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Combinatorial Landscapes


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Title: Combinatorial Landscapes

Combinatorial Landscapes
  • Giuseppe Nicosia
  • University of Catania
  • Department of Mathematics and Computer Science

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
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.

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

(No Transcript)
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.

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.

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

Phase Transition, ?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 problem with n10 variables.
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.
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
  • Evolutionary Algorithms Genetic Algorithms,
    Genetic Programming, Evolution Strategies,
    Evolution Programming, Immune Algorithms.

  • 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,