1 / 10

On Algorithms for Decomposable Constraints

On Algorithms for Decomposable Constraints. Kostas Stergiou Ian Gent, Patrick Prosser, Toby Walsh A . P . E . S . Research Group. Decomposable non-binary constraints. they can be represented by binary constraints on the same set of variables some decomposable constraints

linh
Télécharger la présentation

On Algorithms for Decomposable Constraints

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. On Algorithms for Decomposable Constraints Kostas Stergiou Ian Gent, Patrick Prosser, Toby Walsh A.P.E.S. Research Group

  2. Decomposable non-binary constraints • they can be represented by binary constraints on the same set of variables • some decomposable constraints • all-different -> binary “not-equals” constraints • monotonicity -> binary ordering constraints All-different(v0,...,v7)

  3. Forward Checking • forward checking (FC) • remove from future variablesthe values that are inconsistent with the current instantiation • generalized FC • various definitions of FC for non-binary constraints (nFC0-nFC5) see Bessiere et. al., CP’99 • each generalization achieves a different level of consistency between past and future variables 1,2 future variable  future variable 1  current variable 0,1,2

  4. Forward Checking on Decomposable Constraints • FC > nFC0 in visited nodes • the difference can be exponential • nFC1 > FC in visited nodes • the difference can be exponential • nFC1 ~ FC ~ nFC0 in consistency checks • but there can be the same exponential differences nFC4 nFC5 nFC2 nFC1 FC nFC0 nFC3

  5. Arcconsistency • Arcconsistency (AC) • remove froma variable the values that are incompatible with all values of the other variable in a binary constraint • Generalized AC • remove froma variable the values that are incompatible with all n-tuples of the other variables in a n-ary constraint • Maintaing AC (GAC) is the most popular complete search algorithm 1,2    1 0,1,2

  6. The complexity of arcconsistency • The complexity of AC on e “not-equal” constaints with d domain size is O(e) • the previous bound was O(ed) • each edge is processed at most once • process is done in constant time • This generalizes to all antifunctional constraints • each value in a variable is unsupported by at most one value of the other variable in the constraint • All-different constraint on k variables • AC -> O(k2) GAC -> O(k2d2)

  7. GAC on Decomposable Constraints • GAC stronger than AC on binary decomposition • Exponential differences in node visits between MAC - MGAC • GAC incomparable to • strong path consistency (PC) • restricted PC (RPC) • singleton AC (SAC) • path inverse consistency (PIC) • neighbourhood inverse consistency (NIC) binary not-equals are AC ternary all-different is not GAC GAC strong PC SAC PIC RPC AC NIC

  8. Singleton arcconsistency • Singleton arcconsistency(SAC) • remove a value a of variable x if the instantiation xa results in a problem that is not AC • SAC is stronger than AC • SingletonGAC (SGAC) • SAC for non-binary constraints • SGAC is stronger than GAC 0,1   x  0,1,2 0,1 0,1    x0 0,1 SAC removes 0 from x

  9. SGAC on decomposable constraints • SGAC stronger than SAC, PIC, RPC on binary decomposition • SGAC incomparable to strongPC, NIC • SGAC is useful for preprocessing • Expensive to maintain during search SGAC strong PC SAC PIC RPC AC NIC

  10. Conclusions • Generalizations of FC, AC and SAC can achieve high levels of consistency • The complexity of AC on antifunctional constraints is O(e) • Non-binary representations offer considerable advantages over binary ones • decompositions can significantly reduce the level of consistency • Representation of problems can have large effect on efficiency of search

More Related