Case study 3: orthogonal Latin squares

1. Case study 3: orthogonal Latin squares Modelled by Barbara Smith

2. Modelling decisions • Many different ways to model even simple problems • Combining models can be effective • Channel between models • Need additional constraints • Symmetry breaking • Implied (but logically) redundant

3. Orthogonal Latin squares • Find a pair of Latin squares • Every cell has a different pair of elements • Generalized form: • Find a set of m Latin squares • Each possible pair is orthogonal

4. 1 2 3 4 1 2 3 4 2 1 4 3 3 4 1 2 3 4 1 2 4 3 2 1 4 3 2 1 2 1 4 3 11 22 33 44 23 14 41 32 34 43 12 21 42 31 24 13 Two 4 by 4 Latin squares No pair is repeated Orthogonal Latin squares

5. History of (orthogonal) Latin squares • Introduced by Euler in 1783 • Also called Graeco-Latin or Euler squares • No orthogonal Latin square of order 2 • There are only 2 (non)-isomorphic Latin squares of order 2 and they are not orthogonal

6. History of (orthogonal) Latin squares • Euler conjectured in 1783 that there are no orthogonal Latin squares of order 4n+2 • Constructions exist for 4n and for 2n+1 • Took till 1900 to show conjecture for n=1 • Took till 1960 to show false for all n>1 • 6 by 6 problem also known as the 36 officer problem “… Can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 6 by 6 array such that no row or column duplicates a rank or a regiment?”

7. More background • Lam’s problem • Existence of finite projective plane of order 10 • Equivalent to set of 9 mutually orthogonal Latin squares of order 10 • In 1989, this was shown not to be possible after 2000 hours on a Cray (and some major maths) • Orthogonal Latin squares are used in experimental design • To ensure no dependency between independent variables

8. A simple 0/1 model • Suitable for integer programming • Xijkl = 1 if pair (i,j) is in row k column l, 0 otherwise • Avoiding advice never to use more than 3 subscripts! • Constraints • Each row contains one number in each square Sum_jl Xijkl = 1 Sum_il Xijkl = 1 • Each col contains one number in each square Sum_jk Xijkl = 1 Sum_ik Xijkl = 1

9. A simple 0/1 model • Additional constraints • Every pair of numbers occurs exactly once Sum_kl Xijkl = 1 • Every cell contains exactly one pair of numbers Sum_ij Xijkl = 1 Is there any symmetry?

10. Symmetry removal • Important for solving CSPs • Especially for proofs of optimality? • Orthogonal Latin square has lots of symmetry • Permute the rows • Permute the cols • Permute the numbers 1 to n in each square • How can we eliminate such symmetry?

11. Symmetry removal • Fix first row 11 22 33 … • Fix first column 11 23 32 .. • Eliminates all this symmetry?

12. What about a CSP model? • Exploit large finite domains possible in CSPs • Reduce number of variables • O(n^4) -> ? • Exploit non-binary constraints • Problem states that squares contain pairs that are all different • All-different is a non-binary constraint our solvers can reason with efficiently

13. CSP model • 2 sets of variables • Skl = i if the 1st element in row k col l is i • Tkl = j if the 2nd element in row k col l is j • How do we specify all pairs are different? • All distinct (k,l), (k’,l’) if Skl = i and Tkl = j then Sk’l’=/ i or Tk’l’ =/ j O(n^4) loose constraints, little constraint propagation! What can we do?

14. CSP model • Introduce auxiliary variables • Fewer constraints, O(n^2) • Tightens constraint graph => more propagation • Pkl = i*n + j if row k col l contains the pair i,j • Constraints • 2n all-different constraints on Skl, and on Tkl • All-different constraint on Pkl • Channelling constraint to link Pkl to Skl and Tkl

15. CSP model 3n^2 variables Domains of size n, n and n^2+n O(n^2) constraints Large and tight non-binary constraints 0/1 model n^4 variables Domains of size 2 O(n^4) constraints Loose but linear constraints Use IP solver! CSP model v O/1 model

16. Solving choices for CSP model • Variables to assign • Skl and Tkl, or Pkl? • Variable and value ordering • How to treat all-different constraint • GAC using Regin’s algorithm O(n^4) • AC using the binary decomposition

