Equivalence Class Subset Algorithm

The equivalence class subset algorithm is a powerful tool for solving a wide variety of constraint satisfaction problems and is based on the use of a decision function which has a very high but not perfect accuracy. Perfect accuracy is not required in the decision function as even a suboptimal solution contains valuable information that can be used to help find an optimal solution. In the hardest problems, the decision function can break down leading to a suboptimal solution where there are more equivalence classes than are necessary and which can be viewed as a mixture of good decision and bad decisions. By choosing a subset of the decisions made in reaching a suboptimal solution an iterative technique can lead to an optimal solution, using series of steadily improved suboptimal solutions. The goal is to reach an optimal solution as quickly as possible. Various techniques for choosing the decision subset are evaluated.

Authors:

• Jeffrey L. Duffany

Keywords:

• np-complete
• complexity
• algorithm.

References:

[1] S. Cook and D. Mitchell. (1997). "Finding Hard Instances of the
Satisfiability Problem: A Survey", Satisfiability Problems: Theory and
Applications. American Mathematical Society, Providence, Rhode
Island.
[2] C.H. Papadimitrou and K. Steiglitz, "Combinatorial Optimization:
Algorithms and Complexity", Dover, ISBM 0-486-40258-4, pp. 344.
[3] Duffany, J.L., "Statistical Characterization of NP-Complete Problems",
Foundations of Computer Science Conference, World Computer
Congress, Las Vegas, Nevada, July 14-17, 2008.
[4] Duffany, J.L. "Systems of Inequations", 4th LACCEI Conference,
Mayaguez, PR, June 21-23, 2006.
[5] Duffany, J.L. "Generalized Decision Function and Gradient Search
Technique for NP-Complete Problems", XXXII CLEI Conference,
Santiago Chile, August 20-23, 2006.
[6] E. Horowitz, S. Sahni, "Fundamentals of Computer Algorithms",
Computer Science Press, Maryland, 1978.
[7] R. M. Karp, "Reducibility Among Combinatorial Problems", In
Complexity of Computer Computation, pages 85-104. Plenum Press,
New York, 1972.
[8] Weisstein, E. W., "NP-Hard Problem." From MathWorld--A Wolfram
Web Resource. http://mathworld.wolfram.com/NP-HardProblem.html.
[9] Weisstein, E. W., "NP-Complete Problem." From MathWorld--A
Wolfram Web Resource: http:// mathworld.wolfram.com/NPCompleteProblem.
html.
[10] Weisstein, E. W., "Inequation." From MathWorld--A Wolfram Web
Resource. http:// mathworld.wolfram.com /Inequation.html.
[11] Wikipedia - Inequation page: http://en.wikipedia.org /wiki/Inequation.
[12] Duffany, J.L., "Optimal Solution of Constraint Satisfaction Problems",
International Conference on Applied Computer Science, Sharm el Sheik,
Egypt, January, 2009.
[13] http://mat.gsia.cmu.edu/COLOR/instances.html
[14] http://www.r-project.org