The Satisfiability Threshold For Random Linear Equations


Let A be a random m × n matrix over the finite field \(\mathbb{F}_q\) with precisely k non-zero entries per row and let \(y\in\mathbb{F}_q^m\) be a random vector chosen independently of A. We identify the threshold m/n up to which the linear system Ax = y has a solution with high probability and analyse the geometry of the set of solutions. In the special case q = 2, known as the random k-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002 for k = 3, Pittel and Sorkin 2016 for k > 3], and the proof technique was subsequently extended to the cases q = 3,4 [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to q > 4. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.

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  1. [1]

    D. Achlioptas, A. Chtcherba, G. Istrate and C. Moore: The phase transition in 1-in-k SAT and NAE 3-SAT, Proc. 12th SODA (2001), 721–722.

    Google Scholar 

  2. [2]

    D. Achlioptas and A. Coja-oghlan: Algorithmic barriers from phase transitions, Proc. 49th FOCS (2008), 793–802.

    Google Scholar 

  3. [3]

    D. Achlioptas and M. Molloy: The solution space geometry of random linear equations, Random Structures and Algorithms46 (2015), 197–231.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    D. Achlioptas and C. Moore: Random fc-SAT: two moments suffice to cross a sharp threshold, SIAM Journal on Computing36 (2006), 740–762.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    D. Achlioptas and A. Naor: The two possible values of the chromatic number of a random graph, Annals of Mathematics162 (2005), 1333–1349.

    MathSciNet  MATH  Google Scholar 

  6. [6]

    D. Achlioptas, A. Naor and Y. Peres: Rigorous location of phase transitions in hard optimization problems, Nature435 (2005), 759–764.

    Google Scholar 

  7. [7]

    M. Alzenman, R. Sims and S. Starr: An extended variational principle for the SK spin-glass model, Phys. Rev. B68 (2003), 214403.

  8. [8]

    V. Bapst and A. Coja-Oghlan: Harnessing the Bethe free energy, Random Structures and Algorithms49 (2016), 694–741.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    E. Ben-Spasson and A. Wlgderson: Short proofs are narrow-resolution made simple, Journal of the ACM48 (2001), 149–169.

    MathSciNet  MATH  Google Scholar 

  10. [10]

    P. Cheeseman, B. Kanefsky and W. Taylor: Where the really hard problems are, Proc. IJCAI (1991), 331–337.

    Google Scholar 

  11. [11]

    V. Chvátal and B. Reed: Mick gets some (the odds are on his side), Proc. 33th FOCS (1992), 620–627.

    Google Scholar 

  12. [12]

    S. Cocco, O. Dubois, J. Mandler and R. Monasson: Rigorous decimation-based construction of ground pure states for spin glass models on random lattices, Phys. Rev. Lett.90 (2003), 047205.

    Google Scholar 

  13. [13]

    A. Coja-oghlan, C. Efthymiou, N. Jaafari, M. Kang and T. Kapetanopou- LOS: Charting the replica symmetric phase, Communications in Mathematical Physics359 (2018), 603–698.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    A. Coja-oghlan, F. Krzakala, W. Perkins and L. Zdeborova: Information-theoretic thresholds from the cavity method, Proc. 48th STOC (2017), 146–157.

    Google Scholar 

  15. [15]

    A. Coja-OGHLAN and K. Panagiotou: The asymptotic k-SAT threshold, Advances in Mathematics288 (2016), 985–1068.

    MathSciNet  MATH  Google Scholar 

  16. [16]

    H. Connamacher and M. Molloy: The satisfiability threshold for a seemingly intractable random constraint satisfaction problem, SIAM J. DISCRETE Mathematics26 (2012), 768–800.

    MathSciNet  MATH  Google Scholar 

  17. [17]

    C. Cooper, A. Frieze and W. Pegden: On the rank of a random binary matrix, Proc. 30th SODA (2019), 946–955.

    Google Scholar 

  18. [18]

    M. Dietzfelbinger, A. Goerdt, M. Mltzenmacher, A. Montanari, R. Pagh and M. Rink: Tight thresholds for cuckoo hashing via XORSAT, Proc. 37th ICALP (2010), 213–225.

    Google Scholar 

  19. [19]

    J. Ding, A. Sly and N. Sun: Satisfiability threshold for random regular NAE-SAT, Communications in Mathematical Physics341 (2016), 435–489.

    MathSciNet  MATH  Google Scholar 

  20. [20]

    J. Ding, A. Sly and N. Sun: Proof of the satisfiability conjecture for large k, Proc. 47th STOC (2015), 59–68.

    Google Scholar 

  21. [21]

    O. Dubois and J. Mandler: The 3-XORSAT threshold, Proc. 43rd FOCS (2002), 769–778.

    Google Scholar 

  22. [22]

    M. Dyer, A. Frieze and C. Greenhill: On the chromatic number of a random hypergraph, J. Comb. Theory Series B113 (2015), 68–122.

    MathSciNet  MATH  Google Scholar 

  23. [23]

    P. Erdős and A. Renyi: On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutato Int. Közl.5 (1960), 17–61.

    MathSciNet  MATH  Google Scholar 

  24. [24]

    A. Frieze and N. Wormald: Random k-Sat: a tight threshold for moderately growing k, Combinatorica25 (2005), 297–305.

    MathSciNet  MATH  Google Scholar 

  25. [25]

    A. Galanis, D. Stefankovic and E. Vlgoda: Inapproximability for antiferromag-netic spin systems in the tree nonuniqueness region, J. A CM62 (2015), 50.

