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
• .

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Lets start with the easy case!

• Decomposable constraints
• Non-binary constraints that can be represented by
  binary constraints with introducing new variables
• Its a special case that sometimes occurs about
  which we can be (theoretically) quite precise

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Binary decompositions

• Certain non-binary constraints decompose into
  binary constraints on same vars
• Sometimes called network decomposable

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Binary decompositions

• Two examples
• all-different(x1,x2,x3) is x1\x2, x1\x3,
  x2\x3
• monotone(x1,x2,x3) is x1 lt x2 lt x3
• One non-example
• even(x1x2x3)
• Can you see why not?

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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

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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

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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

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Binary decompositions

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

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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

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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

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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

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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

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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

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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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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

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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

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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

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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!

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Binary encodings

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

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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

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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!

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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

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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

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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

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Future directions?

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