Global%20Constraints

1
Global Constraints

• Toby Walsh
• National ICT Australia and
• University of New South Wales
• www.cse.unsw.edu.au/tw

2
Course outline

• Introduction
• All Different
• Lex ordering
• Value precedence
• Complexity
• GAC-Schema
• Soft Global Constraints
• Global Grammar Constraints
• Roots Constraint
• Range Constraint
• Slide Constraint
• Global Constraints on Sets

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Global constraints

• Hardcore algorithms
• Data structures
• Graph theory
• Flow theory
• Combinatorics
• Computational complexity
• Global constraints are often balanced on the
  limits of tractability!

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Global constraints are NP-hard

• Can solve 3SAT using a single global constraint!
• Given 3SAT problem in N vars and M clauses
• 3SAT(X1,Xn) where nN3M2
• Constraint holds iff X1N, X2M,
• X_2i is 0/1 representing value assigned to xi
• X_2N3j, X_2N3j1 and X_2N3j2 represents
• jth clause

5
Our hammer

• Use tools of computational complexity to study
  global constraints

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Limits of Global Constraints

• Enforce lesser consistency
• Constraints cannot be combined
• Constraints cannot be generalized
• Decomposition will hurt pruning

Also study limits of reasoning with symmetries,
nogoods, meta-constraints like cardinality
7
Questions to ask?

• GACSupport? is NP-complete
• Does this value have support?
• Basic question asked within many propagators
• MaxGAC? is DP-complete
• Are these domains the maximal generalized
  arc-consistent domains?
• Termination test for a propagator
• DP NP u coNP
• Propagation harder than solving the problem!

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Questions to ask?

• IsItGAC? is NP-complete
• Are the domains GAC?
• Wakeup test for a propagator
• NoGACWipeOut? is NP-complete
• If we enforce GAC, do we not get a wipeout?
• Used in many reductions
• GACDomain? is NP-hard
• Return the maximal GAC domains
• What a propagator actually does!

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Relationships between questions

• NoGACWipeOut GACSupport GACDomain
• NoGACWipeOut in P lt-gt GACDomain in P
• NoGACWipeOut in NP lt-gt GACDomain in NP
• GACDomain in P gt MaxGAC in P gt IsItGAC in P
• IsItGAC in NP gt MaxGAC in NP gt GACDomain in NP
• Open if arrows cannot be reversed!

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Constraints in practice

• Some constraints proposed in the past are
  intractable
• NValues(N,X1,..Xn)
• AtMost1(S1,..,Sn)
• Card(C1,..,Cn,N)
• CardPath(N,X1,..,Xn,C)

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NValues

• NValues(N,X1,..,Xm)
• N values used in X1,,Xm
• Useful for resource allocation

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NValues

• NValues(Y,X1,..,Xn)
• Reduction of 3SAT to NValues
• 3SAT problem in N vars, M clauses
• Xi in i,-i for 1 i N
• XNs in i,-j,k if s-th clause is (i or -j or
  k)
• Y N
• Hence 3SAT has a solution gt NoGACWipeOut answers
  yes

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NValues

• NValues(N,X1,..,Xm)
• Reduction of 3SAT to NValues
• 3SAT problem in n vars, l clauses
• Xi in i,-i for 1 i n
• Xns in i,-j,k if s-th clause is (i or -j or
  k)
• N n
• Hence 3SAT has a solution ltgt NoGACWipeOut
  answers yes
• Enforce lesser level of local consistency (e.g.
  BC)

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Composing constraints

• Take some tractable constraints
• E.g. Intersect(S1,,Sn,k), S1c, ,Snc
• Could we combine them?
• E.g. FixedCardIntersect(S1,,Sn,k,c)
• No, NP-hard to enforce BC!
• AtMost1 with cardinality bounds on set variables
  is FixedCardIntersect
• Used in social golfers problem

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Generalizing constraints

• Take a tractable constraint
• GCC(X1,..,Xn,l1,..,lm,u1,..,um)
• Value j occurs between lj and uj times in
  X1,..,Xn
• Generalize some constants to variables
• E.g. GCC(X1,..,Xn,O1,..,Om)
• NP-hard to enforce GAC!

