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On the Parameterized Parallel Complexity and the Vertex Cover Problem

  • Faisal N. Abu-Khzam
  • Shouwei LiEmail author
  • Christine Markarian
  • Friedhelm Meyer auf der Heide
  • Pavel Podlipyan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10043)

Abstract

Efficiently parallelizable parameterized problems have been classified as being either in the class FPP (fixed-parameter parallelizable) or the class PNC (parameterized analog of NC), which contains FPP as a subclass. In this paper, we propose a more restrictive class of parallelizable parameterized problems called fixed-parameter parallel-tractable (FPPT). For a problem to be in FPPT, it should possess an efficient parallel algorithm not only from a theoretical standpoint but in practice as well. The primary distinction between FPPT and FPP is the parallel processor utilization, which is bounded by a polynomial function in the case of FPPT. We initiate the study of FPPT with the well-known k-vertex cover problem. In particular, we present a parallel algorithm that outperforms the best known parallel algorithm for this problem: using \(\mathcal {O}(m)\) instead of \(\mathcal {O}(n^2)\) parallel processors, the running time improves from \(4\log n + \mathcal {O}(k^k)\) to \(\mathcal {O}(k\cdot \log ^3 n)\), where m is the number of edges, n is the number of vertices of the input graph, and k is an upper bound of the size of the sought vertex cover. We also note that a few P-complete problems fall into FPPT including the monotone circuit value problem (MCV) when the underlying graphs are bounded by a constant Euler genus.

Keywords

Vertex Cover Input Graph Processor Utilization Maximum Match Parallel Processor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

We wish to thank the anonymous referees for their valuable comments to improve the quality and presentation of this paper.

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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Faisal N. Abu-Khzam
    • 1
    • 4
  • Shouwei Li
    • 2
    Email author
  • Christine Markarian
    • 3
  • Friedhelm Meyer auf der Heide
    • 2
  • Pavel Podlipyan
    • 2
  1. 1.Department of Computer Science and MathematicsLebanese American UniversityBeirutLebanon
  2. 2.Heinz Nixdorf Institute & Department of Computer SciencePaderborn UniversityPaderbornGermany
  3. 3.Department of Mathematical SciencesHaigazian UniversityBeirutLebanon
  4. 4.School of Engineering and Information TechnologyCharles Darwin UniversityDarwinAustralia

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