Symmetry Breaking Constraints in Constraint Programming

1. Symmetry Breaking Constraints in Constraint Programming Barbara Smith

2. Outline • Constraint satisfaction problems • Symmetry in CSPs • Symmetry breaking approaches • Symmetry breaking constraints • Lex leader constraints • Partial symmetry breaking • Matrix models Oberwolfach August 2010

3. Introduction to Constraint Satisfaction Problems • A CSP consists of: • A set of variables • Each with a finite set of possible values, its domain • A set of constraints: each constraint acts on a subset of the variables and restricts the possible assignments to those variables • Formally, a constraint is a subset of the possible assignments to the variables in its scope • In practice, constraints are usually stated intensionally, e.g. x < y • A solution to a CSP is an assignment of a value from its domain to every variable, that satisfies all the constraints • Optimization Oberwolfach August 2010

4. Constraint Examples • ax < by + c • arithmetic constraints involving variables and constants • either x1 < y1or x2 < y2 • logical constraints can express the logic of the problem directly • x = (y < z) • reified constraints • ixi jyi≥ dj(xij , yi , djconstants or variables) • constraints on arrays of variables • allDifferent(x1, x2, …, xn) • (the number of variables in x1, x2, …, xnwith value c) < a Oberwolfach August 2010

5. A = set of assignments made so far var = val var ≠ val Solving CSPs • Find a solution by combining systematic search and constraint propagation • At each search node, choose a variable not yet assigned (var) and assign it a value from its domain (val) • Create a choice between var = val and var ≠ val • On the left branch, propagate the constraintvar = val • domain pruning: remove values from the domains of future variables that are no longer valid • consider all problem constraints in turn • If any future variable has an empty domain, backtrack to try the choice var ≠ val • or if a solution has been found but we want all of them Oberwolfach August 2010

6. Symmetry • A (constraint) symmetry of a CSP P is a permutation of the variable-value pairs that preserves (the constraints of) P • and hence also preserves the solutions of P • A symmetry σ maps any variable-value pair (xi, j) to another, σ(xi, j) • A variable symmetry affects only the variables, so that (xi, j) is mapped to (xσ(i), j) • if the variables are x1, x2 …, xn, σ permutes the index set {1,2,…,n} • Here, I assume that we know the symmetry of the CSP (through inspection, intuition, inspiration, …) Oberwolfach August 2010

7. Symmetry Breaking Approaches • reformulate the problem (symmetry avoidance?) • modify the search algorithm (dynamic symmetry breaking) • e.g. add constraints on backtracking to a choice point to rule out symmetric equivalents in the future • often uses computational group theory • add constraints to the CSP before search Oberwolfach August 2010

8. Symmetry Breaking During Search (SBDS) • A symmetry can be eliminated by describing its effect on a variable-value assignment • e.g. for n-queens, we can completely eliminate all symmetry by describing the 7 symmetries • we can ignore the 8th symmetry (identity) • Variable r[i] represents the queen on row i • its values represent the columns (1 to n) x(r[i]=j) → r[n−i+1]=j y(r[i]=j) → r[i]=n−j+1 d1(r[i]=j) → r[j]=i d2(r[i]=j) → r[n−j+1]=n−i+i r90(r[i]=j) → r[j] = n−i+1 r180(r[i]=j) → r[n−i+1]=n−j+1 r270(r[i]=j) → r[n−j+1]=i Oberwolfach August 2010

9. A = set of assignments made so far var = val var ≠ val Symmetry Breaking During Search (SBDS) + g(var!= val) for any unbroken symmetry g, i.e. if g(A) is (or will be) true Oberwolfach August 2010

10. r[2]=4 r[2] ≠ 4 SBDS Example – 8 queens x: r[8]=2 y: r[1]=7 d1: r[2]=1 d2: r[7]=8 r90: r[2]=8 r180: r[8]=7 r270: r[7]=1 x x x x r[1]=2 r90: if r[2]=8 then r[4] ≠ 7 r180: if r[8]=7 then r[7] ≠ 5 r270: if r[7]=1 then r[5] ≠ 2 Oberwolfach August 2010