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Generalizing constraints

• GCC(X1,..,Xn,O1,..,Om)
• Reduction from 1in3SAT on positive clauses
• If jth clause is (x or y or z) then Xj in x,y,z
• If x occurs k times in all clauses then Ox in
  0,k
• Hence 1in3SAT has a solution iff NoGACWipeOut
  answers yes
• Thus enforcing GAC is NP-hard

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Decomposing constraints

• Consider a global constraint that is NP-hard to
  propagate
• E.g. AtMost1(S1,,Sn) with cardinality constraint
  on sets
• Decompose into polynomial number of constraints
• Si intersect Sj 1 for all iltj
• Since it is polynomial to propagate this
  decomposition
• Decomposition hinders propagation (assuming P?NP)

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Decomposing constraints

• Consider a global constraint that is NP-hard
• E.g. AtMost1(S1,,Sn)
• BC on sets GAC on characteristic function
• Reduction from 3SAT
• Sets have cardinality 2
• If jth clause is x or -y or z, introduce
• mj subseteq Sj subseteq mj, xj, -yj,
  zj
• Sj mj, xj if x satisfied jth clause
• Suppose kth clause contains -x, introduce two set
  vars
• mj subseteq Tjk subseteq mj, xj, -xk
• -xk subseteq Ujk subseteq mj, mk, -xk

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Decomposing constraints

• Consider a global constraint that is NP-hard
• E.g. AtMost1(S1,,Sn)
• BC on sets GAC on characteristic function
• Reduction from 3SAT
• Sets have cardinality 2
• If jth clause is x or -y or z, introduce
• mj subseteq Sj subseteq mj, xj, -yj,
  zj
• Sj mj, xj if x satisfied jth clause
• Suppose kth clause contains -x, introduce two set
  vars
• mj subseteq Tjk subseteq mj, xj, -xk
• -xk subseteq Ujk subseteq mj, mk, -xk
• Tjkmj,-xk, Ujkmk,-xk and
  Sk/mk,xk

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Symmetry breaking

• Symmetry is a major problem
• Symmetric machines, symmetric orders
• Often matrix of decision vars
• Row and Cols symmetric

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Symmetry breaking

• Add (global) constraints to eliminate symmetries
• Prune symmetric branches of search tree
• E.g. lex order rows, lex order cols
• Can we break all row col symmetry this way?

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Symmetry breaking

• Add (global) constraints to eliminate symmetries
• Prune symmetric branches of search tree
• E.g. lex order rows, lex order cols
• Can we break all row col symmetry this way?
• Suppose we had a global constraint that broke all
  row and column symmetry
• Enforcing GAC on such a constraint is NP-hard
• Actually coNP-hard

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Meta-constraints

• Put together existing constraints
• Card(N,C1,,Cm) iff N of C1,..Cm are satisfied
• implements conjunction, disjunction, negation
• conjunction is Nm
• disjunction is Ngt0
• negation is N0 and m1
• But NP-hard to propagate
• Typically delayed in most solvers

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Meta-constraints

• Special case of Card is CardPath
• Useful for sequencing problems
• CardPath(C,X1,..Xn,N) iff C(Xi,..Xik) holds N
  times
• E.g. number of changes is CardPath(/,X1,..Xn,N
  )
• Fixed parameter tractable
• k fixed, GAC takes O(ndk) time
• k O(n), GAC is NP-hard even when C is
  polynomial to test

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Meta-constraints

• CardPath(C,X1,..Xn,N) iff C(Xi,..Xik) holds N
  times
• Reduce 3SAT in N variables and M clauses to
  CardPath where kN2
• NM vars Xi to represent repeated truth assignment
• M vars Yj to represent jth clause
• C(X1,..,XN,Yj,X1) iff Yjk and Xk1 and X1X1
• or Yj-k and Xk0 and X1X1
• C(X2,..,XM.Yj,X1,X2) iff X2X2
• ..

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Learning

• Arc-consistency is essentially learning (unary)
  nogoods
• Same tools apply to learning
• NoGood? is coNP-complete
• MinimalNoGood? is DP-complete

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Conclusions

• Computational complexity is a useful hammer to
  study global constraints
• Uncovers fundamental limits of reasoning with
  global constraints
• Lesser consistency needs to be enforced
• Decomposition hurts pruning
• Composition intractable
• Generalization intractable
• ..