Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

Jansen, Bart M. P.; Pilipczuk, Marcin; Wrochna, Marcin

doi:10.1007/s00453-019-00614-4

Cited by 7 publications

(4 citation statements)

References 30 publications

(57 reference statements)

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“…Proof of Theorem A.1. We use the result of Jansen, Pilipczuk, and Wrochna [20] who gave a polynomial Turing kernel for the Longest Path problem in H-topologicalminor-free for every fixed H. That is, they showed how to solve Longest Path in H-topological-minor-free graphs by a polynomial-time algorithm that has access to an oracle solving the problem on H-topological-minor-free graphs which have \ell \scrO (1) vertices.…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

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

Nederlof

Pilipczuk

Swennenhuis

et al. 2023

SIAM J. Discrete Math.

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Section: Conclusion and Further Researchmentioning

confidence: 99%

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

Nederlof

Pilipczuk

Swennenhuis

et al. 2023

SIAM J. Discrete Math.

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“…Proof. We first recall that Jansen et al [19] gave a polynomial Turing kernel for the Longest Path problem in H-topological-minor-free, for every fixed H. That is, they showed how to solve Longest Path in H-topological-minor-free graphs by a polynomial-time algorithm that has access to an oracle solving the problem on H-topological-minor-free graphs which have O(1) vertices.…”

Section: B Problems Definitionsmentioning

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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“…More than ten years ago Binkele-Raible et al [2] found the first problem, namely leaf out-tree(k), which has a polynomial Turing compression but does not admit a polynomial compression unless coNP ⊆ NP/poly. However, by now, people only found limited problems of this kind [1,3,6,14,15,16,17,19].…”

Section: Introductionmentioning

confidence: 99%

Polynomial Turing Compressions for Some Graph Problems Parameterized by Modular-Width

Luo¹

2022

Preprint

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In this paper we investigate the parameterized complexity for NP-hard graph problems parameterized by a structural parameter modular-width. We develop a recipe that is able to simplify the process of obtaining polynomial Turing compressions for a class of graph problems parameterized by modular-width. Moreover, we prove that several problems, which include chromatic number, independent set, hamiltonian cycle, etc. have polynomial Turing compressions parameterized by modular-width. In addition, under the assumption that P = NP, we provide tight kernels for a few problems such as steiner tree parameterized by modular-width. Meanwhile, we demonstrate that some problems, which includes dominating set, odd cycle transversal, connected vertex cover, etc. are fixed-parameter tractable parameterized by modular-width.

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Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

Cited by 7 publications

References 30 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

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

Polynomial Turing Compressions for Some Graph Problems Parameterized by Modular-Width

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