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Combinatorial Landscapes

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Combinatorial Landscapes Giuseppe Nicosia University of Catania Department of Mathematics and Computer Science nicosia_at_dmi.unict.it www.dmi.unict.it/~nicosia – PowerPoint PPT presentation

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Title: Combinatorial Landscapes


1
Combinatorial Landscapes
  • Giuseppe Nicosia
  • University of Catania
  • Department of Mathematics and Computer Science
  • nicosia_at_dmi.unict.it
  • www.dmi.unict.it/nicosia

2
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.
3
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.

4
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 V2 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

5
(No Transcript)
6
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.

7
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.

8
Hypercubes
9
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,

10
Phase Transition, ?4.256
11
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 n10 variables.
12
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.
13
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 search algorithm.
  • 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.

14
References
  • 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 2003.
  • 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 2003.
  • G. Nicosia, V. Cutello, Multiple Learning using
    Immune Algorithms, 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 Combinatorial Optimization Problems,Lecture
    Notes in Computer Science, LNAI 2527 pp. 361-370,
    2002.
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