )-free graphs such that one of the following holds:…”

Section: Graph Colouringmentioning

confidence: 99%

Clique-width for hereditary graph classes

Dabrowski¹,

Johnson²,

Paulusma³

2019

Surveys in Combinatorics 2019

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“…In the same paper they reduced this number to 7 and gave the following state-of-the-art summary; recall that K + 1,t and K ++ 1,t are the graphs obtained from K 1,t by subdividing one edge once or twice, respectively. Theorem 5.8 ( [19]) For a class G of graphs defined by two forbidden induced subgraphs, the following holds: 11 to a class of (H 1 , H 2 )-free graphs such that one of the following holds:…”

Section: Graph Colouringmentioning

confidence: 99%

Clique-Width for Hereditary Graph Classes

Dabrowski,

Johnson,

Paulusma

2019

Preprint

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Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class G is bounded by a constant, a wide range of problems that are NP-complete in general can be shown to be polynomial-time solvable on G. For this reason, the boundedness or unboundedness of clique-width has been investigated and determined for many graph classes. We survey these results for hereditary graph classes, which are the graph classes closed under taking induced subgraphs. We then discuss the algorithmic consequences of these results, in particular for the Colouring and Graph Isomorphism problems. We also explain a possible strong connection between results on boundedness of clique-width and on well-quasi-orderability by the induced subgraph relation for hereditary graph classes.

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Fast exact algorithms for some connectivity problems parameterized by clique-width

Cited by 19 publications

References 19 publications

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Clique-width for hereditary graph classes

Clique-Width for Hereditary Graph Classes

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