Eliminating non-binary constraints

1. Eliminating non-binary constraints Toby Walsh Cork Constraint Computation Center

2. Eliminating non-binary constraints • Two methods • Encodings: Replace with binary constraints by introducing new vars • Decompositions: (For restricted class of non-binary constraints) Replace with binary constraints on same variables

3. Meta-motivation • Theoretical results informative • Comparing non-binary constraint propagation with binary • Suggests where non-binary constraints are valuable

4. Bibliography • Lots of valuable results! 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 Gent, Stergiou & Walsh, Artificial Intelligence 2000 Bacchus, Chen, van Beek, Walsh, Artificial Intelligence 2002 ….

5. Let’s start with the easy case! Decomposable constraints: Non-binary constraints that can be represented by binary constraints with introducing new variables It’s a special case that sometimes occurs about which we can be (theoretically) quite precise

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) is x1\=x2, x1\=x3, x2\=x3 • monotone(x1,x2,x3) is x1 < x2 < x3 • One non-example: • even(x1+x2+x3) • Can you see why not?

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 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

10. 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

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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

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

25. 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

26. 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

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

28. 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

29. 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

30. 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

31. Binary encodings • Dual v hidden encoding BT on dual = FC on hidden AC on dual > AC on hidden SAC on dual > SAC on hidden strongPC on dual = strongPC on hidden • Path consistency is enough to get through the hidden star!

32. Binary encodings • Experimental results • crossword puzzles • Golomb rulers • random CSPs • random 3-SAT

33. Binary encodings • Crossword puzzles • “Original” model • vars for letters, domains {A-Z} • Dual model • vars for words, domains = dictionary • Dual at least 1,000 times faster on larger problems

34. Binary encodings • Golomb ruler • set of ticks, xi on a ruler • all inter-tick distances, dij different • Ternary constraints, dij=xi-xj • encoded using hidden var/double encoding • Competitive with non-binary model • runs on any binary CSP solver!

35. Binary encodings • Random CSPs and SAT • Bacchus and van Beek plot contours to show constraint tightness where encoding pays • gives good predictions for performance on structured problems like crosswords, … • number of satisfying tuples in constraint is most important factor in such predictions

36. Conclusions • Non-binary v binary decompositions • GAC on non-binary can be stronger than PIC on decomposition • Non-binary v binary encodings • GAC on non-binary = AC on hidden • AC on dual > GAC on non-binary

37. Conclusions • Non-binary v binary decompositions • decomposition can add significantly to search cost • Non-binary v binary encodings • encoding pays in practice on tight constraints

38. Future directions? • Mixed models • e.g. dual encoding for only some of the (non-binary) constraints • Optimization • objective function typically non-binary