Structure of random r-SAT below the pure literal threshold

It is well known that there is a sharp density threshold for a random $r$-SAT formula to be satisfiable, and a similar, smaller, threshold for it to be satisfied by the pure literal rule. Also, above the satisfiability threshold, where a random formula is with high probability (whp) unsatisfiable, the unsatisfiability is whp due to a large "minimal unsatisfiable subformula" (MUF). By contrast, we show that for the (rare) unsatisfiable formulae below the pure literal threshold, the unsatisfiability is whp due to a unique MUF with smallest possible "excess", failing this whp due to a unique MUF with the next larger excess, and so forth. In the same regime, we give a precise asymptotic expansion for the probability that a formula is unsatisfiable, and efficient algorithms for satisfying a formula or proving its unsatisfiability. It remains open what happens between the pure literal threshold and the satisfiability threshold. We prove analogous results for the $k$-core and $k$-colorability thresholds for a random graph, or more generally a random $r$-uniform hypergraph.

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

[2]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

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

[4]  Michael Molloy.  A gap between the appearances of a k‐core and a (k + 1)‐chromatic graph , 1996 .

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

[6]  Amin Coja-Oghlan.  A Better Algorithm for Random k-SAT , 2009, ICALP.

[7]  E. Friedgut,et al.  Sharp thresholds of graph properties, and the -sat problem , 1999 .

[8]  J. W. P. Hirschfeld,et al.  Thresholds for colourability and satisfiability in random graphs and boolean formulae , 2001 .

[9]  Michael Krivelevich,et al.  Why almost all satisfiable $k$-CNF formulas are easy , 2007 .

[10]  Michael Molloy.  A gap between the appearances of a k-core and a (k+1)-chromatic graph , 1996, Random Struct. Algorithms.

[11]  Michael Molloy,et al.  The pure literal rule threshold and cores in random hypergraphs , 2004, SODA '04.

[12]  M. Mitzenmacher.  Tight Thresholds for The Pure Literal Rule , 1997 .

[13]  Alan M. Frieze,et al.  On the satisfiability and maximum satisfiability of random 3-CNF formulas , 1993, SODA '93.

[14]  Alex D. Scott,et al.  Solving Sparse Random Instances of Max Cut and Max 2-CSP in Linear Expected Time , 2006, Combinatorics, Probability and Computing.

[15]  Amin Coja-Oghlan,et al.  Exact and approximative algorithms for coloring G(n,p) , 2004, Random Struct. Algorithms.

[16]  Amin Coja-Oghlan.  A Better Algorithm for Random k-SAT , 2010, SIAM J. Comput..

[17]  alcun K. grafo.  ASYMPTOTIC ENUMERATION OF Kn-FREE GRAPHS , 2004 .

[18]  Hans Jürgen Prömel,et al.  On the structure of clique‐free graphs , 2001, Random Struct. Algorithms.

[19]  Yuval Peres,et al.  The threshold for random k-SAT is 2k (ln 2 - O(k)) , 2003, STOC '03.

[20]  Dimitris Achlioptas,et al.  THE THRESHOLD FOR RANDOM k-SAT IS 2k log 2 O(k) , 2004, FOCS 2004.

[21]  Bruce A. Reed,et al.  Mick gets some (the odds are on his side) (satisfiability) , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

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

[23]  Michael Mitzenmacher,et al.  Analysis of random processes via And-Or tree evaluation , 1998, SODA '98.