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Fractional Edge Cover Number of Model RB

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Frontiers in Algorithmics (FAW 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9711))

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Abstract

Model RB is a random constraint satisfaction problem with a growing domain size, which exhibits exact phase transition phenomena. Many hard instances with planted solutions can be generated via Model RB, to be used as benchmarks for algorithmic competitions and researches. In the past, some structural parameters of constraint hypergraphs are analyzed to show hardness of Model RB, such as hinge width, decycling number, treewidth, and hypertree width. In this paper, one more structural parameter of constraint hypergraphs of Model RB, namely the fractional edge cover number, is analyzed. We show upper and lower bounds on the fractional edge cover number of Model RB. In particular, the fractional edge cover number of Model RB is shown to be asymptotically linear in the number of variables, like hinge width, decycling number, treewidth and hypertree width. These results together provide further evidences on the hardness of Model RB.

Partially supported by Natural Science Foundation of China (Grant Nos. 61370052 and 61370156).

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References

  1. Adler, I., Gottlob, G., Grohe, M.: Hypertree width and related hypergraph invariants. Eur. J. Comb. 28, 2167–2181 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheeseman, P., Kanefsky, R., Taylor, W.: Where the really hard problems are. In: Proceedings of IJCAI 1991, pp. 163–169 (1991)

    Google Scholar 

  3. Cook, S.A., Mitchell, D.G.: Finding hard instances of the satisfiability problem: a survey. DIMACS Ser. 35, 1–17 (1997)

    MathSciNet  MATH  Google Scholar 

  4. Dubois, O., Boufkhad, Y., Mandler, J.: Typical random 3-SAT formulae and the satisfiability threshold. In: Proceedings of SODA 2000, pp. 126–127 (2000)

    Google Scholar 

  5. Fan, Y., Shen, J.: On the phase transitions of random \(k\)-constraint satisfaction problems. Artif. Intell. 175, 914–927 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan, Y., Shen, J., Xu, K.: A general model and thresholds for random constraint satisfaction problems. Artif. Intell. 193, 1–17 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gao, Y.: Phase transition of tractability in constraint satisfaction and Bayesian network inference. In: Proceedings of UAI, pp. 265–271 (2003)

    Google Scholar 

  8. Gao, Y.: On the threshold of having a linear treewidth in random graphs. In: Chen, D.Z., Lee, D.T. (eds.) COCOON 2006. LNCS, vol. 4112, pp. 226–234. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  9. Gao, Y.: Treewidth of Erdos-Renyi random graphs, random intersection graphs, and scale-free random graphs. Discrete Appl. Math. 160(4–5), 566–578 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gao, Y., Culberson, J.: Consistency and random constraint satisfaction problems. J. Artif. Intell. Res. 28, 517–557 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Grohe, M., Marx, D.: Constraint solving via fractional edge covers. ACM Trans. Alg. 11(1), Article 4 (2014)

    Google Scholar 

  12. Jiang, W., Liu, T., Ren, T., Xu, K.: Two hardness results on feedback vertex sets. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 233–243. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  13. Kloks, T.: Treewidth: Computations and Approximations, pp. 18–55. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  14. Lee, C., Lee, J., Oum, S.: Rank-width of random graphs. J. Graph. Theor. 70(3), 339–347 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, T., Lin, X., Wang, C., Su, K., Xu, K.: Large hinge width on sparse random hypergraphs. In: Proceedings of IJCAI, pp. 611–616 (2011)

    Google Scholar 

  16. Liu, T., Wang, C., Xu, K.: Large hypertree width for sparse random hypergraphs. J. Comb. Optim. 29, 531–540 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Marx, D.: Tractable hypergraph properties for constraint satisfaction and conjunctive queries. J. ACM 60(6), 42 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mezard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297(5582), 812–815 (2002)

    Article  Google Scholar 

  19. Mitchell, D.G., Selman, B., Levesque, H.J.: Hard and easy distributions of sat problems. In: Proceedings of AAAI 1992, pp. 459–465 (1992)

    Google Scholar 

  20. Selman, B., Mitchell, D.G., Levesque, H.J.: Generating hard satisfiability problems. Artif. Intell. 81, 17–29 (1996)

    Article  MathSciNet  Google Scholar 

  21. Rucinski, A., Janson, S., Luczak, T.: Random Graphs. Wiley, New York (2000)

    MATH  Google Scholar 

  22. Wang, C., Liu, T., Cui, P., Xu, K.: A note on treewidth in random graphs. In: Wang, W., Zhu, X., Du, D.-Z. (eds.) COCOA 2011. LNCS, vol. 6831, pp. 491–499. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  23. Xu, K., Li, W.: Exact phase transitions in random constraint satisfaction problems. J. Artif. Intell. Res. 12, 93–103 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Xu, K.: BHOSLIB: Benchmarks with Hidden Optimum Solutions for Graph Problems. http://www.nlsde.buaa.edu.cn/kexu/benchmarks/graph-benchmarks.htm

  25. Xu, K., Li, W.: Many hard examples in exact phase transitions. Theor. Comput. Sci. 355, 291–302 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xu, K., Boussemart, F., Hemery, F., Lecoutre, C.: Random constraint satisfaction: easy generation of hard (satisfiable) instances. Artif. Intell. 171, 514–534 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xu, W., Zhang, P., Liu, T., Gong, F.: The solution space structure of random constraint satisfaction problems with growing domains. J. Stat. Mech. Theor. Exp. 2015, P12006 (2015)

    Article  Google Scholar 

  28. Zhao, C., Zhang, P., Zheng, Z., Xu, K.: Analytical and belief-propagation studies of random constraint satisfaction problems with growing domains. Phys. Rev. E 85, 016106 (2012)

    Article  Google Scholar 

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Acknowledgments

We thank Ms. Yu Song for drafting an earlier version of this paper.

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Authors and Affiliations

  1. Key Laboratory of High Confidence Software Technologies, Ministry of Education, Peking University, Beijing, 100871, China

    Tian Liu

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  1. Tian Liu
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Correspondence to Tian Liu .

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Editors and Affiliations

  1. Shandong University, Jinan, China

    Daming Zhu

  2. MS EC-31, University of Texas at Dallas, Richardson, Texas, USA

    Sergey Bereg

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Liu, T. (2016). Fractional Edge Cover Number of Model RB. In: Zhu, D., Bereg, S. (eds) Frontiers in Algorithmics. FAW 2016. Lecture Notes in Computer Science(), vol 9711. Springer, Cham. https://doi.org/10.1007/978-3-319-39817-4_11

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  • DOI: https://doi.org/10.1007/978-3-319-39817-4_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-39816-7

  • Online ISBN: 978-3-319-39817-4

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