The Phase Transition in Exact Cover

We study EC3, a variant of Exact Cover which is equivalent to Positive 1-in-3 SAT. Random instances of EC3 were recently used as benchmarks for simulations of an adiabatic quantum algorithm. Empirical results suggest that EC3 has a phase transition from satisfiability to unsatisfiability when the number of clauses per variable r exceeds some threshold r* ~= 0.62 +- 0.01. Using the method of differential equations, we show that if r <= 0.546 w.h.p. a random instance of EC3 is satisfiable. Combined with previous results this limits the location of the threshold, if it exists, to the range 0.546 < r* < 0.644.

[1]  Andrew M. Childs,et al.  Robustness of adiabatic quantum computation , 2001, quant-ph/0108048.

[2]  Cristopher Moore,et al.  Almost all graphs with average degree 4 are 3-colorable , 2002, STOC '02.

[3]  Wenjin Mao,et al.  Solving Satisfiability Problems by the Ground-State Quantum Computer , 2005, ArXiv.

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

[5]  Cristopher Moore,et al.  The phase transition in 1-in-k SAT and NAE 3-SAT , 2001, SODA '01.

[6]  Chu Min Li,et al.  Heuristics Based on Unit Propagation for Satisfiability Problems , 1997, IJCAI.

[7]  Dimitris Achlioptas,et al.  Lower bounds for random 3-SAT via differential equations , 2001, Theor. Comput. Sci..

[8]  W. Mao.  Quantum Algorithm to Solve Satisfiability Problems , 2004, quant-ph/0411194.

[9]  N. Wormald.  Differential Equations for Random Processes and Random Graphs , 1995 .

[10]  Edward Farhi,et al.  A Numerical Study of the Performance of a Quantum Adiabatic Evolution Algorithm for Satisfiability , 2000, ArXiv.

[11]  Sharad Malik,et al.  Chaff: engineering an efficient SAT solver , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[12]  Lefteris M. Kirousis,et al.  The probabilistic analysis of a greedy satisfiability algorithm , 2002, Random Struct. Algorithms.

[14]  Ming-Te Chao,et al.  Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k satisfiability problem , 1990, Inf. Sci..

[15]  Umesh V. Vazirani,et al.  How powerful is adiabatic quantum computation? , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[16]  Michael Molloy,et al.  The analysis of a list-coloring algorithm on a random graph , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[17]  Alan M. Frieze,et al.  Analysis of Two Simple Heuristics on a Random Instance of k-SAT , 1996, J. Algorithms.

[18]  E. Farhi,et al.  A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem , 2001, Science.