Random Constraint Satisfaction: Flaws and Structure

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Random Constraint Satisfaction
Flaws and Structure

• I.P. Gent, E. MacIntyre, P. Prosser, B. M. Smith,
  and T. Walsh
• Presented by Qin Wang
• Nov. 27, 2003

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Outline

• 1. Motivation of this paper
• 2. Introduction of CSP
• 3. Flaws
• 4. Structure
• 5. Conclusions

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1 Motivation

• Achlioptas provided a negative result for all
  four random models
• They prove that if then , as
  ,
• there almost always exists a flawed
  variable.
• Flawed variable Every value for it is flawed
• Flawed value the value is inconsistent with
  every value of an adjacent variable

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1 Motivation (Cont.)

• A problem with a flawed variable cannot have a
  solution.
• This paper studies the impact of Achlioptas
  theoretical result on experimental basis

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2.1 Constraint Satisfaction Problem

• Set of variables X1, X2, , Xn
• Each variable Xi has a domain Di of possible
  values
• Usually Di is discrete and finite
• Set of constraints C1, C2, , Cp
• Each constraint Ck involves a subset of variables
  and specifies the allowable combinations of
  values of these variables
• Specifically, a binary Constraint Satisfaction
  Problem satisfies that each constraint defines
  the allowable values of a pair of variables
• Decide if there is an assignment of values to
  variables such that all constraints are satisfied

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2.2 Conflict Matrix and Constraint Graph for
Binary CSP

• Conflict Matrix (0-1)
• Describe the constraint between the variables x
  and y
• The (i, j) entry is 0, iff the i th value for x
  is incompatible with the j th value for y and 1
  otherwise
• Constraint graph G
• Vertices variables
• Edges two variables appear together in a
  constraint

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2.3 Four models of random problems

• Model A Independently select each one of the
  n(n-1)/2 possible edges with pr. p1, and for each
  selected edge, pick one of m2 possible pairs of
  values, independently with pr. p2, as being
  incompatible
• Model B Randomly select exactly p1n(n-1)/2
  edges, and for each selected edge, randomly pick
  exactly p2m2 pairs of values as incompatible
• Model C Independently select each one of the
  n(n-1)/2 possible edges with pr. p1, and for each
  selected edge, randomly pick exactly p2m2 pairs
  of values as incompatible
• Model D Randomly select exactly p1n(n-1)/2
  edges, and for each selected edge, pick one of m2
  possible pairs of values, independently with pr.
  p2, as being incompatible

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2.3 Four models of random problems (Cont.)

• Step1 Generating a constraint graph G
• Step2 Generating conflict matrices for edges
  in G
• The 2 steps of the four models differ in how to
  generate the constraint graph and how to choose
  incompatible values.

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2.3 Four models of random problems (Cont.)

• Using tuple ltn, m, p1, p2gt to describe problems
• n number of variables
• m uniform domain size
• p1 the density of the constraint graph
• p2 tightness of the constraints

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2.4 Model E a model can overcome the deficiency
of model A to D, but

• Achlioptas propose another random problem class,
  model E, which has better asymptotic properties
  than models A to D
• In model E the constraint graph emerges from the
  nogoods selected, and cannot be independently
  controlled
• Models A to D generate the constraint graph and
  constraint matrices separately. Thus they give
  much better flexibility in the range of instance
  types that can be generated

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3.1 Past Experimental Practice

• The survey of the literature from 1994 to 1997
• showing that many past studies may have been
  compromised by flaws, which is consistent with
  Achlioptass result

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3.2 Probability of Flawed Variables

• Assumption
• Each variable is connected to exactly p1(n-1)
  others
• The probabilities that the different variables
  have at least one unflawed value are independent

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3.2 Probability of Flawed Variables (Cont.)

