Non-binary constraints

1. Non-binary constraints Toby Walsh Dept of CS University of York England

2. Non-binary constraints • Bibliography Bacchus & van Beek, AAAI-98 Chen, PhD thesis, UAlberta, 2000 Dechter, AAAI-90 Mohr & Masini, ECAI-88 Regin, AAAI-94 Stergiou & Walsh, AAAI-99 & IJCAI-99 ….

3. Non-binary constraints • Major contribution to competitiveness of CP • despite academic focus on binary CSPs! • Some classic examples: all-different, cardinality, ...

4. Non-binary constraints • Compact & declarative problem specifications • that often run with minimal “tweaking” • Efficient algorithms for enforcing high levels of consistency • Regin’s specialized algorithm for a k-ary all-different constraint • O(k^2d^2) compared compared to O(d^k) for a generic algorithm

5. Non-binary constraints • Comparison with binary constraints: • binary models without additional vars (binary decompositions) • binary models with additional hidden vars (binary encodings) • Theoretical and empirical results

6. Binary decompositions • Certain non-binary constraints decompose into binary constraints on same vars • Sometimes called “network decomposable”

7. Binary decompositions • Two examples: • all-different(x1,x2,x3) = x1<> x2, x1<>x3, x2<>x3 • monotone(x1,x2,x3) = x1 < x2 < x3 • One non-example: • even(x1+x2+x3)

8. Binary decompositions • Theoretical comparison direct • compare pruning of vars in binary decomposition with that in non-binary • Empirical experiments reinforce theory • decomposing non-binary constraints can add orders of magnitude to solution cost

9. Binary consistencies • Arc-consistency (AC) • any assignment can be extended to a second • Path-consistency (PC) • any pair of assignments can be extended to a third • Strong path-consistency (strong PC) • AC and PC

10. Binary consistencies • Path-consistency rarely used in practice • time complexity • adds new (binary) constraints • Path-consistency can be achieved with a (generalized) arc-consistency algorithm • see later

11. Binary consistencies • Between AC and PC • inverse and singleton consistencies • just prune values (no additional binary constraints) • Experiments suggest can be useful • good pruning/time

12. Binary consistencies • Path inverse consistency (PIC) • any assignment can be extended to two more vars • Neighbourhood inverse consistency (NIC) • any assignment can be extended to all vars in immediate neighbourhood

13. Binary consistencies • Singleton arc-consistency (SAC) • after any assignment, problem can be made AC • Restricted path-consistency (RPC) • problem is AC, and if any assignment is consistent with just a single value for a 2nd var then any 3rd var has a consistent value

14. Binary consistencies • Simple relationships between them strongPC > SAC > PIC > RPC > AC NIC > PIC NIC ~ SAC NIC ~ strongPC

15. Non-binary consistencies • Generalized arc-consistency (GAC) • any assignment can be extended to all other vars in same constraint • Efficient algorithms exist for GAC on certain non-binary constraints • e.g. Regin’s all-different algorithm

16. Non-binary consistencies • GAC v AC {1,2} x1 {1,2} {1,2} x3 x2 x1<>x2, x1<>x3, x2<>x3 AC all-different(x1,x2,x3) not GAC

17. Non-binary consistencies • GAC can be used to enforce path-consistency • given a binary CSP • construct a ternary CSP’ • CSP is PC iff CSP’ is GAC

18. Non-binary consistencies • Given CSP has: • vars xi with values a,b,c,... • binary constraints R(a,b) • Construct CSP’ with: • vars xij with values <a,b>,... • ternary constraints R’(<a,b>,<b,c>,<c,a>) iff R(a,b) & R (b,a) & R(c,a)

19. Non-binary consistencies • Forward checking • forward checking (FC) on binary constraints enforces limited form of AC • most recently instantiated var and all futures vars are made AC • Choices about how to generalize this to non-binary constraints

20. Non-binary consistencies • nFC0 • any constraint with one un-instantianted var made AC • nFC1 • apply one pass of AC to each constraint and projection involving current var and one future var • … nFC5

21. Binary decompositions • Upper and lower bound on FC nFC1 on non-binary > FC on decomposition > nFC0 on non-binary • Gaps can be exponential Consider n-ary all-different with n-1 values nFC1 takes (n-1) branches FC on decomposition takes (n-1)! branches

22. Binary decompositions • GAC lower bound GAC on non-binary > AC on decomposition Gap again can be exponential But if we decompose too much, GAC=AC! • GAC upper bound In general, GAC ~ NIC, GAC ~ PIC .. BUT if decomposition to clique, NIC > GAC

23. Binary decompositions • Tighter results provable for stricter classes • Tree decomposable constraints • constraint graph is tree • Triangle preserving constraints • non-binary constraints on all triangles

24. Binary decompositions • Tree decomposable constraints • e.g. monotone(x1,x2,x3) • GAC=AC • not surprising as AC enough to solve! • Decomposition here doesn’t lose us anything • but even one cycle is enough for GAC>AC

