A Relation-Algebraic View on Evolutionary Algorithms for Some Graph Problems

We take a relation-algebraic view on the formulation of evolutionary algorithms in discrete search spaces. First, we show how individuals and populations can be represented as relations and formulate some standard mutation and crossover operators for this representation using relation-algebra. Evaluating a population with respect to their constraints seems to be the most costly step in one generation for many important problems. We show that the evaluation process for a given population can be sped up by using relation-algebraic expressions in the process. This is done by examining the evaluation of possible solutions for three of the best-known NP-hard combinatorial optimization problems on graphs, namely the vertex cover problem, the computation of maximum cliques, and the determination of a maximum independent set. Extending the evaluation process for a given population to the evaluation of the whole search space we get exact methods for the considered problems, which allow to evaluate the quality of solutions obtained by evolutionary algorithms.

[1]  Frank Neumann,et al.  RelView - An OBDD-Based Computer Algebra System for Relations , 2005, CASC.

[2]  Dr. Zbigniew Michalewicz,et al.  How to Solve It: Modern Heuristics , 2004 .

[3]  C. R. Subramanian,et al.  Almost Optimal (on the average) Combinatorial Algorithms for Boolean Matrix Product Witnesses, Computing the Diameter (Extended Abstract) , 1998, RANDOM.

[4]  Gunther Schmidt,et al.  Relations and Graphs: Discrete Mathematics for Computer Scientists , 1993 .

[5]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[6]  Bryant A. Julstrom,et al.  Edge sets: an effective evolutionary coding of spanning trees , 2003, IEEE Trans. Evol. Comput..

[7]  Xin Yao,et al.  A comparative study of three evolutionary algorithms incorporating different amounts of domain knowledge for node covering problem , 2005, IEEE Trans. Syst. Man Cybern. Part C.

[8]  Dirk Beyer,et al.  Efficient relational calculation for software analysis , 2005, IEEE Transactions on Software Engineering.

[9]  Rudolf Berghammer,et al.  Implementation of Relational Algebra Using Binary Decision Diagrams , 2001, RelMiCS.

[10]  Frank Neumann,et al.  Relational Implementation of Simple Parallel Evolutionary Algorithms , 2005, RelMiCS.

[11]  I. Wegener.  Branching Programs and Binary Deci-sion Diagrams-Theory and Applications , 1987 .

[12]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

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

[14]  David W. Corne,et al.  A Powerful New Encoding for Tree-Based Combinatorial Optimisation Problems , 2004, PPSN.

[15]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .