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Random Instances of Problems in NP – Algorithms and Statistical Physics

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Algorithms, Probability, Networks, and Games

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9295))

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Abstract

One of the most intriguing discoveries made by Erdös and Rényi in the course of their investigating random graphs is the so-called phase transition phenomenona, like the sudden emergence of the giant component. Since then, this kind of phenomena have been observed in many, diverse, areas of combinatorics and discrete mathematics in general. Typically, the notion of phase transition in combinatorics is related to a sudden change in the structural properties of a combinatorial construction, e.g. a (hyper)graph, arithmetic progressions e.t.c. However, it seems that the study of phase transitions goes much further. There is an empirical evidence that certain phase transition phenomena play a prominent role in the performance of algorithms for a lot of natural computational problems. That is, phase transitions are related to the, somehow elusive, notion of computational intractability. The last fifteen-twenty years, there has been serious attempts to put this relation on a mathematically rigorous basis. Our aim is to highlight some of the most central problems that arise in this endeavor.

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Notes

  1. 1.

    As local greedy algorithms.

  2. 2.

    The interested reader can find rapid mixing bound fro the hard-core model (weighted independent sets), too.

  3. 3.

    The paper in [31] is for G(n, p) model for \(p=d/n\), the result for G(n, m) follow by using standard arguments.

  4. 4.

    There are cases where the dynamics remains ergodic beyond non-reconstruction, e.g. hard-core model. In these cases the non-ergodicity is substituted by “low conductance”, which implies slow mixing.

  5. 5.

    Since this search usually gets stuck in a local but not a global optimum, it is customary to carry out the process several times, starting from different configurations, and save the best result.

  6. 6.

    Independent set of a graph is any subset of its vertices which do not span any edge with each other.

  7. 7.

    Somehow the problem of computing marginals turns out to be easier than sampling.

  8. 8.

    The algorithm in [30] is for the related G(n, p) where \(p=d/n\). result for G(n, m) follows by just using standard arguments.

  9. 9.

    To be more precise the colour remains asymptotically random.

  10. 10.

    This justifies the name message passing algorithm.

References

  1. Achlioptas, D., Coja-Oghlan, A.: Algorithmic Barriers from Phase Transitions. In: Proceedings of 49th IEEE Symposium on Foundations of Computer Science, FOCS (2008)

    Google Scholar 

  2. Achlioptas, D., Coja-Oghlan, A., Ricci-Tersenghi, F.: On the solution-space geometry of random constraint satisfaction problems. Random Struct. Algorithms 38(3), 251–268 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Achlioptas, D., Friedgut, E.: A sharp threshold for \(k\)-colorability. Random Struct. Algorithms 14(1), 63–70 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Achlioptas, D., Moore, C.: Random \(k\)-SAT: two moments suffice to cross a sharp threshold. SIAM J. Comput. 36(3), 740–762 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Achlioptas, D., Naor, A.: The two possible values of the chromatic number of a random graph. Ann. Math. 162(3), 1333–1349 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Achlioptas, D., Peres, Y.: The threshold for random \(k\)-SAT is \(2^k \log 2 - O(k)\). J. AMS 17, 947–973 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Alon, N., Krivelevich, M., Sudakov, B.: Finding a large hidden clique in a random graph. In: Proceedings of 9th ACM-SIAM Symposium on Discrete Algorithms, SODA 1998 (1998)

    Google Scholar 

  8. Bapst, V., Coja-Oghlan, A., Hetterich, S., Rassmann, F., Vilenchik, D.: The condensation phase transition in random graph coloring. In: proceedings of APPROX-RANDOM 2014, pp. 449–464 (2014)

    Google Scholar 

  9. van den Berg, J., Maes, C.: Disagreement percolation in the study of Markov fields. Ann. Probab. 22, 749–763 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bhatnagar, N., Sly, A., Tetali, P.: Decay of Correlations for the Hardcore Model on the \(d\)-regular Random Graph. http://arxiv.org/abs/1405.6160

  11. Braunstein, A., Mézard, A., Zecchina, R.: Survey propagation: an algorithm for satisfiability. Random Struct. Algorithms 27, 201–226 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brightwell, G., Winkler, P.: A second threshold for the hard-core model on a Bethe lattice. Random Struct. Algorithms 24, 303–314 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in markov chains. In: proceedings of 38th FOCS, pp 223–231 (1997)

    Google Scholar 

  14. Coja-Oghlan, A.: A Better Algorithm for Random \(k\)-SAT. In: proceedings of ICALP (1) 2009: 292–303. SIAM J. Comput. 39(7) 2823–2864 (2010)

    Google Scholar 

  15. Coja-Oghlan, A.: The asymptotic \(k\)-SAT threshold. In: proceedings of STOC 2014: 804–813 (2014)

    Google Scholar 

  16. Coja-Oghlan, A., Efthymiou, C.: On independent sets in Random Graphs. In: Proceedings of 22nd ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp 136–144 (2011)