25. Binary decompositions • Triangle preserving decomposition e.g. all-different(x1,x2,x3), quasigroups, ... GAC > PIC, gap can again be exponential GAC ~ SAC, strongPC • PIC is very strong consistency to be achieving at each node • GAC can do even better than this! • decomposition carries a very large price

26. Binary decompositions • Experimental results • quasigroup completion • quasigroup existence • Quasigroup is a Latin square • completion is completing partially filled square • existence is finding one with additional properties

27. Binary decompositions • Modelling the quasigroup problem • n^2 vars, each with domain of size n • Non-binary model • 2n all-different constraints (one for each row and column • Binary decomposition • 2n cliques of not-equals constraints

28. Binary decompositions • Quasigroup completion • Gomes & Selman report “heavy-tailed” distributions • Maintaining AC on binary decomposition • problems often take long time to solve • Maintaining GAC on all-different • almost all problems trivial

29. Binary decompositions • Quasigroup existence • best paper at IJCAI-93 • of interest to design theory • Open results first proved by computer • in some cases, only ever proved by computer • Maintaining GAC very competitive • compared to specialized model finders like FINDER, SEM

30. Intermission • Mid-point of talk • Short commercial break CSPLib: a benchmark library for constraints http://csplib.cs.strath.ac.uk

32. Why build CSPLib? • Benchmarking • shift field from random to more realistic problems • Modelling • Show-case • Challenges

33. Why not build CSPLib? • Over-fitting • CSPLib needs to be large & extensible • Unrepresentative • CSPLib needs your “realistic” problems • No standard modelling language • CSPLib uses natural language • Hard work • CSPLib needs to be a “community” effort

34. What does CSPLib contain? • Over 20 problem types in 5 categories • bin packing (e.g. vessel loading) • scheduling (e.g. basketball tournament) • frequency assignment (e.g. Golomb rulers) • combinatorics (e.g. perfect square) • puzzles (e.g. nonagrams) • Other useful links • solvers, other benchmark libraries, ...

35. What does CSPLib contain? • Example entry: • NL specification • results • references • (data) for instances • code • anything else of use “A Golomb ruler is a set of m integers, 0=a1<a2<..<am such that the m(m-1)/2 differences, aj-ai distinct. Such a ruler is of length am. The objective is to find the smallest length ruler…”

36. How to add to CSPLib? • Send us your entries. NOW! • it takes just 1/2 hour • Everyone has at least one problem to contribute • get others to solve your open problems • a problem shared is a problem solved • Email problems to csplib@cs.strath.ac.uk

37. End of intermission Now back to ... Binary encodings

38. Binary encodings • Not all non-binary constraint decompose into binary constraints • on the same set of variables • Consider again • even(x1+x2+x3) • But binary CSPs NP-complete

39. Binary encodings • Every non-binary constraint can be encoded into binary constraints • using polynomial number of additional (hidden) variables • Two popular encodings • hidden variable encoding • dual encoding

40. Binary encodings • Hidden variable encoding • add hidden var for each non-binary constraint • Dual encoding • add hidden var for each non-binary constraint • throw away original variables

41. Binary encodings • Dual encoding • consider c1:even(x1+x2), c2:odd(x2+x3) {00,11} c1 R21 R21=<00,01> or <11,10> {01,10} c2

42. Binary encodings • Hidden variable encoding • consider c1:even(x1+x2), c2:odd(x2+x3) {00,11} c1 r1 r2 {0,1} {0,1} {0,1} x1 x2 x3 r1=<0*,0> or <1*,1> r2=<*0,0> or <*1,1> r1 r2 {01,10} c2

43. Binary encodings • Hidden var encoding -> dual encoding • compose relations, discard original vars {00,11} c1 R21 = r2 + r1 {0,1} {0,1} x1 R21 x3 {01,10} c2

44. Binary encodings • Double encoding • dual + hidden var encoding • original vars + hidden vars • all constraints of dual and of hidden • Also called “combined encoding”

45. Binary encodings • Theoretical analysis complicated by: • encoding “builds” in GAC for hidden vars • must translate between (original and hidden) vars • pruning in dual can infer large arity nogoods in original

46. Binary encodings • Hidden var encoding FC on hidden ~ nFC0 on original each can be exponentially better than the other • FC+ propogates through hidden vars FC+ on hidden = nFC1 on original

47. Binary encodings • Hidden var encoding AC on hidden = GAC on original • Before looking for efficient (specialized) GAC algorithm • try AC on hidden var encoding

48. Binary encodings • No point doing NIC on hidden var encoding NIC on hidden = AC on hidden due to star shaped topology of constraint graph • Higher consistencies remain distinct strongPC on hidden > SAC on hidden > NIC, AC on hidden

49. Binary encodings • Dual encoding FC on dual ~ nFC0 on original each can be exponentially better than the other • Dual better for tight constraints • domains for hidden vars then small

50. Binary encodings • Dual encoding AC on dual > GAC on original • BUT domains of hidden vars very large when non-binary constraints loose • AC on dual prohibitively expensive