The solution space geometry of random linear equations

We consider random systems of linear equations over GF(2) in which every equation binds k variables. We obtain a precise description of the clustering of solutions in such systems. In particular, we prove that with probability that tends to 1 as the number of variables, n, grows: for every pair of solutions \sigma, \tau, either there exists a sequence of solutions \sigma,...,\tau, in which successive elements differ by O(log n) variables, or every sequence of solutions \sigma,...,\tau, contains a step requiring the simultaneous change of \Omega(n) variables. Furthermore, we determine precisely which pairs of solutions are in each category. Our results are tight and highly quantitative in nature. Moreover, our proof highlights the role of unique extendability as the driving force behind the success of Low Density Parity Check codes and our techniques also apply to the problem of so-called pseudo-codewords in such codes.

[1]  Andrea Montanari,et al.  Reconstruction and Clustering in Random Constraint Satisfaction Problems , 2011, SIAM J. Discret. Math..

[2]  Amin Coja-Oghlan,et al.  On independent sets in random graphs , 2010, SODA '11.

[3]  Jeong Han Kim,et al.  Poisson Cloning Model for Random Graph , 2004 .

[4]  Riccardo Zecchina,et al.  Alternative solutions to diluted p-spin models and XORSAT problems , 2002, ArXiv.

[5]  Amin Coja-Oghlan,et al.  Algorithmic Barriers from Phase Transitions , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[6]  Boris G. Pittel,et al.  On Tree Census and the Giant Component in Sparse Random Graphs , 1990, Random Struct. Algorithms.

[7]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[8]  Andrea Montanari,et al.  Tight Thresholds for Cuckoo Hashing via XORSAT , 2009, ICALP.

[9]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[10]  Olivier Dubois,et al.  The 3-XORSAT threshold , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[11]  Federico Ricci-Tersenghi,et al.  On the solution-space geometry of random constraint satisfaction problems , 2006, STOC '06.

[12]  Svante Janson,et al.  A simple solution to the k-core problem , 2007, Random Struct. Algorithms.

[13]  Amin Coja-Oghlan,et al.  On belief propagation guided decimation for random k-SAT , 2010, SODA '11.

[14]  Endre Szemerédi,et al.  Many hard examples for resolution , 1988, JACM.

[15]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[16]  Joel H. Spencer,et al.  Sudden Emergence of a Giantk-Core in a Random Graph , 1996, J. Comb. Theory, Ser. B.

[17]  Jeong Han Kim,et al.  Poisson Cloning Model for Random Graphs , 2008, 0805.4133.

[18]  M. Mézard,et al.  Information, Physics, and Computation , 2009 .

[19]  George Havas,et al.  A Family of Perfect Hashing Methods , 1996, Comput. J..

[20]  Daniel A. Spielman,et al.  Analysis of low density codes and improved designs using irregular graphs , 1998, STOC '98.

[21]  Michael Molloy,et al.  Cores in random hypergraphs and Boolean formulas , 2005, Random Struct. Algorithms.

[22]  Daniel Fernholz.  Cores and Connectivity in Sparse Random Graphs , 2004 .

[23]  Thierry Mora,et al.  Pairs of SAT-assignments in random Boolean formulæ , 2005, Theor. Comput. Sci..

[24]  Mohammad R. Salavatipour,et al.  The resolution complexity of random constraint satisfaction problems , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[25]  Marco Guidetti,et al.  Complexity of several constraint satisfaction problems using the heuristic, classical, algorithm, WalkSAT , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Colin Cooper,et al.  The cores of random hypergraphs with a given degree sequence , 2004, Random Struct. Algorithms.

[27]  Béla Bollobás,et al.  Random Graphs , 1985 .

[28]  Thierry Mora,et al.  Clustering of solutions in the random satisfiability problem , 2005, Physical review letters.

[29]  M. Mézard,et al.  Pairs of SAT Assignments and Clustering in Random Boolean Formulae , 2005 .

[30]  M. Mézard,et al.  Two Solutions to Diluted p-Spin Models and XORSAT Problems , 2003 .

[31]  Federico Ricci-Tersenghi,et al.  Random Formulas Have Frozen Variables , 2009, SIAM J. Comput..

[32]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[33]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[34]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[35]  Michael Molloy,et al.  The exact satisfiability threshold for a potentially intractable random constraint satisfaction problem , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[36]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[37]  Tomasz Łuczak.  Component behavior near the critical point of the random graph process , 1990 .

[38]  Michael Molloy,et al.  The Satisfiability Threshold for a Seemingly Intractable Random Constraint Satisfaction Problem , 2012, SIAM J. Discret. Math..

[39]  Michael Molloy,et al.  A sharp threshold in proof complexity yields lower bounds for satisfiability search , 2004, J. Comput. Syst. Sci..

[40]  Boris G. Pittel,et al.  The Satisfiability Threshold for k-XORSAT , 2012, Combinatorics, Probability and Computing.

[41]  Yashodhan Kanoria,et al.  The set of solutions of random XORSAT formulae , 2011, SODA.