Instability of one-step replica-symmetry-broken phase in satisfiability problems

We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two random combinatorial problems: k-XORSAT and k-SAT. We present a general method for establishing the stability of these solutions with respect to further steps of replica-symmetry breaking. Our approach extends the ideas of Montanari and Ricci-Tersenghi (2003 Eur. Phys. J. B 33 339) to more general combinatorial problems. It turns out that 1RSB is always unstable at sufficiently small clause density α or high energy. In particular, the recent 1RSB solution to 3-SAT is unstable at zero energy for α < αm, with αm ≈ 4.153. On the other hand, the SAT–UNSAT phase transition seems to be correctly described within 1RSB.

[1]  Riccardo Zecchina,et al.  Simplest random K-satisfiability problem , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Rémi Monasson,et al.  THE EUROPEAN PHYSICAL JOURNAL B c○ EDP Sciences , 1999 .

[3]  Simona Cocco,et al.  Approximate analysis of search algorithms with "physical" methods , 2003, ArXiv.

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

[5]  M. Mézard,et al.  The Bethe lattice spin glass revisited , 2000, cond-mat/0009418.

[6]  Nadia Creignou,et al.  Satisfiability Threshold for Random XOR-CNF Formulas , 1999, Discret. Appl. Math..

[7]  Rémi Monasson,et al.  Determining computational complexity from characteristic ‘phase transitions’ , 1999, Nature.

[8]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[9]  O. Nicrosini,et al.  Higher-order QED corrections to single-W production in electron–positron collisions , 2000, hep-ph/0005121.

[10]  A. Crisanti,et al.  The 3-SAT problem with large number of clauses in the ∞-replica symmetry breaking scheme , 2001, cond-mat/0108433.

[11]  Brendan J. Frey,et al.  Factor graphs and the sum-product algorithm , 2001, IEEE Trans. Inf. Theory.

[12]  E. Gardner.  Spin glasses with p-spin interactions , 1985 .

[13]  G. Parisi,et al.  The K-SAT Problem in a Simple Limit , 2000, cond-mat/0007364.

[14]  Giorgio Parisi,et al.  On the survey-propagation equations for the random K-satisfiability problem , 2002, ArXiv.

[15]  Giorgio Parisi,et al.  The role of the Becchi–Rouet–Stora–Tyutin supersymmetry in the calculation of the complexity for the Sherrington–Kirkpatrick model , 2003 .

[16]  R. Monasson.  Optimization problems and replica symmetry breaking in finite connectivity spin glasses , 1997, cond-mat/9707089.

[17]  M. Mézard,et al.  Random K-satisfiability problem: from an analytic solution to an efficient algorithm. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  M. A. Moore,et al.  Comment on ``On the formal equivalence of the TAP and thermodynamic methods in the SK model'' , 2003 .

[19]  A. Montanari,et al.  On the nature of the low-temperature phase in discontinuous mean-field spin glasses , 2003, cond-mat/0301591.

[20]  R. Monasson,et al.  Rigorous decimation-based construction of ground pure states for spin-glass models on random lattices. , 2002, Physical review letters.

[21]  Michele Leone,et al.  Replica Bounds for Optimization Problems and Diluted Spin Systems , 2002 .

[22]  M A Moore,et al.  Complexity of Ising spin glasses. , 2004, Physical review letters.

[23]  A. Montanari,et al.  Cooling-schedule dependence of the dynamics of mean-field glasses , 2004, cond-mat/0401649.

[24]  Monasson.  Structural glass transition and the entropy of the metastable states. , 1995, Physical Review Letters.

[25]  T Rizzo,et al.  Spin-glass complexity. , 2004, Physical review letters.

[26]  Joachim Rosenthal,et al.  Codes, systems, and graphical models , 2001 .

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