Abstract
We introduce a highly structured family of hard satisfiable 3-SAT formulas corresponding to an ordered spin-glass model from statistical physics. This model has provably “glassy” behavior; that is, it has many local optima with large energy barriers between them, so that local search algorithms get stuck and have difficulty finding the true “ground state,” i.e., the unique satisfying assignment. We test the hardness of our formulas with two Davis-Putnam solvers, Satz and zChaff, the recently introduced Survey Propagation (SP), and two local search algorithms, WalkSAT and Record-to-Record Travel (RRT). We compare our formulas to random 3-XOR-SAT formulas and to two other generators of hard satisfiable instances, the minimum disagreement parity formulas of Crawford et al., and Hirsch’s hgen2. For the complete solvers the running time of our formulas grows exponentially in \({\sqrt n}\), and exceeds that of random 3-XOR-SAT formulas for small problem sizes. SP is unable to solve our formulas with as few as 25 variables. For WalkSAT, our formulas appear to be harder than any other known generator of satisfiable instances. Finally, our formulas can be solved efficiently by RRT but only if the parameter d is tuned to the height of the barriers between local minima, and we use this parameter to measure the barrier heights in random 3-XOR-SAT formulas as well.
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References
Achlioptas, D., Gomes, C., Kautz, H., Selman, B.: Generating satisfiable problem instances. In: Proc. AAAI 2000, pp. 256–261 (2000)
Achlioptas, D., Jia, H., Moore, C.: Hiding satisfying assignments: two are better than one (Submitted)
Asahiro, Y., Iwama, K., Miyano, E.: Random generation of test instances with controlled attributes. [8], op. cit
Barthel, W., Hartmann, A.K., Leone, M., Ricci-Tersenghi, F., Weigt, M., Zecchina, R.: Hiding solutions in random satisfiability problems: a statistical mechanics approach. Phys. Rev. Lett. 88, 188701 (2002)
Baxter, R.J.: Hard hexagons: exact solution. J. Physics A 13, 1023–1030 (1980)
Cocco, S., Dubois, O., Mandler, J., Monasson, R.: Rigorous decimation-based construction of ground pure states for spin glass models on random lattices. Phys. Rev. Lett. 90(4), 47205 (2003)
Crawford, J.M., Kearns, M.J.: The Minimal Disagreement Parity Problem as a Hard Satisfiability Problem. In: Crawford, J.M., Kearns, M.J., Schapire, R.E. (eds.) The minimal disagreement parity problem as a hard satisfiability problem. Technical report, CIRL (1994), ftp://dimacs.rutgers.edu/pub/challenge/satisfiability/benchmarks/cnf/
Johnson, D., Trick, M. (eds.): Second DIMACS Implementation Challenge, 1993. DIMACS Series in Disc. Math. and Theor. Comp. Sci. AMS, Washington DC (1996)
Dueck, G.: New optimization heuristics: the great deluge algorithm and the record-to-record travel. J. Comp. Phys. 104, 86–92 (1993)
Franz, S., Mézard, M., Ricci-Tersenghi, F., Weigt, M., Zecchina, R.: A ferromagnet with a glass transition. Europhys. Lett. 55, 465 (2001)
Garrahan, J.P., Newman, M.E.J.: Glassiness and constrained dynamics of short-range non-disordered spin model. Phys. Rev. E 62, 7670–7678 (2000)
Hirsch, E.A.: hgen2 formula generator source site, http://logic.pdmi.ras.ru/hirsch/
Holroyd, A.E.: Sharp metastability threshold for two-dimensional bootstrap percolation. Prob. Theory and Related Fields 125, 195–224 (2003)
Hoos, H.H.: SATLIB, A collection of SAT tools and data, http://www.informatik.tu-darmstadt.de/AI/SATLIB
Kautz, H., Ruan, Y., Achlioptas, D., Gomes, C., Selman, B., Stickel, M.: Balance and Filtering in Structured Satisfiable Problems. In: Proc. IJCAI 2001, pp. 351–358 (2001)
Mézard, M., Ricci-Tersenghi, F., Zecchina, R.: Alternative solutions to diluted p-spin models and XORSAT problems. J. Stat. Phys. 111, 505 (2003)
Mézard, M., Zecchina, R.: Random K-satisfiability: from an analytic solution to a new efficient algorithm. Phys. Rev. E 66 (2002), See also Braunstein, A., Mézard, M.M., Zecchina, R., Survey propagation: an algorithm for satisfiability. Preprint, (2002), http://www.ictp.trieste.it/~zecchina/SP/
Newman, M.E.J., Moore, C.: Glassy Dynamics in an Exactly Solvable Spin Model. Phys. Rev. E 60, 5068–5072 (1999)
Seitz, S., Orponen, P.: An efficient local search method for random 3-satisfiability. In: LICS 2003, workshop on Typical Case Complexity and Phase Transitions. Electronic Notes in Discrete Mathematics, vol. 16 (2003)
Selman, B., Kautz, H.A., Cohen, B.: Noise strategies for improving local search. In: Proc. AAAI 1994 (1994)
Shaw, P., Stergiou, K., Walsh, T.: Arc consistency and quasigroup completion. In: ECAI 1998. workshop on binary constraints (1998)
Ricci-Tersenghi, F., Weigt, M., Zecchina, R.: Simplest random K-satisfiability problem. Phys. Rev. E 63, 26702 (2001)
Competition result site. In: SAT 2003 (2003), http://www.satlive.org/SATCompetition/2003/results.html
Van Gelder, A.: Problem generator mkcnf.c contributed to the DIMACS, Challenge archive (1993),
Zhang, L.: zchaff source site, http://ee.princeton.edu/~chaff/zChaff.php
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Jia, H., Moore, C., Selman, B. (2005). From Spin Glasses to Hard Satisfiable Formulas. In: Hoos, H.H., Mitchell, D.G. (eds) Theory and Applications of Satisfiability Testing. SAT 2004. Lecture Notes in Computer Science, vol 3542. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11527695_16
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DOI: https://doi.org/10.1007/11527695_16
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