Exploring Langford's Problem: Models and Solutions

1. Case study 7: Langford’s problem Model due to Barbara Smith

2. Outline • Introduction • Langford’s problem • Modelling it as a CSP • Basic model • Refined model • Experimental Results • Conclusions

3. Recipe • Create a basic model • Decide on the variables • Introduce auxiliary variables • For messy/loose constraints • Consider dual, combined or 0/1 models • Break symmetry • Add implied constraints • Customize solver • Variable, value ordering

4. Langford’s problem • Prob024 @ www.csplib.org • Find a sequence of 8 numbers • Each number [1,4] occurs twice • Two occurrences of i are i numbers apart • Unique solution • 41312432

5. Langford’s problem • L(k,n) problem • To find a sequence of k*n numbers [1,n] • Each of the k successive occrrences of i are i apart • We just saw L(2,4) • Due to the mathematician Dudley Langford • Watched his son build a tower which solved L(2,3)

6. Langford’s problem • L(2,3) and L(2,4) have unique solutions • L(2,4n) and L(2,4n-1) have solutions • L(2,4n-2) and L(2,4n-3) do not • Computing all solutions of L(2,19) took 2.5 years! • L(3,n) • No solutions: 0<n<8, 10<n<17, 20, .. • Solutions: 9,10,17,18,19, .. A014552 Sequence: 0,0,1,1,0,0,26,150,0,0,17792,108144,0,0,39809640,326721800, 0,0,256814891280,2636337861200

7. Basic model • What are the variables?

8. Basic model • What are the variables? Variable for each occurrence of a number X11 is 1st occurrence of 1 X21 is 1st occurrence of 2 .. X12 is 2nd occurrence of 1 X22 is 2nd occurrence of 2 .. • Value is position in the sequence

9. Basic model • What are the constraints? • Xij in [1,n*k] • Xij+1 = i+Xij • Alldifferent([X11,..Xn1,X12,..Xn2,..,X1k,..Xnk])

10. Recipe • Create a basic model • Decide on the variables • Introduce auxiliary variables • For messy/loose constraints • Consider dual, combined or 0/1 models • Break symmetry • Add implied constraints • Customize solver • Variable, value ordering

11. Break symmetry • Does the problem have any symmetry?

12. Break symmetry • Does the problem have any symmetry? • Of course, we can invert any sequence!

13. Break symmetry • How do we break this symmetry?

14. Break symmetry • How do we break this symmetry? • Many possible ways • For example, for L(3,9) • Either X92 < 14 (2nd occurrence of 9 is in 1st half) • Or X92=14 and X82<14 (2nd occurrence of 8 is in 1st half)

15. Recipe • Create a basic model • Decide on the variables • Introduce auxiliary variables • For messy/loose constraints • Consider dual, combined or 0/1 models • Break symmetry • Add implied constraints • Customize solver • Variable, value ordering

16. What about dual model? • Can we take a dual view?

17. What about dual model? • Can we take a dual view? • Of course we can, it’s a permutation!

18. Dual model • What are the variables? • Variable for each position i • What are the values?

19. Dual model • What are the variables? • Variable for each position i • What are the values? • If use the number at that position, we cannot use an all-different constraint • Each number occurs not once but k times

20. Dual model • What are the variables? • Variable for each position i • What are the values? • Solution 1: use values from [1,n*k] with the value i*n+j standing for the ith occurrence of j • Now want to find a permutation of these numbers subject to the distance constraint

21. Dual model • What are the variables? • Variable for each position i • What are the values? • Solution 2: use as values the numbers [1,n] • Each number occurs exactly k times • Fortunately, there is a generalization of all-different called the global cardinality constraint (gcc) for this

22. Global cardinality constraint • Gcc([X1,..Xn],l,u) enforces values used by Xi to occur between l and u times • All-different([X1,..Xn]) = Gcc([X1,..Xn],1,1) • Regin’s algorithm enforces GAC on Gcc in O(n^2.d) • Regin’s papers are tough to follow but this seems to beat his algorithm for all-different!?

23. Dual model • What are the constraints? • Gcc([D1,…Dk*n],k,k) • Distance constraints?

24. Dual model • What are the constraints? • Gcc([D1,…Dk*n],k,k) • Distance constraints: • Di=j then Di+j+1=j

25. Combined model • Primal and dual variables • Channelling to link them • What do the channelling constraints look like?