17. Good choices for the CSP model • Experience and small instances suggest: • Assign the Skl and Tkl variables • Choose variable to assign with Fail First (smallest domain) heuristic • Break ties by alternating between Skl and Tkl • Use GAC on all-different constraints for Skl and Tkl • Use AC on binary decomposition of large all-different constraint on Pkl

18. Performance

19. Dual CSP model • 4 subscripts in 0/1 model are interchangeable • Suggests a dual model • DSij = k if the pair (i,j) occurs in row k • DTij = l if the pair (i,j) occurs in the row l • DPij = k*n + l if the pair (i,j) occurs in row i col j • Dual constraints to the primal model

20. Combined model • Primal and dual model together • Channelling constraints to link them • But new search decisions • Do we assign both primal and dual variables? • How do handle dual constraints (AC, GAC …)? • … • Other dualities • Any choice of 2 subscripts from 4 • Diminishing returns

21. Dual performance

22. Conclusions • Many ways to model even simple problems • Introduce auxiliary variables • Reduce number of constraints, improve propagation • Combining models often beneficial • Channelling constraints link models • Need to deal with symmetry • Don’t always use GAC on all-different constraints

23. General methodology? • Choose a basic model • Consider auxiliary variables • To reduce number of constraints, improve propagation • Consider combined models • Channel between views • Break symmetries • Add implied constraints • To improve propagation

24. Case study 4: template design Again model due to Barbara Smith

25. Introduction • Prob002 at www.csplib.org • Problem comes from a printing firm • Cat food labels that need to be printed using templates • Several designs (tuna, chicken, …) go on each template • Different demand for each flavour • Aside: where did cats get the taste for tuna?

26. Template design problem • For Francesca’s benefit • How else can I get a cat picture into my talk?

27. Basic CSP model • Assume number of templates is fixed • Variables • Pij = number of slots on template i for design j • Ri = run length for template i • Constraints • Sum_j Pij = s, number of slots on each template • Sum_i Pij * Ri >= dj, total production equals demand

28. Basic CSP model • Optimization problem • Introduce variable to minimize • Production = Sum_i Ri • Solved as sequence of decision problems Production < l1, Production < l2 … l1 set to minimum number of printings with 1 template

29. Symmetry • Does the model have any symmetry? • If so, how can we eliminate it?

30. Symmetry • The templates are indistinguishable and can be permuted • Swap all designs on one template with all those on a second template • Break this symmetry by distinguishing the templates • R1 <= R2 <= R3 …

31. Symmetry • Designs j, k with the same demand are indistinguishable • We can break this symmetry • [P1j,P2j,P3j,…] <lex [P1k,P2k,P3k,…] • Efficient GAC algorithm for lex ordering constraint

32. Sort of symmetry? • Symmetries can be subtle to spot! • Consider designs j and j’ with demand for j less than for j’ • Suppose we produce more of j than j’ • We could swap j and j’ and still have solution • Prevent this with constraint on production • Sum_i Pij Ri <= Sum_i Pij’ Ri

33. Implied constraints • We should always look for implied constraints we can add to model • Encourage constraint propagation

34. Implied constraints • 2 templates • R1+R2 = Production • R1 <= R2 • Hence R1 <= Production/2 R2 >= Production/2

35. Implied constraints • 3 templates • R1 + R2 + R3 = Production • R1 <= R2 <= R3 • Hence R1 <= Production/3 R2 <= Production/2 R3 >= Production/3 …

36. Solving choices • Variable ordering • As with Golomb ruler, assign variables to construct solution systematically • Assign all designs on one template before moving on to a second template • Encourages constraint propagation on runlength constraints

37. Performance • Basic model • Difficult to solve 2 or 3 template problems • Full model • Problem solved quickly • Can solve much larger problems than feasible with the basic model • Optimality can still be tough!

38. Conclusions • Basic model often obvious • To refine such a model we need: • Consider dual/combined models • Symmetries eliminated • Implied constraints • Variable ordering heuristics • Hopefully you can start to see patterns in what we do!