Combinatorial Landscapes & Evolutionary Algorithms

1. Combinatorial Landscapes &Evolutionary Algorithms Prof. Giuseppe Nicosia University of Catania Department of Mathematics and Computer Science nicosia@dmi.unict.it www.dmi.unict.it/~nicosia DMI - Università di Catania

2. Talk Outline • Combinatorial Landscapes • Evolutionary Computing DMI - Università di Catania

3. 1. Combinatorial Landscapes The notion of landscape is among the rare existing concepts which help to understand the behaviour of search algorithms and heuristics and to characterize the difficulty of a combinatorial problem. DMI - Università di Catania

4. Search Space Given a combinatorial problem P, a search space associated to a mathematical formulation of P is defined by a couple (S,f) • where S is a finite set of configurations (or nodes or points) and • f a cost function which associates a real number to each configurations of S. For this structure two most common measures are the minimum and the maximum costs.In this case we have the combinatorial optimization problems. DMI - Università di Catania

5. Example: K-SAT An instance of the K-SAT problem consists of a set V of variables, a collection C of clauses over V such that each clause c  C has |c|= K. The problem is to find a satisfying truth assignment for C. The search space for the 2-SAT with |V|=2 is (S,f) where • S={ (T,T), (T,F), (F,T), (F,F) } and • the cost function for 2-SAT computes only the number of satisfied clauses fsat (s)= #SatisfiedClauses(F,s), s  S DMI - Università di Catania

6. An example of Search Space Let we consider F = (A  B)  ( A B) DMI - Università di Catania

7. Search Landscape • Given a search space (S,f), a search landscape is defined by a triplet (S,n,f) where n is a neighborhood function which verifies n : S  2S -{ 0} • This landscape, also called energy landscape, can be considered as a neutral one since no search process is involved. • It can be conveniently viewed as weighted graph G=(S, n , F) where the weights are defined on the nodes, not on the edges. DMI - Università di Catania

8. Example and relevance of Landscape The search Landscape for the K-SAT problem is a N dimensional hypercube with N = number of variables = |V| . • Combinatorial optimization problems are often hard to solve since such problems may have huge and complex search landscape. DMI - Università di Catania

9. Hypercubes DMI - Università di Catania

10. Solvable & Impossible • The New York Times, July 13, 1999 “Separating Insolvable and Difficult”. • B. Selman, R. Zecchina, et al.“Determing computational complexity from characteristic ‘phase transitions’ ”, Nature, Vol. 400, 8 July 1999, DMI - Università di Catania

11. Phase Transition, =4.256 DMI - Università di Catania

12. Characterization of the Landscape in terms of Connected Components Number of solutions, number of connected components and CCs' cardinality versus  for #3-SAT problem with n=10 variables. DMI - Università di Catania

13. CC's cardinality at phase transition (3)=4.256 Number of Solutions, number of connected components and CC's cardinality at phase transition (3)=4.256 versus number of variables n for #3-SAT problem. DMI - Università di Catania

14. Process Landscape Given a search landscape (S, n, f), a process landscape is defined by a quadruplet (S, n, f, ) where  is a search process. • The process landscape represents a particular view of the neutral landscape (S, n, f) seen by a searchalgorithm. • Examples of search algorithms: • Local Search Algorithms. • Complete Algorithms (e. g. Davis-Putnam algorithm). • Evolutionary Algorithms: Genetic Algorithms, Genetic Programming, Evolution Strategies, Evolution Programming, Immune Algorithms. DMI - Università di Catania

15. 2. Evolutionary Algorithms EAs are optimization methods based on an evolutionary metaphor that showed effective in solving difficult problems. “Evolution is the natural way to program”Thomas Ray DMI - Università di Catania

16. Evolutionary Algorithms 1. Set of candidate solutions (individuals): Population. 2. Generating candidates by: • Reproduction: Copying an individual. • Crossover:  2 parents   2 children. • Mutation: 1 parent  1 child. 3. Quality measure of individuals: Fitness function. 4. Survival-of-the-fittest principle. DMI - Università di Catania

17. Main components of EAs 1. Representation of individuals: Coding. 2. Evaluation method for individuals: Fitness. 3. Initialization procedure for the 1st generation. 4. Definition of variation operators (mutation and crossover). 5. Parent (mating) selection mechanism. 6. Survivor (environmental) selection mechanism. 7. Technical parameters (e.g. mutation rates, population size). Experimental tests, Adaptation based on measured quality, Self-adaptation based on evolution. DMI - Università di Catania

18. Mutation and Crossover EAs manipulate partial solutions in their search for the overall optimal solution . These partial solutions or `building blocks' correspond to sub-strings of a trial solution - in our case local sub-structures within the overall conformation. DMI - Università di Catania

19. Algorithm Outline procedure EA; { t = 0; initialize population (P(t), d); evaluate P(t); until (done) { t = t + 1; parent_selection P(t); recombine (P(t), pcross); mutate ( P(t), pmut); evaluate P(t); survive P(t); } } DMI - Università di Catania