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

26/04/05DMI - Università di Catania 1

Combinatorial Landscapes

Giuseppe Nicosia

University of Catania

Department of Mathematics and Computer Science

[email protected]

www.dmi.unict.it/~nicosia


1. Combinatorial LandscapesThe notion of landscape is among the rare existing concepts which help to understand the behaviour of search algorithmsthe behaviour of search algorithms and heuristics and to characterize the difficultyto characterize the difficulty of a combinatorial problem.


Search SpaceGiven a combinatorial problem PP, a search search spacespace associated to a mathematical formulation of PP is defined by a couple (S,f)(S,f)

– where SS is a finite set of configurations (or nodes or points) and

– ff a cost functioncost function which associates a real number to each configurations of SS.

For this structure two most common measures are the minimum and the maximum coststhe minimum and the maximum costs.In this case we have the combinatorial combinatorial optimization problemsoptimization problems.


Example: K-SATAn 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

– SS={ (T,T), (T,F), (F,T), (F,F) } and

– the cost functionthe cost function for 2-SAT computes only the number of satisfied clauses

ffsatsat (s)= #SatisfiedClauses(F,s), s (s)= #SatisfiedClauses(F,s), s S S


An example of Search Space

Let we consider F = (A B) ( A B)

1F T

2F F

2T F

1T Tfsat(F,s) A B


Search Landscape• Given a search space (S,f)(S,f), a search search

landscapelandscape is defined by a triplet (S,n,f)(S,n,f) where nn is a neighborhood functionneighborhood function which verifies

n : S n : S 2 2SS -{ 0} -{ 0}

• This landscape, also called energy landscapeenergy landscape, can be considered as a neutralneutral one since no search process is involved.

• It can be conveniently viewed as weighted weighted graphgraph G=(S, n , F) where the weights are defined on the nodes, not on the edges.


Example and relevance of Landscape

The search Landscape for the K-SAT problem is a N dimensional hypercubeN dimensional hypercube with

N = number of variables = |V| .

• Combinatorial optimization problems are often hard to solvehard to solve since such problems may have huge and complex search landscapehuge and complex search landscape.


Hypercubes


Solvable & Impossible•The New York Times, July 13, 1999 “Separating Separating Insolvable and DifficultInsolvable and Difficult”.

• B. Selman, R. Zecchina, et al.“Determing computational complexity from characteristic ‘phase transitions’ ”, Nature, Vol. 400, 8 July 1999,


Phase Transition, =4.256=4.256


Characterization of the Landscape in terms of Connected Components

Number of solutions, number of connected components and CCs' cardinality versus for #3$-SAT#3$-SAT problem with n=10n=10 variables.


CC's cardinality at phase transition (3)=4.256(3)=4.256

Number of Solutions, number of connected components and CC's cardinality at phase transition (3)=4.256(3)=4.256 versus number of variables nn for #3-SAT problem#3-SAT problem.


Process LandscapeProcess LandscapeGiven a search landscape (S, n, f), a process process landscapelandscape is defined by a quadruplet (S, n, f, (S, n, f, )) where is a search processsearch 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 AlgorithmsEvolutionary Algorithms: Genetic Algorithms, Genetic Programming, Evolution Strategies, Evolution Programming, Immune Algorithms.


References• G. Nicosia, V. Cutello, Noisy Channel and Reaction-Diffusion Noisy Channel and Reaction-Diffusion

Systems: Models for Artificial Immune SystemsSystems: Models for Artificial Immune Systems, to appear in Lecture Notes in Computer Science LNCS/LNAI 2003.

• G. Nicosia, V. Cutello, M. Pavone, A Hybrid Immune Algorithm A Hybrid Immune Algorithm with Information Gain for the Graph Coloring Problemwith Information Gain for the Graph Coloring Problem, to appear in Lecture Notes in Computer Science LNCS/LNAI 2003.

• G. Nicosia, V. Cutello, Multiple Learning using Immune Multiple Learning using Immune AlgorithmsAlgorithms, Proceedings of the 4th International Conference on Recent Advances in Soft Computing, RASC 2002, pp. 102-107, Nottingham, UK, 12 -13 December 2002.

• G. Nicosia, V. Cutello, An Immunological approach to An Immunological approach to Combinatorial Optimization ProblemsCombinatorial Optimization Problems,Lecture Notes in Computer Science, LNAI 2527 pp. 361-370, 2002.

Documents

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