    Google Scholar 

  26. [26]

    P. Gao and M. Molloy: The stripping process can be slow: part I, arXiv:1501.02695 (2015).

  27. [27]

    A. Goerdt: A threshold for unsatisfiability, Proc. 17th MFCS (1992), 264–274.

    Google Scholar 

  28. [28]

    A. Goerdt and L. Falke: Satisfiability thresholds beyond k-XORSAT, Proc. 7th International Computer Science Symposium in Russia (2012), 148–159.

    Google Scholar 

  29. [29]

    M. Ibrahimi, Y. Kanoria, M. Kraning and A. Montanari: The set of solutions of random XORSAT formulae, Annals of Applied Probability25 (2015), 2743–2808.

    MathSciNet  MATH  Google Scholar 

  30. [30]

    Y. Kabashima and D. Saad: Statistical mechanics of error correcting codes, Euro-phys. Lett.45 (1999), 97–103.

    Google Scholar 

  31. [31]

    S. Klrkpatrick and B. Selman: Critical behavior in the satisfiability of random Boolean expressions, Science264 (1994), 1297–1301.

    MathSciNet  MATH  Google Scholar 

  32. [32]

    V. Kolchin: Random graphs and systems of linear equations in finite fields, Random Structures and Algorithms5 (1995), 425–436.

    MathSciNet  Google Scholar 

  33. [33]

    F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian and L. Zde-Borová: Gibbs states and the set of solutions of random constraint satisfaction problems, Proc. National Academy of Sciences104 (2007), 10318–10323.

    MathSciNet  MATH  Google Scholar 

  34. [34]

    R. Mceliece and J. Cheng: Some high-rate near capacity codecs for the Gaussian channel, Proc. ALLERTON (1996), 494–503.

    Google Scholar 

  35. [35]

    M. Mezard and A. Montanari: Information, physics and computation, Oxford University Press 2009.

    MATH  Google Scholar 

  36. [36]

    M. Mezard, G. Parisi and R. Zecchina: Analytic and algorithmic solution of random satisfiability problems, Science297 (2002), 812–815.

    Google Scholar 

  37. [37]

    M. Mezard, F. Ricci-Tersenghi and R. Zecchina: TWO solutions to diluted p-spin models and XORSAT problems, Journal of Statistical Physics111 (2003), 505–533.

    MathSciNet  MATH  Google Scholar 

  38. [38]

    D. Mitchell, B. Selman and H. Levesque: Hard and easy distribution of SAT problems, Proc. 10th AAAI (1992), 459–465.

    Google Scholar 

  39. [39]

    M. Molloy: Cores in random hypergraphs and Boolean formulas, Random Structures and Algorithms27 (2005), 124–135.

    MathSciNet  MATH  Google Scholar 

  40. [40]

    C. Moore: The phase transition in random regular exact cover, Annales l’institut Henri Poincare D, combinatorics, physics and their Interactions3 (2016), 349–362.

    MathSciNet  MATH  Google Scholar 

  41. [41]

    A. Montanari: Estimating random variables from random sparse observations, European Transactions on Telecommunications19 (2008), 385–403.

    Google Scholar 

  42. [42]

    B. Pittel and G. Sorkin: The satisfiability threshold for k-XORSAT, Combinatorics, Probability and Computing25 (2016), 236–268.

    MathSciNet  MATH  Google Scholar 

  43. [43]

    T. Richardson and R. Urbanke: Modern coding theory, Cambridge University Press (2008).

    MATH  Google Scholar 

  44. [44]

    R. Robinson and N. Wormald: Almost all cubic graphs are Hamiltonian, Random Structures and Algorithms3 (1992), 117–125.

    MathSciNet  MATH  Google Scholar 

  45. [45]

    A. Sly: Computational transition at the uniqueness threshold, Proc. 51st FOCS (2010), 287–296.

    Google Scholar 

  46. [46]

    L. Valiant and V. Vazirani: NP is as easy as detecting unique solutions, Theoretical Computer Science47 (1986), 85–93.

    MathSciNet  MATH  Google Scholar 

  47. [47]

    M. J. Wainwright, E. Maneva and E. Martinian: Lossy source compression using low-density generator matrix codes: Analysis and algorithms, IEEE Transactions on Information theory56 (2010), 1351–1368.

    MathSciNet  MATH  Google Scholar 

  48. [48]

    L. Zdeborová and F. Krzakala: Statistical physics of inference: thresholds and algorithms, Advances in Physics65 (2016), 453–552.

    Google Scholar 

  49. [49]

    L. Zdeborová and M. Mézard: Locked constraint satisfaction problems, Phys. Rev. Lett.101 (2008), 078702.

    MATH  Google Scholar 

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We thank Charilaos Efthymiou, Alan Frieze, Mike Molloy and Wesley Pegden for helpful discussions.

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Corresponding author

Correspondence to Pu Gao.

Additional information

Gao’s research is supported by ARC DE170100716 and ARC DP160100835.

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Ayre, P., Coja-Oghlan, A., Gao, P. et al. The Satisfiability Threshold For Random Linear Equations. Combinatorica 40, 179–235 (2020).

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Mathematics Subject Classification (2010)

  • 05C80
  • 05C50