Thierry PETIT

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Generalization of Constructive Disjunction for
Over Constrained Scheduling Problems
Thierry PETIT ILOG - LIRMM Jean-Charles
REGIN ILOG Christian BESSIERE LIRMM
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Over Constrained Problems

• No solution satisfies all the constraints
• Some constraints have to be relaxed
• Hard constraints must be satisfied
• Soft constraints can be relaxed

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Over Constrained Problems

• Activities A1, A2, A3 require a unary resource R
• Temporal constraints
• The duration of each Ai is at least 5
• A1 and A2 start before 3
• A3 ends at 12

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5
Time
3
12
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Over Constrained Problem

• Relaxation
• OC aspect in the
• propagation
• All the soft constraints
• are relaxed
• Global objective function
• on the relaxed constraints
• ? 1 over-constrained
• problem is solved

• Decomposition
• OC aspect in the search
• The relaxation is handled
• by hand
• - add Ci while fail
• - if fail then relax Ci
• ? n satisfaction
• problems are solved

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Over Constrained Problem

• Decomposition
• Realistic approach but no dedicated algorithms

• Relaxation
• Good existing algorithms but unrealistic
  approach how to characterize and provide
  solutions that satisfy the user ?

? mix the two approaches ?
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Applications

• Applications involving global soft constraints
• Each soft constraint affects widely the problem
• Example car-sequencing

Options of each car (HARD)
Sequence constraints (SOFT)
Hard
Soft
Number of cars (HARD)
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Applications

• Applications involving global soft constraints
• Difficult to solve with a pure relaxation
  approach
• Decomposition methods are more adapted

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Applications

• Application involving Local soft constraints
• Example scheduling

Resources (SOFT)
Hard
Soft
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Applications

• Application involving Local soft constraints
• Decomposition methods can be used
• Relaxation methods can be used

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Simple Relaxation Framework Max-CSP

• Goal to find a solution minimizing the number of
  violated constraints
• Example
• Activities A1, A2, A3
• C1 A2 starts after end of A1
• C2 A3 starts after end of A2
• C3 A1 starts after end of A3
• Optimal Solution 1 violation

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Constructive Disjunction

• Applications
• Over-constrained Scheduling Problems (generalized
  C.D.)
• General scheduling problems alternative
  resources
• Principle
• C1 ? ... ? Cn
• At least one constraint must be satisfied
• Values violating the n constraints can be removed

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Constructive Disjunction

• Example D(x)a,b,c
• C1 DC1(x) D(x)\a,b c
• C2 DC2(x) D(x)\a b,c
• C3 DC3(x) D(x)\a,b c
• C1 ? C2 ? C3
• D(x) D(x)\a,b?a?a,b DC1(x) ?
  DC2(x) ? DC3(x) b,c

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Generalized Constructive Disjunction

• C1 ? ... ? Cn
• At least k constraints must be satisfied (?
  Max-CSP)
• Values violating a number of constraints v gt
  n-k can be removed

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Generalized Constructive Disjunction

• Example D(x)a,b,c
• C1 DC1(x) D(x)\a,b c
• C2 DC2(x) D(x)\a b,c
• C3 DC3(x) D(x)\a,b c
• C1 ? C2 ? C3 and k 2
• v(x, a) 3 a can be removed from D(x)
• v(x, b) 2 b can be removed from D(x)
• D(x) c

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Generalized Constructive Disjunction

• v(x,2) 1
• 1 ? n-k ? 2 is not removed from D(x)
• Although
• Any solution violates either C2 or C3
• (x,2) violates C1

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How to Improve this Filtering?

• Basic idea to take into account constraints
  between all the variables different from x when
  filtering D(x)
• Lower bound LB of the number of violations
  involved by these constraints
• Global LB nothing to do with SAC!

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How to Improve this Filtering?