    Google Scholar 

  17. Coja-Oghlan, A., Efthymiou, C., Hetterich, S.: On the chromatic number of random regular graphs. http://arxiv.org/abs/1308.4287

  18. Coja-Oghlan, A., Efthymiou, C., Jaafari, N.: Local convergence of random graph colorings. http://arxiv.org/abs/1501.06301

  19. Coja-Oghlan, A., Panagiotou, K.: Going after the k-SAT threshold. In: procedings of STOC 2013: 705–714 (2013)

    Google Scholar 

  20. Coja-Oghlan, A., Panagiotou, K.: Catching the k-NAESAT threshold. In: proceedings of STOC 2012: 899–908 (2012)

    Google Scholar 

  21. Coja-Oghlan, A., Vilenchik, D.: Chasing the k-colorability threshold. In: Proceedings of 54th IEEE Symposium on Foundations of Computer Science, FOCS 2013, pp 380–389 (2013)

    Google Scholar 

  22. Chvátal, V., Reed, B.: Mick gets some (the odds are on his side). In: Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, pp. 620–627 (1992)

    Google Scholar 

  23. Dani, V., Moore, C.: Independent sets in random graphs from the weighted second moment method. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) RANDOM 2011 and APPROX 2011. LNCS, vol. 6845, pp. 472–482. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  24. Ding, J., Sly, A., Sun, N.: Maximum independent sets on random regular graphs. http://arxiv.org/abs/1310.4787

  25. Ding, J., Sly, A., Sun, N.: Satisfiability threshold for random regular NAE-SAT. In: proceendings of STOC 2014: 814–822 (2014)

    Google Scholar 

  26. Ding, J., Sly, A., Sun, N.: Proof of the satisfiability conjecture for large \(k\). To appear in STOC 2015 (2015)

    Google Scholar 

  27. Dyer, M., Flaxman, A., Frieze, A.M., Vigoda, E.: Random colouring sparse random graphs with fewer colours than the maximum degree. Random Struct. Algorithms 29, 450–465 (2006)

    Article  MATH  Google Scholar 

  28. Dyer, M.E., Frieze, A.M.: The solution of some random np-hard problems in polynomial expected time. J. Algorithms 10(4), 451–489 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dyer, M., Frieze, A.M., Hayes, A., Vigoda, E.: Randomly colouring constant degree graphs. In proceedings of 45th FOCS, pp 582–589 (2004)

    Google Scholar 

  30. Efthymiou, C.: A simple algorithm for random colouring \(G(n, d/n)\) using \((2 + \epsilon )d\) colours. In: Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 (2012)

    Google Scholar 

  31. Efthymiou, C.: MCMC sampling colourings and independent sets of G(n, d/n) near the uniqueness threshold. In: proceedings of Symposium on Discrete Algorithms, SODA (2014)

    Google Scholar 

  32. Efthymiou, C.: Switching colouring of G(n,d/n) for sampling up to gibbs uniqueness threshold. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 371–381. Springer, Heidelberg (2014)

    Google Scholar 

  33. Efthymiou, C.: Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees. CoRR abs/1406.3617 (2014)

    Google Scholar 

  34. Freeman, W.T., Paztor, E.C., Carmichael, O.T.: Learning low-level vision. Int. J. Comput. Vis. 40, 25–47 (2000)

    Article  MATH  Google Scholar 

  35. Friedgut, E.: Necessary and sufficient conditions for sharp thresholds of graph properties, and the k-SAT problem. J. Amer. Math. Soc. 12, 1017–1054 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Frieze, A.M.: On the independence number of random graphs. Discrete Math. 81(183), 171–175 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. Frieze, A.M., Vera, J.: On randomly colouring locally sparse graphs. Discrete Math. & Theor. Comput. Sci. 8(1), 121–128 (2006)

    MathSciNet  MATH  Google Scholar 

  38. Georgii, H.O.: Gibbs Measures and Phase Transitions, de Gruyter Stud. Math. 9, de Gruyter, Berlin (1988)

    Google Scholar 

  39. Goldberg, L.A., Martin, R.A., Paterson, M.: Strong spatial mixing with fewer colors for lattice graphs. SIAM J. Comput. 35(2), 486–517 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Grimmett, G.R., McDiarmid, C.J.H.: On colouring random graphs. Math. Proc. Camb. Phil. Soc. 77(02), 313–332 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  41. Hayes, T., Vera, J., Vigoda, E.: Randomly coloring planar graphs with fewer colors than the maximum degree. In: proceedings of 39th STOC, pp 450–458 (2007)

    Google Scholar 

  42. Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18(6), 1138–1162 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  43. Jerrum, M.R.: Large cliques elude the Metropolis process. Random Struct. Algorithms 3, 347–359 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  44. Jerrum, M.R., Sinclair, A.: The Markov chain Monte Carlo method: an approach to approximate counting and integration. In: Approximation Algorithms for NP-hard Problems, (Dorit Hochbaum, ed.), PWS (1996)