26. Combined model • Primal and dual variables • Channelling to link them • Xij=k implies Dk=i

27. Solving choices? • Which variables to assign? • Xij or Di

28. Solving choices? • Which variables to assign? • Xij or Di, doesn’t seem to matter • Which variable ordering heuristic? • Fail First or Lex?

29. Solving choices? • Which variables to assign? • Xij or Di, doesn’t seem to matter • Which variable ordering heuristic? • Fail First very marginally better than Lex • How to deal with the permutation constraint? • GAC on the all-different • AC on the channelling • AC on the decomposition

30. Solving choices? • Which variables to assign? • Xij or Di, doesn’t seem to matter • Which variable ordering heuristic? • Fail First very marginally better than Lex • How to deal with the permutation constraint? • AC on the channelling is often best for time

31. Conclusions • Modelling is an art but there are patterns • Develop basic model • Decide on the variables and their values • Use auxiliary variables to represent constraints compactly/efficiently • Consider dual, combined and 0/1 models • Break symmetry • Add implied constraints • Customize solver for your model

32. Case study 8: social golfers problem Model again due to Barbara Smith

33. Outline • Introduction • Social golfers problem • Modelling it as a CSP • Basic model • Refined model • Experimental Results • Conclusions

34. Social golfers problem • Prob013 @ www.csplib.org • 32 golfers wish to play in 8 groups of 4 each week • No two play in the same group more than once • How many weeks can they play?

35. Social golfers problem • Prob013 @ www.csplib.org • 32 golfers wish to play in 8 groups of 4 each week • No two play in the same group more than once • How many weeks can they play? • 9 weeks and this is optimal

36. Social golfers problem • Of course, generalize problem to g groups of s players over w weeks • Kirkman’s schoolgirls’ problem [Lady’s & Gentleman’s Diary 1850] “… a schoolmistress was in the habit of taking her girls for a daily walk. The girls were 15 in number, and were arranged in 5 rows of 3 each, so that each girl might have 2 companions. The problem is to so dispose them so that for 7 consecutive days no girl will walk with any of her school-fellows in a triplet twice …” This is equivalent to social golfers problem of 5 groups of 3 players over 7 weeks

37. Recipe • Create a basic model • Decide on the variables • Introduce auxiliary variables • For messy/loose constraints • Consider dual, combined or 0/1 models • Break symmetry • Add implied constraints • Customize solver • Variable, value ordering

38. Basic model • What are the variables?

39. Basic model • What are the variables? • Group_ij is the set of s golfers assigned to group i in period j • But I haven’t shown you set variables before!

40. Set variables • CSP variables can range over (finite) domains like integers • X1 is 1, 2, 3 or 4 • Or over sets of (finite) domains • Y1 is {}, {1}, {2}, or {1,2}

41. Set variables • CSP variables can range over (finite) domains like integers • X1 is 1, 2, 3 or 4 • Or over sets of (finite) domains • Y1 is {}, {1}, {2}, or {1,2} • Usually set operations can be posted as constraints on these set variables • Y1 subset Z1, Y1 intersect Z1 = {}, 1 in Y1, …

42. Set variables • Set variables are potentially expensive to reason about • If X1 is a subset of Y1, then X1 has exponentially many possible values • Compromise • CSP solvers just maintain upper and lower bounds on set variable • {} subseteq X1 subseteq {1,2}

43. Set variables • Set variables are potentially expensive to reason about • If X1 is a subset of Y1, then X1 has exponentially many possible values • Compromise • CSP solvers just maintain upper and lower bounds on set variable • {} subseteq X1 subseteq {1,2} • We loose the ability to represent disjunction E.g. X1= {1} or X1={2} but X1=/ {1,2}

44. Basic model • What are the variables? • Group_ij is the set of s golfers assigned to group i in period j • What are constraints?

45. Basic model • What are the variables? • Group_ij is the set of s golfers assigned to group i in period j • What are constraints? • Size of group, |Group_ij|=s • Groups do not overlap, Group_ij intersect Group_i’j={} • Never meet twice, for j<l . | Group_ij intersect Group_kl | <= 1

46. Break symmetry • What symmetry does the problem have?

47. Break symmetry • What symmetry does the problem have? • Lots! • Players are symmetrical • Groups are symmetrical • Weeks are symmetrical

48. Break symmetry • What symmetry does the problem have? • Lots! • Players are symmetrical • Groups are symmetrical • Weeks are symmetrical • Set variables saved us worrying about order within group

49. Break symmetry • We can assign some values and break some of this symmetry

50. Break symmetry • We can assign some values and break some of this symmetry • Make first week: {1,2,..s}, {s+1,s+2,..2s},…