1.2 SATISFIABILITY AND HARD-PROBLEM INSTANCES....... iv

“NP-complete”problemsareat thecoreof many computational tasksof practical interest,whichmeansthatthecostof solutionwill grow exponentiallywith problem sizein the worst case(providedP  NP). Althoughheuristicscanbe quite effective on suchproblemsin most cases,thereis a growing appreciationthat theseproblemscontainphasetransitions,andat thephaseboundariesexponential complexity becomesthetypical outcome,not just theworstcase.Usingmethods from statisticalphysics,a much betterunderstandingof suchphasetransition phenomenain computationalproblemshasbeenobtainedin recentyears. We will review severalkey resultsin this area,therebyillustratingsomeof thedeep connectionsbetweencomputerscienceandstatisticalphysics.Theseminalwork by Fu andAnderson[13, 14] providedtheinitial impetusfor muchof this work.

[1]  David B. Wilson,et al.  The empirical values of the critical k-SAT exponents are wrong , 2000 .

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

[3]  L. Kirousis,et al.  Approximating the unsatisfiability threshold of random formulas , 1998, Random Struct. Algorithms.

[4]  Yacine Boufkhad,et al.  A General Upper Bound for the Satisfiability Threshold of Random r-SAT Formulae , 1997, J. Algorithms.

[5]  Brian Hayes,et al.  CAN’T GET NO SATISFACTION , 1997 .

[6]  Hector J. Levesque,et al.  Generating Hard Satisfiability Problems , 1996, Artif. Intell..

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

[8]  James M. Crawford,et al.  Implicates and Prime Implicates in Random 3-SAT , 1996, Artif. Intell..

[9]  BarnhartCynthia,et al.  The fleet assignment problem , 1995 .

[10]  Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture , 1995, Random Struct. Algorithms.

[11]  Tad Hogg,et al.  Expected Gains from Parallelizing Constraint Solving for Hard Problems , 1994, AAAI.

[12]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

[13]  James M. Crawford,et al.  Experimental Results on the Crossover Point inSatis ability , 1993 .

[14]  Yumi K. Tsuji,et al.  EVIDENCE FOR A SATISFIABILITY THRESHOLD FOR RANDOM 3CNF FORMULAS , 1992 .

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

[16]  Tad Hogg,et al.  Using Deep Structure to Locate Hard Problems , 1992, AAAI.

[17]  Hector J. Levesque,et al.  A New Method for Solving Hard Satisfiability Problems , 1992, AAAI.

[18]  John V. Franco,et al.  Elimination of Infrequent Variables Improves Average Case Performance of Satisfiability Algorithms , 1991, SIAM J. Comput..

[19]  B A Huberman,et al.  Cooperative Solution of Constraint Satisfaction Problems , 1991, Science.

[20]  Peter C. Cheeseman,et al.  Where the Really Hard Problems Are , 1991, IJCAI.

[21]  Daniel L. Stein,et al.  Lectures In The Sciences Of Complexity , 1989 .

[22]  Paul Walton Purdom,et al.  Polynomial-average-time satisfiability problems , 1987, Inf. Sci..

[23]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[24]  John Franco,et al.  Probabilistic analysis of the Davis Putnam procedure for solving the satisfiability problem , 1983, Discret. Appl. Math..

[25]  Stephen Wolfram,et al.  Universality and complexity in cellular automata , 1983 .

[26]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[27]  Hilary Putnam,et al.  A Computing Procedure for Quantification Theory , 1960, JACM.