Global Constraints

1. Toby Walsh National ICT Australia and University of New South Wales www.cse.unsw.edu.au/~tw Global Constraints

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

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

4. 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 n=N+3M+2 • Constraint holds iff X1=N, X2=M, • X_2+i is 0/1 representing value assigned to xi • X_2+N+3j, X_2+N+3j+1 and X_2+N+3j+2 represents jth clause

5. Our hammer • Use tools of computational complexity to study global constraints

6. 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!

8. 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!

9. Relationships between questions • NoGACWipeOut = GACSupport = GACDomain • NoGACWipeOut in P <-> GACDomain in P • NoGACWipeOut in NP <-> GACDomain in NP • GACDomain in P => MaxGAC in P => IsItGAC in P • IsItGAC in NP => MaxGAC in NP => GACDomain in NP • Open if arrows cannot be reversed!

10. 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) • …

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

12. 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 • XN+s in {i,-j,k} if s-th clause is: (i or -j or k) • Y= N • Hence 3SAT has a solution => NoGACWipeOut answers “yes”

13. 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 • Xn+s in {i,-j,k} if s-th clause is: (i or -j or k) • N = n • Hence 3SAT has a solution <=> NoGACWipeOut answers “yes” • Enforce lesser level of local consistency (e.g. BC)

14. Composing constraints • Take some tractable constraints • E.g. Intersect(S1,…,Sn,k), |S1|=c,… ,|Sn|=c • 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

15. 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!

16. 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

17. 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 i<j • Since it is polynomial to propagate this decomposition • Decomposition hinders propagation (assuming P≠NP)

18. 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}

19. 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} • Tjk={mj,-xk}, Ujk={mk,-xk} and Sk/={mk,xk}

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

21. 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?

22. 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 …

23. Meta-constraints • Put together existing constraints • Card(N,[C1,…,Cm]) iff N of C1,..Cm are satisfied • implements conjunction, disjunction, negation … • conjunction is N=m • disjunction is N>0 • negation is N=0 and m=1 • But NP-hard to propagate • Typically delayed in most solvers

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

25. Meta-constraints • CardPath(C,[X1,..Xn],N) iff C(Xi,..Xi+k) holds N times • Reduce 3SAT in N variables and M clauses to CardPath where k=N+2 • NM vars Xi to represent repeated truth assignment • M vars Yj to represent jth clause • C(X1,..,XN,Yj,X1’) iff Yj=k and Xk=1 and X1=X1’ or Yj=-k and Xk=0 and X1=X1’ • C(X2,..,XM.Yj,X1’,X2’) iff X2=X2’ • ..

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

27. 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 • ..