• The probability that a problem has a flawed
  variable is (for model A)
• 1-(1-(1-(1-(Pr value inconsistent with a value
  of adjacent variable )m) p1(n-1) )m)n
• 1-(1-(1-(1-(p2)m) p1(n-1) )m)n

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3.3 Occurrence of Flawed Variable(1) in
model B
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3.3 Occurrence of Flawed Variable(2) in
model B
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4.1 Flawless Random Problem Generation

• Standard models of random CSP allow flawed
  values
• Flawed values can cause flawed variables
• Flawed variables cause trivial insolubility

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4.1.1 Flawless model

• Add structure to generation models to eliminate
  flaws
• Basic idea each value is supported by at least
  one unique value

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4.1.1 Flawless model (Cont.)

• A conflict matrix is flawless if there is a
  permutation of 1,2,,m such that all the
  pairs of values
  are allowed
• A flawless matrix must be arc consistent, since
  the value always supports value i (But
  the converse is not true!)

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4.1.2 Using flawless method on existing models

• For models B and C, we choose a random
  permutation of 1,2,3,m. The set of goods
  based on this permutation is
  . A conflict matrix
  that contains these goods cannot give a flawed
  value
• For models A and D, the process is similar,
  except that having removed a set of goods, we
  increase p2 to
• mp2 / (m-1) before selecting conflicts

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4.2 Theory of Flawlessness

• Theorem
• If a binary CSP with uniform domain size
  contains only flawless constraints with p2lt1/2,
  and each component in the constraint graph
  contains at most one cycle, the instance is
  soluble

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Proof of the Theorem

• 1. If the constraint graph is acyclic, using
  Lemma 1 to prove it.
• 2. Otherwise, consider a constraint graph
  containing a single component which contains
  exactly one cycle.
• 3. Then apply 2 to each component of a graph in
  turn

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4.2.1 Two Corollaries

• Corollary 1. Problems generated according to
  flawless model B or C at any value of p2lt1/2 do
  not suffer asymptotically from trivial
  insolubility
• Corollary 2. Problems generated according to
  standard model B or C at any value of p2lt1/m do
  not suffer asymptotically from trivial
  insolubility

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4.2.2 Two surprising results

• p2 1/m characterizes the region of trivial
  insolubility in standard models B and C
• P2 1/2 is the higher bound of trivial
  insolubility for flawless methods

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4.3 Experimental Comparison of Flawless and
Flawed Models (1)
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4.3 Experimental Comparison of Flawless and
Flawed Models (2)
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4.3 Experimental Comparison of Flawless and
Flawed Models (3)
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4.3 Experimental Comparison of Flawless and
Flawed Models (4)
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4.4 Structured Constraint Graphs

• Real problems are different from the Random
  problems
• Real problems can contain structures that occur
  very rarely in the random models
• Example
• 1994 exam time-tabling problem (59 nodes/485
  edges)
• Quasigroup completion (n2 nodes/ n2(n-1)edges )
  2n cliques, size n

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4.4.1 Quasigroup problem vs unstructured flawless
model B problems(1)
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4.4.1 Quasigroup problem vs unstructured flawless
model B problems(2)
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4.4.2 Exam timetabling prob. vs unstructured
flawless model B problems(1)
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4.4.2 Exam time-tabling prob. vs unstructured
flawless model B problems(2)
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4.5 Ideas inspired from experiment

• The search cost of structured problems is very
  different from that seen with existing random
  models
• Quasigroup constraint graph is harder than purely
  random problems, while the time-tabling graph
  gives easier problems than the random problems
  (Why?)

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5. Conclusion

• Achlioptas et al. point that if
  then , as , there almost always
  exists a flawed variable. This result is tight
  for model B and C
• Structures can be introduced into the conflict
  matrices to make them flawless. Thus we can
  generate problems that are not trivially
  insoluble
• Experiments and theory results can benefit
  greatly from each other

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References

• I.P. Gent, E. MacIntyre, P. Prosser, B. M. Smith,
  and T. Walsh. Random Constraint Satisfaction
  Flaws and Structure. 2001
• D. Achlioptas, L.M. Kirousis, E. Kranasis, D.
  Krizanc, et al. Random Constraint Satisfaction A
  More Accurate Picture. In Proc. CP97. Springer,
  1997

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End

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