On the Analysis of Backtrack Procedures for the Colouring of Random Graphs

Backtrack search algorithms are procedures capable of deciding whether a decision problem has a solution or not through a sequence of trials and errors. Analysis of the performances of these procedures is a long-standing open problem in theoretical computer science. I present some statistical physics ideas and techniques to attack this problem. The approach is illustrated on the colouring of random graphs, and some current limitations and perspectives are presented.

[1]  Jonathan S. Turner,et al.  Almost All k-Colorable Graphs are Easy to Color , 1988, J. Algorithms.

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

[3]  Riccardo Zecchina,et al.  Coloring random graphs , 2002, Physical review letters.

[4]  Hector J. Levesque,et al.  Hard and Easy Distributions of SAT Problems , 1992, AAAI.

[5]  R. Monasson,et al.  Statistical physics analysis of the computational complexity of solving random satisfiability problems using backtrack algorithms , 2000, cond-mat/0012191.

[6]  Donald E. Knuth,et al.  Selected papers on analysis of algorithms , 2000, CSLI lecture notes series.

[7]  Cristopher Moore,et al.  How Much Backtracking Does It Take to Color Random Graphs? Rigorous Results on Heavy Tails , 2004, CP.

[8]  David G. Mitchell,et al.  The resolution complexity of random graph k-colorability , 2005, Discret. Appl. Math..

[9]  Cristopher Moore,et al.  Almost all graphs with average degree 4 are 3-colorable , 2003, J. Comput. Syst. Sci..

[10]  Paul L. Krapivsky,et al.  Extreme Value Statistics and Traveling Fronts: An Application to Computer Science , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  J. Langer,et al.  Relaxation Times for Metastable States in the Mean-Field Model of a Ferromagnet , 1966 .

[12]  S Cocco,et al.  Trajectories in phase diagrams, growth processes, and computational complexity: how search algorithms solve the 3-satisfiability problem. , 2001, Physical review letters.

[13]  D. Knuth.  Estimating the efficiency of backtrack programs. , 1974 .

[14]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[15]  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.

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

[17]  Liat Ein-Dor,et al.  The dynamics of proving uncolourability of large random graphs: I. Symmetric colouring heuristic , 2003 .

[18]  Donald W. Loveland,et al.  A machine program for theorem-proving , 2011, CACM.

[19]  D. Achlioptas,et al.  A sharp threshold for k-colorability , 1999 .

[20]  R. Monasson,et al.  Field-theoretic approach to metastability in the contact process. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[22]  Simona Cocco,et al.  Heuristic average-case analysis of the backtrack resolution of random 3-satisfiability instances , 2004, Theor. Comput. Sci..

[23]  Joseph C. Culberson,et al.  Frozen development in graph coloring , 2001, Theor. Comput. Sci..