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Phase Transition for Maximum Not-All-Equal Satisfiability

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Frontiers in Algorithmics (FAW 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10336))

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Abstract

Phase transition is a dramatic transition from one state to another state when a particular parameter varies. This paper aims to study the phase transition of maximum not-all-equal satisfiability problem (Max NAE SAT), an optimization of not-all-equal satisfiability problem (NAE SAT). Given a conjunctive normal formula (CNF) F with n variables and rn k-clauses (the clause exactly contains k literals), we use first-moment method to obtain an upper bound for f(n, rn) the expectation of the maximum number of NAE-satisfied clauses of random Max NAE k-SAT. In addition, we also consider the phase transition of decision version of random Max NAE k-SAT—bounded not-all-equal satisfiability problem (NAE k-SAT(b)). We demonstrate that there is a phase transition point \(r_{k,b}\) separating the region where almost all NAE k-SAT(b) instances can be solved from the region where almost all NAE k-SAT(b) instances can’t be solved. Furthermore, we analyze the upper bound and lower bound for \(r_{k,b}\).

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Acknowledgement

The authors of this paper wish to extend their sincere gratitude to all anonymous reviewers for their efforts. This work was supported in part by NSFC (under Grant Nos.61503074, 61403076, 61402070, and 61403077), the Natural Science Foundation for Youths of JiLin Province (20160520104JH) and (NCET-13-0724).

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Authors and Affiliations

  1. College of Computer Science and Information Technology, Northeast Normal University, Changchun, 130117, China

    Junping Zhou, Shuli Hu & Minghao Yin

  2. College of Information Science and Technology, Dalian Maritime University, Dalian, 116026, China

    Tingting Zou

Authors
  1. Junping Zhou
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  2. Shuli Hu
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  3. Tingting Zou
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  4. Minghao Yin
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Corresponding author

Correspondence to Minghao Yin .

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Editors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, China

    Mingyu Xiao

  2. University of Bergen, Bergen, Norway

    Frances Rosamond

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Zhou, J., Hu, S., Zou, T., Yin, M. (2017). Phase Transition for Maximum Not-All-Equal Satisfiability. In: Xiao, M., Rosamond, F. (eds) Frontiers in Algorithmics. FAW 2017. Lecture Notes in Computer Science(), vol 10336. Springer, Cham. https://doi.org/10.1007/978-3-319-59605-1_24

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  • DOI: https://doi.org/10.1007/978-3-319-59605-1_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59604-4

  • Online ISBN: 978-3-319-59605-1

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