The Graph Coloring Problem by James Pate Williams, Jr.

The Graph Coloring Problem and N-Queens Puzzle are both NP-Complete constraint satisfaction problems. There does not exist polynomial time algorithms to solve large instances of either problem. The Graph Coloring Problem is given a graph of n-vertices and m-edges find the minimum number of colors such that no two connected vertices are of the same color. The N-Queens Puzzle is given a n-by-n chessboard and n-queens place the queens such that no two queens are attacking one another. A queen in chess can move any number of squares diagonally, horizontally, or vertically.

As a Master of Software Engineering graduate student in the era 1998 to 2000, I majored in the subdiscipline of artificial intelligence known as constraint satisfaction. I developed evolutionary hill-climbing algorithms to solve constraint satisfaction problems. I specifically solved instances of the Graph Coloring Problem and the N-Queens Puzzle. I compared my algorithms with other researchers’ algorithms. One of the algorithms tested was Makato Yokoo’s Weak-Commitment Search Algorithm (WCSA). Yakoo wrote a very nice book named “Distributed Constraint Satisfaction” which has excellent flow-charts of several algorithms including the Min-Conflicts Search Algorithm (MCSA) and his WCSA. I implemented both algorithms in the C++ and Java computer languages.

I rewrote the preceding two algorithms in the C# computer language sometime in the epoch 2008 – 2009. Below are three instances of the Graph Coloring Problem with four, eight, and twelve vertices solved by the Yakoo’s WCSA.

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Author: jamespatewilliamsjr

My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises.

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