11. SBDS Pros and Cons • For complete symmetry breaking, SBDS needs a separate function for each symmetry • The symmetry group can be enormous • In a few special cases, complete symmetry breaking can be achieved with a small set of SBDS functions • e.g. if the variables represent n interchangeable objects, we can eliminate the symmetry with SBDS functions for the transpositions of pairs of objects, rather than all n! elements of the group • Otherwise, we can specify a subset of the symmetry functions and get partial symmetry breaking • Still useful to reduce search • Or we can link SBDS to something like GAP Oberwolfach August 2010

12. Symmetry Breaking Constraints • Add constraints to the CSP that will be satisfied by (ideally) only one assignment in any equivalence class • Creates a new CSP, with fewer solutions • Disadvantages: • Dangerous, unless done systematically • Can conflict with the search order • Advantage: • Constraints are useful & can lead to domain pruning • Anyone can do it Oberwolfach August 2010

13. Lex-Leader Constraints • Crawford, Ginsberg, Luks & Roy (KR96) introduced a symmetry-breaking method (in SAT) that is easily adapted to variable symmetries • Choose an order of the variables, say x1, x2 …, xn • and the values, but here we assume that domains are integer • Defines a lexicographic order on complete assignments to the variables • Add constraints to allow only the smallest element in each symmetry equivalence class, the lex-leader • or the largest Oberwolfach August 2010

14. Lex-Leader Constraints (II) • For each variable symmetry σ, add a constraint: • the assignment to x1, x2 …, xn is lexicographically smaller than the assignment to xσ(1) , xσ(2) ,…, xσ(n) , i.e. • x1≤ xσ(1) ; • if x1 =xσ(1) then x2≤ xσ(2) ; • if x1 = xσ(1) and x2 = xσ(2) then x3≤ xσ(3) ;….. • if xi = xσ(i) for 1 ≤ i < n then xn≤ xσ(n) • x1 x2 … xn≤lexxσ(1)xσ(2) … xσ(n) • Frisch et al. (CP02, AIJ 2006) give a propagator for the ≤lex constraint • Linear in the length of the constraint Oberwolfach August 2010

15. Lex Constraint Example • Suppose a CSP has variables x1, x2 …, x6 • A variable symmetry σ is a permutation of {1,2,3,4,5,6}, say (1 3) (2 6 5) • Choose the variable order x1, x2 …, x6 • The lex-leader constraint for σ is: • x1 x2 … x6≤lexxσ(1)xσ(2) … xσ(6) • x1 x2 x3 x4 x5 x6≤lex x3 x6 x1 x4 x2 x5 • x1 x2 x5≤lex x3 x6 x2 Oberwolfach August 2010

16. Simplifying Lex-Leader Constraints • The lex-leader constraint for σ can be reduced (Flener & Pearson, SymCon02) • If a variable is not affected by σ, remove it • If σ contains a cycle of variable indices, remove the last pair of variables from this cycle • NB If σ contains only 2-cycles, the reduced lex-leader constraint has one pair of variables for each 2-cycle • Reduction rules for sets of lex-leader constraints as well Oberwolfach August 2010

17. Consistency and Completeness • If we add a lex-leader constraint for every symmetry of the CSP, we are guaranteed: • Consistency: at least one solution in every equivalence class of solutions • Completeness: at most one solution in every equivalence class • But if the symmetry group is too big, we can add some lex-leader constraints - partial symmetry breaking • May have to sacrifice completeness • Sometimes, a small set of lex-leader constraints does give complete symmetry breaking, e.g. if variables x1 , x2, … , xn are interchangeable, add x1 ≤ x2≤ … ≤ xn Oberwolfach August 2010

18. Matrix Models • Many problems can be modelled as CSPs with a matrix of variables • e.g. the covering array problem • In a b× 5 array, every subset of 3 columns must contain every combination of 0s and 1s • Minimize b • The rows and columns can be permuted • A typical case for partial symmetry breaking Oberwolfach August 2010