• Replace
• if v(x,a) gt n-k then D(x)?
  D(x)\a
• By
• if v(x,a) LB gt n-k then D(x)?
  D(x)\a

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Variable Based Lower Bound

• Binary case Freuder and Wallace, 1992 Larrosa,
  Meseguer and Schiex, 1999
• Direct violations of constraints by values
• a directly violates C means a is not consistent
  with C
• For each a in D(x)
• inc(x,a) number of direct violations of value
  a
• inc(x) min(inc(x,a)), a ? D(x)
• LB particular sum of all inc(x)

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Variable Based Lower Bound

• Problem
• When the two variables x and y involved in a
  constraint are future the same violation must not
  be counted twice when including inc(x) and inc(y)
  into the sum
• Solution
• Directed graph of constraints

C is taken into account only in inc(x)
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Variable Based Lower Bound

• v(x,2)1
• LB 1
• inc(y) 1
• inc(z) inc(t) 0
• v(x,2) LB 2 gt n-k
• 2 can be removed from D(x)

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Variable Based Lower Bound

• Non binary case Régin, Petit, Bessière and
  Puget, 2000-2001
• Binary case directed constraint graph
• Non-binary case variable-based partition of the
  constraint set
• P p(x),p(y),...
• p(x) C1,C3,...
• ? x,y p(x) ? p(y) ?
• A constraint C ? p(x) is taken into account only
  when computing inc(x)

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Recapitulation

• inc(y) is computed only with constraints of p(y)
• LB ? inc(y), y ? x
• Filtering theorem
• if v(x,a) LB gt n-k then D(x)? D(x)\a

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Ignored Constraints Example 1
Value 2 of each domain does not directly violate
any constraint inc(x)inc(y)inc(z) 0
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Conflict Set Based Lower Bound Régin, Petit,
Bessière and Puget, 2000-2001

• A conflict set is a part of the constraint
  network which is not arc-consistent
• It is not possible to satisfy simultaneously all
  the constraints

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Conflict Set Based Lower Bound

• One conflict set (at least) one violation
• LBcs size of a set of disjoint conflict sets

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Conflict Set Based Lower Bound

• Smaller conflict sets ? greater LB
• Given a set of constraints Q
• Step 1 LB ? 0
• Step 2 Find a conflict set K in Q, if there is
  not return LB
• Step 3 K ? minimal conflict set from K
• LB
• Q ? Q - K
• goto step 2

Polynomial algorithm for computing a minimal
conflict set from a conflict set e.g. De
Siquiera and Puget, 1988, improved by Junker,
2001
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Combination of BoundsRégin, Petit, Bessière and
Puget, 2001

• Basic Idea
• Compute a variable-based bound LB
• Extract the set of constraints that are ignored
  in LB
• Compute a conflict-set based lower bound LBcs(I)
  on a subset I of these ignored constraints

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Combination of Bounds

• Ignored Constraints
• Ignored constraints are constraints that are not
  violated by a value a such that inc(x,a) inc(x)
• Problem
• Not trivial computation of the set I we cannot
  use all the ignored constraints for computing
  LBcs(I)
• I must be an independent set of ignored
  constraints

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Independent Set Of Ignored Constraints
p(x)C1,C2,C3,C4 inc(x) inc(x,a) inc(x,b)
1 C3,C4 is not independent if we consider
p(x) p(x)- C3,C4 then inc(x) 0 ? The
value of inc(x)changes! ? LB does not remain
valid without C3 and C4!
x
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Independent Set Of Ignored Constraints Greedy
Algorithm

• Idea to pick and remove ignored constraints one
  by one while this does not affect the
  variable-based lower bound LB
• For each set p(x) of the partition
• I ? ?
• While ? C ? p(x) such that p(x) p(x)\C
  satisfies inc(x)gt inc(x), add C to I and remove
  C from p(x)

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Summary

• Existing results (binary constraints)
• if v(x,a) LB gt n-k then
  D(x)? D(x)\a
• New result (general case)
• if v(x,a) LB LBcs(I) gt n-k then
  D(x)? D(x)\a