    Google Scholar 

  45. Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. ACM 51(4), 671–697 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. Karp, R., Sipser, M.: Maximum matchings in sparse random graphs. In: proceedings of FOCS 1981, pp. 364 375 (1981)

    Google Scholar 

  47. Kelly, F.P.: Loss networks. Ann. Appl. Probab. 1(3), 319–378 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimisation by simulated annealing. Science 220, 671–680 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kirousis, L.M., Kranakis, E., Krizanc, D., Stamatiou, Y.C.: Approximating the unsatisfiability threshold of random formulas. Random Struct. Algor. 12(3), 253–269 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  50. Krzakala, F., Montanari, A., Ricci-Tersenghi, F., Semerjianc, G., Zdeborova, L.: Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. 104, 10318–10323 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  51. Kschischang, F., Frey, B., Loeliger, H.A.: Factor graphs and the sum-product algorithm. IEEE Trans. Inf. Theory 47, 498519 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  52. Levin, D., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. American Mathematical Society (2008)

    Google Scholar 

  53. Lovász, L., Vempala, S.: Simulated annealing in convex bodies and an \(O*(n^4)\) volume algorithm. In: Proceedings of the 44th IEEE Foundations of Computer Science (FOCS 2003) (2003). Also in JCSS (FOCS 2003 special issue)

    Google Scholar 

  54. Lucier, B., Molloy, M., Peres, Y.: The Glauber Dynamics for Colourings of Bounded Degree Trees. In: proceedings of RANDOM 2009, pp 631–645 (2009)

    Google Scholar 

  55. Martinelli, F., Sinclair, A., Weitz, D.: Fast mixing for independent sets, colorings and other models on trees. In: proceedings of 15th SODA, pp 456–465 (2004)

    Google Scholar 

  56. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)

    Article  Google Scholar 

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

    Article  Google Scholar 

  58. Molloy, M.: The Glauber dynamics on the colourings of a graph with large girth and maximum degree. SIAM J. Comput. 33, 721–737 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  59. Montanari, A., Restrepo, R., Tetali, P.: Reconstruction and clustering in random constraint satisfaction problems. SIAM J. Discrete Math. 25(2), 771–808 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  60. Montanari, A., Shah, D.: Counting good truth assignments of random \(k\)-SAT formulae. In: proceedings of the 18th Annual ACM-SIAM, SODA 2007, pp 1255–1264 (2007)

    Google Scholar 

  61. Mossel, E., Sly, A.: Gibbs rapidly samples colorings of \(G_{n, d/n}\). J. Probab. Theory Relat. fields 148, 1–2 (2010)

    Article  MATH  Google Scholar 

  62. Pearl, J.: Probabilistic reasoning in intelligent systems: Networks of plausible inference. Morgan-Kaufmann, Palo Alto (1988)

    MATH  Google Scholar 

  63. Restrepo, R., Stefankovic, D., Vera, J.C., Vigoda, E., Yang, L.: Phase transition for glauber dynamics for independent sets on regular trees. In: proceedings SODA 2011, pp 945–956 (2011)

    Google Scholar 

  64. Richardson, T., Urbanke, R.: The capacity of low-density parity check codes under message passing deconding. IEEE Trans. Inf. Theory 47, 599–618 (2001)

    Article  MATH  Google Scholar 

  65. Schöning, U.: A probabilistic algorithm for \(k\)-SAT and constraint satisfaction problems. In: proceedings of Symposium on Foundations of Computer Science, FOCS 1999, pp 410–419 (1999)

    Google Scholar 

  66. Tetali, P., Vera, J.C., Vigoda, E., Yang, L.: Phase transition for the mixing time of the glauber dynamics for coloring regular trees. In: proceedings of SODA 2010, pp 1646–1656 (2010)

    Google Scholar 

  67. Vigoda, E.: Improved bounds for sampling colorings. J. Math. Phys. 41(3), 1555–1569 (2000). A preliminary version appears in FOCS 1999

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

Most of the material in the introduction come from discussions with Amin Coja-Oghlan. For this reason I would like to thank him.

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  1. Georgia Institute of Technology, College of Computing, 266 Ferst Dr. Atlanta, 30332, Atlanta, GA, USA

    Charilaos Efthymiou

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Correspondence to Charilaos Efthymiou .

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  1. University of Patras, Patras, Greece

    Christos Zaroliagis

  2. Technological Educational Institute of Athens, Athens, Greece

    Grammati Pantziou

  3. University of Ioannina, Ioannina, Greece

    Spyros Kontogiannis

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Efthymiou, C. (2015). Random Instances of Problems in NP – Algorithms and Statistical Physics. In: Zaroliagis, C., Pantziou, G., Kontogiannis, S. (eds) Algorithms, Probability, Networks, and Games. Lecture Notes in Computer Science(), vol 9295. Springer, Cham. https://doi.org/10.1007/978-3-319-24024-4_13

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