19. Row and Column Symmetries • For an m× n matrix with row and column symmetry there are m!n! symmetries • Too many to introduce a ≤lex constraint for each one • A practical approach: introduce ≤lex constraints for some of the symmetries • Which ones? • Need to strike a balance between • Breaking more symmetry → reducing search • Adding more constraints → increasing propagation effort Oberwolfach August 2010

20. Double Lex Ordering • Choose the symmetries that swap adjacent rows or adjacent columns • Choose row-wise variable order Oberwolfach August 2010

21. Double Lex Ordering • Choose the symmetries that swap adjacent rows or adjacent columns • Choose row-wise variable order • Swap rows i and i+1: • xi1…xij xi,j+1 …xin ≤lexxi+1,1…xi+1,j xi+1,j+1…xi+1,n • i.e. row i≤lex row i+1 Oberwolfach August 2010

22. Double Lex Ordering • Choose the symmetries that swap adjacent rows or adjacent columns • Choose row-wise variable order • Swap columns j and j+1: • x1j…xij xi+1,j… xnj ≤lexx1,j+1 … xi,j+1xi+1,j+1… xn,j+1 • i.e. column j ≤lex column j+1 Oberwolfach August 2010

23. Double Lex Ordering • i.e. the rows must be in increasing lexicographic order and so must the columns • Lex order is transitive, so constraints from the symmetries that swap non-adjacent rows or columns would be redundant • An example of reducing a set of constraints • Double-lex (lex2) constraints are commonly used in matrix models with row & column symmetry • They break row and column symmetry separately, but not symmetries that permute both rows and columns Oberwolfach August 2010

24. Lex-Leader Constraints & Partial Symmetry Breaking • Given the symmetry group of a CSP, or a subset of the symmetries… • …and a variable order… • …we can construct the lex-leader constraints systematically • e.g. double–lex constraints for matrix models with row & column symmetry are constructed from • The symmetries that swap adjacent rows or adjacent columns • Row-wise variable order Oberwolfach August 2010

25. Breaking More Symmetry in Matrices • Next step: consider symmetries that combine a row permutation and a column permutation • Which ones? Will it be quicker? • Symmetries that swap pairs of variables give compact lex-leader constraints • swapNext: the set of symmetries that swap a pair of adjacent rows and a pair of adjacent columns • mn− 1 constraints for an m × n matrix • each with m+n− 2 pairs of variables • swapAny: symmetries that swap any pair of rows and any pair of columns • O(m2n2) constraints Oberwolfach August 2010

26. Example: Covering Arrays • In a b× k array, every subset of 3 columns must contain every combination of 0s and 1s • Minimize b • Try increasing values of b until there is a solution Oberwolfach August 2010

27. Are they worthwhile? • Compared to lex2 : • Breaking the swapNext and swapAny symmetries can reduce search by a lot • Can reduce the run-time, if matrices are not too large • Maybe we can be more selective and choose a subset of the swapAny symmetries? Oberwolfach August 2010

28. Variable Order • To construct a set of lex-leader constraints, we have to choose a variable order • With a complete set of lex-leader constraints, we get exactly one solution from every equivalence class • With a partial set, we can get more than one • How many more? • It depends on chosen symmetries • …and on the variable order Oberwolfach August 2010

29. Example • Find symmetrically distinct n × n binary matrices • When n = 4, there are 317 • Use lex2constraints • Symmetries swap a pair of adjacent rows or adjacent columns, row-wise variable order • → 650 matrices • Use same symmetries, a different variable order • same number of lex-leader constraints, same size • → 10,354 matrices • Changing the variable order can give less (or more) symmetry-breaking Oberwolfach August 2010

30. Conclusions • Adding constraints to the CSP is a common way to reduce or eliminate symmetry • Almost all symmetry breaking constraints are lex-leader constraints • If we cannot break all the symmetry, there are choices to be made • Which symmetries to break • Variable order • Lex-smallest v. lex-largest • All can affect the number of solutions to the new CSP • Hence how close we come to complete symmetry breaking • Needs more work to understand these choices THE END Oberwolfach August 2010