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26/04/05 DMI - Università di Catania 1 Combinatorial Landscapes Giuseppe Nicosia University of Catania Department of Mathematics and Computer Science

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26/04/05 DMI - Università di Catania 1 1. Combinatorial Landscapes the behaviour of search algorithms to characterize the difficulty 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.

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26/04/05 DMI - Università di Catania 1 Search Space Psearch space P(S,f) Given a combinatorial problem P, a search space associated to a mathematical formulation of P is defined by a couple (S,f) S – where S is a finite set of configurations (or nodes or points) and – fcost function S – f a cost function which associates a real number to each configurations of S. the minimum and the maximum costs combinatorial optimization problems For this structure two most common measures are the minimum and the maximum costs.In this case we have the combinatorial optimization problems.

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26/04/05 DMI - Università di Catania 1 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 – S={ (T,T), (T,F), (F,T), (F,F) } and – the cost function – the cost function for 2-SAT computes only the number of satisfied clauses f sat (s)= #SatisfiedClauses(F,s), s S

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26/04/05 DMI - Università di Catania 1 An example of Search Space Let we consider F = (A B) ( A B) 1 F T 2 F 2 T F 1 T f sat (F,s) A B

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26/04/05 DMI - Università di Catania 1 Search Landscape (S,f)search landscape (S,n,f)n neighborhood function 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 2 S -{ 0} energy landscape neutral This landscape, also called energy landscape, can be considered as a neutral one since no search process is involved. weighted graph It can be conveniently viewed as weighted graph G=(S, n, F) where the weights are defined on the nodes, not on the edges.

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26/04/05 DMI - Università di Catania 1 Example and relevance of Landscape N dimensional hypercube The search Landscape for the K-SAT problem is a N dimensional hypercube with N = number of variables = |V|. hard to solve huge and complex search landscape Combinatorial optimization problems are often hard to solve since such problems may have huge and complex search landscape.

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26/04/05 DMI - Università di Catania 1 Hypercubes

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26/04/05 DMI - Università di Catania 1 Solvable & Impossible Separating Insolvable and Difficult 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,

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26/04/05 DMI - Università di Catania 1 =4.256 Phase Transition, =4.256

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26/04/05 DMI - Università di Catania 1 Characterization of the Landscape in terms of Connected Components #3$-SATn=10 Number of solutions, number of connected components and CCs' cardinality versus for #3$-SAT problem with n=10 variables.

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26/04/05 DMI - Università di Catania 1 (3)=4.256 CC's cardinality at phase transition (3)=4.256 (3)=4.256n #3-SAT problem 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.

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26/04/05 DMI - Università di Catania 1 Process Landscape process landscape(S, n, f, ) search process 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 search algorithm. Examples of search algorithms: – Local Search Algorithms. – Complete Algorithms (e. g. Davis-Putnam algorithm). – Evolutionary Algorithms – Evolutionary Algorithms: Genetic Algorithms, Genetic Programming, Evolution Strategies, Evolution Programming, Immune Algorithms.

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26/04/05 DMI - Università di Catania 1 References Noisy Channel and Reaction-Diffusion Systems: Models for Artificial Immune Systems G. Nicosia, V. Cutello, Noisy Channel and Reaction-Diffusion Systems: Models for Artificial Immune Systems, to appear in Lecture Notes in Computer Science LNCS/LNAI A Hybrid Immune Algorithm with Information Gain for the Graph Coloring Problem G. Nicosia, V. Cutello, M. Pavone, A Hybrid Immune Algorithm with Information Gain for the Graph Coloring Problem, to appear in Lecture Notes in Computer Science LNCS/LNAI Multiple Learning using Immune Algorithms G. Nicosia, V. Cutello, Multiple Learning using Immune Algorithms, Proceedings of the 4th International Conference on Recent Advances in Soft Computing, RASC 2002, pp , Nottingham, UK, December An Immunological approach to Combinatorial Optimization Problems G. Nicosia, V. Cutello, An Immunological approach to Combinatorial Optimization Problems,Lecture Notes in Computer Science, LNAI 2527 pp , 2002.

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