Synchronizing random automata

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Application

A solution scheme of satisfiability problem by active usage of totally unimodularity property.

by Mei Long.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 93-98).Abstracts in English and Chinese.Table of Contents --- p.vAbstract --- p.viiiAcknowledgements --- p.xChapter 1 --- Introduction --- p.1Chapter 1.1 --- Satisfiability Problem --- p.1Chapter 1.2 --- Motivation of the Research --- p.1Chapter 1.3 --- Overview of the Thesis --- p.2Chapter 2 --- Satisfiability Problem --- p.4Chapter 2.1 --- Satisfiability Problem --- p.5Chapter 2.1.1 --- Basic Definition --- p.5Chapter 2.1.2 --- Phase Transitions --- p.5Chapter 2.2 --- History --- p.6Chapter 2.3 --- The Basic Search Algorithm --- p.8Chapter 2.4 --- Some Improvements to the Basic Algorithm --- p.9Chapter 2.4.1 --- Satz by Chu-Min Li --- p.9Chapter 2.4.2 --- Heuristics and Local Search --- p.12Chapter 2.4.3 --- Relaxation --- p.13Chapter 2.5 --- Benchmarks --- p.14Chapter 2.5.1 --- Specific Problems --- p.14Chapter 2.5.2 --- Randomly Generated Problems --- p.14Chapter 2.6 --- Software and Internet Information for SAT solving --- p.16Chapter 2.6.1 --- Stochastic Local Search Algorithms (incomplete) --- p.16Chapter 2.6.2 --- Systematic Search Algorithms (complete) --- p.16Chapter 2.6.3 --- Some useful Links to SAT Related Sites --- p.17Chapter 3 --- Integer Programming Formulation for Logic Problem --- p.18Chapter 3.1 --- SAT Problem --- p.19Chapter 3.2 --- MAXSAT Problem --- p.19Chapter 3.3 --- Logical Inference Problem --- p.19Chapter 3.4 --- Weighted Exact Satisfiability Problem --- p.20Chapter 4 --- Integer Programming Formulation for SAT Problem --- p.22Chapter 4.1 --- From 3-CNF SAT Clauses to Zero-One IP Constraints --- p.22Chapter 4.2 --- Integer Programming Model for 3-SAT --- p.23Chapter 4.3 --- The Equivalence of the SAT and the IP --- p.23Chapter 4.4 --- Example --- p.24Chapter 5 --- Integer Solvability of Linear Programs --- p.27Chapter 5.1 --- Unimodularity --- p.27Chapter 5.2 --- Totally Unimodularity --- p.28Chapter 5.3 --- Some Results on Recognition of Linear Solvability of IP --- p.32Chapter 6 --- TU Based Matrix Research Results --- p.33Chapter 6.1 --- 2x2 Matrix's TU Property --- p.33Chapter 6.2 --- Extended Integer Programming Model for SAT --- p.34Chapter 6.3 --- 3x3 Matrix's TU Property --- p.35Chapter 7 --- Totally Unimodularity Based Branching-and-Bound Algorithm --- p.38Chapter 7.1 --- Introduction --- p.38Chapter 7.1.1 --- Enumeration Trees --- p.39Chapter 7.1.2 --- The Concept of Branch and Bound --- p.42Chapter 7.2 --- TU Based Branching Rule --- p.43Chapter 7.2.1 --- How to sort variables based on 2x2 submatrices --- p.43Chapter 7.2.2 --- How to sort the rest variables --- p.45Chapter 7.3 --- TU Based Bounding Rule --- p.46Chapter 7.4 --- TU Based Branch-and-Bound Algorithm --- p.47Chapter 7.5 --- Example --- p.49Chapter 8 --- Numerical Result --- p.57Chapter 8.1 --- Experimental Result --- p.57Chapter 8.2 --- Statistical Results of ILOG CPLEX --- p.59Chapter 9 --- Conclusions --- p.61Chapter 9.1 --- Contributions --- p.61Chapter 9.2 --- Future Work --- p.62Chapter A --- The Coefficient Matrix A for Example in Chapter 7 --- p.64Chapter B --- The Detailed Numerical Information of Solution Process for Exam- ple in Chapter 7 --- p.66Chapter C --- Experimental Result --- p.67Chapter C.1 --- "# of variables: 20, # of clauses: 91" --- p.67Chapter C.2 --- "# of variables: 50, # of clauses: 218" --- p.70Chapter C.3 --- # of variables: 75，# of clauses: 325 --- p.73Chapter C.4 --- "# of variables: 100, # of clauses: 430" --- p.76Chapter D --- Experimental Result of ILOG CPLEX --- p.80Chapter D.1 --- # of variables: 20´ة # of clauses: 91 --- p.80Chapter D.2 --- # of variables: 50，#of clauses: 218 --- p.83Chapter D.3 --- # of variables: 75，# of clauses: 325 --- p.86Chapter D.4 --- "# of variables: 100, # of clauses: 430" --- p.89Bibliography --- p.9

On the Greedy Algorithm for Satisfiability

We show that for the vast majority of satisfiable 3CNF formulae, the local search heuristic that starts at a random truth assignment, and repeatedly flips the variable that improves the number of satisfied clauses the most, almost always succeeds in discovering a satisfying truth assignment

On the Stupid Algorithm for Satisfiability

I introduce an extremely stupid algorithm for the satisfiability problem. Where random 3-SAT problems are generated randomly to agree with a predetermined truth assignment, I show that the algorithm almost certainly solves problems with N variables and O(N ln N ) clauses very easily. This helps confirm two pieces of folklore. First, that random problems generated in this way are usuually easy. Second, that extrapolating theoretical results beyond their range of direct application must be done with some care. 1 On "On the greedy algorithm for satisfiability" In [1], Koutsoupias and Papadimitriou study the greedy algorithm for satisfiability (as they call it). Actually it's not an algorithm, but a heuristic. This is a very simple algorithm. Choose a truth assignment at random. Then choose a variable such that flipping it increases the number of satisfied clauses: do not allow sideways moves. Repeat until no improvement is possible. If this is a solution, report success. Otherwise give u..