Non-binary%20constraints

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
• Regins specialized algorithm for a k-ary
  all-different constraint
• O(k2d2) compared compared to O(dk) 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) x1ltgt x2, x1ltgtx3,
  x2ltgtx3
• monotone(x1,x2,x3) x1 lt x2 lt x3
• One non-example
• even(x1x2x3)

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 gt SAC gt PIC gt RPC gt AC
• NIC gt 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. Regins all-different algorithm

16
Non-binary consistencies

• GAC v AC

1,2
x1
1,2
1,2
x3
x2
x1ltgtx2, x1ltgtx3, x2ltgtx3
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 lta,bgt,...
• ternary constraints R(lta,bgt,ltb,cgt,ltc,agt) 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 gt FC on decomposition gt 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 gt AC on decomposition
• Gap again can be exponential
• But if we decompose too much, GACAC!
• GAC upper bound
• In general, GAC NIC, GAC PIC ..
• BUT if decomposition to clique, NIC gt 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)
• GACAC
• not surprising as AC enough to solve!
• Decomposition here doesnt lose us anything
• but even one cycle is enough for GACgtAC

25
Binary decompositions

• Triangle preserving decomposition
• e.g. all-different(x1,x2,x3), quasigroups, ...
• GAC gt 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
• n2 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

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Intermission

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

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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, ...

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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, 0a1lta2lt..
ltam 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
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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_at_cs.strath.ac.uk

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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(x1x2x3)
• 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

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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

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Binary encodings

• Dual encoding
• consider c1even(x1x2), c2odd(x2x3)

00,11
c1
R21
R21lt00,01gt or lt11,10gt
01,10
c2
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Binary encodings

• Hidden variable encoding
• consider c1even(x1x2), c2odd(x2x3)

00,11
c1
r1
r2
0,1
0,1
0,1
x1
x2
x3
r1lt0,0gt or lt1,1gt r2lt0,0gt or
lt1,1gt
r1
r2
01,10
c2
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Binary encodings

• Hidden var encoding -gt dual encoding
• compose relations, discard original vars

00,11
c1
R21 r2 r1
0,1
0,1
x1
x3
R21
01,10
c2
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Binary encodings

• Double encoding
• dual hidden var encoding
• original vars hidden vars
• all constraints of dual and of hidden
• Also called combined encoding

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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

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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

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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 gt SAC on hidden
• gt NIC, AC on hidden

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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 gt GAC on original
• BUT domains of hidden vars very large when
  non-binary constraints loose
• AC on dual prohibitively expensive

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Binary encodings

• Dual v hidden encoding
• BT on dual FC on hidden
• AC on dual gt AC on hidden
• SAC on dual gt SAC on hidden
• strongPC on dual strongPC on hidden
• Path consistency is enough to get through the
  hidden star!

52
Binary encodings

• Experimental results
• crossword puzzles
• Golomb rulers
• random CSPs
• random 3-SAT

53
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

54
Binary encodings

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

55
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

56
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 gt GAC on non-binary

57
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

58
Future directions

• Modelling with non-binary constraints
• many more models possible
• role of auxiliary vars and implied constraints
• Different levels of consistency within a
  non-binay model
• e.g. GAC on all-different, but bounds consistency
  on arithmetic constraints

59
Future directions

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