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D y n amic V irtual A rc C onsistency Hiep Nguyen 1 Christian Bessiere 2 Thomas Schiex 1 1 INRA-BIA UR875, Toulouse, France 2 Universit e de Montpellier Montpellier, France JFPC2013 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A

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SLIDE 1

Dynamic Virtual Arc Consistency

Hiep Nguyen1 Christian Bessiere2 Thomas Schiex1

1INRA-BIA

UR875, Toulouse, France

2Universit´

e de Montpellier Montpellier, France

JFPC’2013

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 1 / 13

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SLIDE 2

Problem

Weighted Constraint Satisfaction Problem (WCSPs)

Objectif

Use Dynamic AC to speed up the Virtual Arc Consistency

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 2 / 13

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SLIDE 3

Overview

Weighted CSPs Virtual Arc Consistency Dynamic Virtual Arc Consistency Experiments Conclusion

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 3 / 13

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SLIDE 4

Weighted CSPs

X = {1, ..., n} variables D = {d1, ..., dn} domains

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 4 / 13

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SLIDE 5

Weighted CSPs

X = {1, ..., n} variables D = {d1, ..., dn} domains W = {wS1, ..., wSe} cost functions

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 4 / 13

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SLIDE 6

Weighted CSPs

X = {1, ..., n} variables D = {d1, ..., dn} domains W = {wS1, ..., wSe} cost functions

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 4 / 13

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SLIDE 7

Weighted CSPs Minimimum cost assignment

X = {1, ..., n} variables D = {d1, ..., dn} domains W = {wS1, ..., wSe} cost functions

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 4 / 13

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SLIDE 8

Weighted CSPs Minimimum cost assignment

X = {1, ..., n} variables D = {d1, ..., dn} domains W = {wS1, ..., wSe} cost functions

◮ w∅ : 0-arity function that defines a LB on the cost of any solution. ◮ useful for Branch and Bound pruning ◮ significantly increased by Virtual Arc Consistency.

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 4 / 13

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SLIDE 9

Virtual Arc Consistency (VAC − [AIJ2010])

Bool(P)

Classic CSP induced by a WCSP P that authorizes

  • nly zero cost values and tuples (ignoring w∅).
  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 5 / 13

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SLIDE 10

Virtual Arc Consistency (VAC − [AIJ2010])

Bool(P)

Classic CSP induced by a WCSP P that authorizes

  • nly zero cost values and tuples (ignoring w∅).

2 2

1 1 1 1 P Bool(P)

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 5 / 13

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SLIDE 11

Virtual Arc Consistency (VAC − [AIJ2010])

Bool(P)

Classic CSP induced by a WCSP P that authorizes

  • nly zero cost values and tuples (ignoring w∅).

VAC

P is VAC iff the AC closure of Bool(P) is non-empty

2 2

1 1 1 1 P Bool(P)

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 5 / 13

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SLIDE 12

Virtual Arc Consistency (VAC − [AIJ2010])

Bool(P)

Classic CSP induced by a WCSP P that authorizes

  • nly zero cost values and tuples (ignoring w∅).

VAC

P is VAC iff the AC closure of Bool(P) is non-empty

If P is not VAC:

enforcing AC in Bool(P) leads to a wipe out ∃ a way of shifting costs in P which leads to an increase of w∅.

2 2

1 1 1 1 P Bool(P)

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 5 / 13

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SLIDE 13

Enforcing VAC

Iterative process

enforcing AC in Bool(P) until a wipe-out occurs transforming P into an equivalent problem with an increased w∅.

Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable.

a b x1 a b x2 a b x3 a b x4 Bool(P)

Constraint revision order : w13, w34, w12, w24

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 6 / 13

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SLIDE 14

Enforcing VAC

Iterative process

enforcing AC in Bool(P) until a wipe-out occurs transforming P into an equivalent problem with an increased w∅.

Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable.

a b x1 a b x2 a b x3 a b x4 Bool(P)

Constraint revision order : w13, w34, w12, w24

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 6 / 13

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SLIDE 15

Enforcing VAC

Iterative process

enforcing AC in Bool(P) until a wipe-out occurs transforming P into an equivalent problem with an increased w∅.

Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable.

2 a b x1 2 a b x2 a b x3 a b x4 1 1 1 1 w∅ = 0

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 6 / 13

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SLIDE 16

Enforcing VAC

Iterative process

enforcing AC in Bool(P) until a wipe-out occurs transforming P into an equivalent problem with an increased w∅.

Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable.

2 a b x1 2 a b x2 a b x3 a b x4 1 1 1 1 w∅ = 0

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 6 / 13

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SLIDE 17

Enforcing VAC

Iterative process

enforcing AC in Bool(P) until a wipe-out occurs transforming P into an equivalent problem with an increased w∅.

Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable.

1 a b x1 2 a b x2 a b x3 a b x4 1 1 1 1 1 1 w∅ = 0

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 6 / 13

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SLIDE 18

Enforcing VAC

Iterative process

enforcing AC in Bool(P) until a wipe-out occurs transforming P into an equivalent problem with an increased w∅.

Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable.

1 a b x1 2 a b x2 a b x3 a b x4 1 1 1 1 1 1 w∅ = 0

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 6 / 13

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SLIDE 19

Enforcing VAC

Iterative process

enforcing AC in Bool(P) until a wipe-out occurs transforming P into an equivalent problem with an increased w∅.

Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable.

1 a b x1 1 2 a b x2 a b x3 a b x4 1 1 1 1 w∅ = 0

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 6 / 13

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SLIDE 20

Enforcing VAC

Iterative process

enforcing AC in Bool(P) until a wipe-out occurs transforming P into an equivalent problem with an increased w∅.

Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable.

1 a b x1 1 a b x2 a b x3 a b x4 1 1 1 1 w∅ = 1

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 6 / 13

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SLIDE 21

Motivation for Dynamic VAC

VAC enforces AC on a sequence of incrementally modified CNs Maintaining Bool(P) by Dynamic AC ⇒ Dynamic VAC

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 7 / 13

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SLIDE 22

Dynamic VAC

Property

Each VAC iteration leads only to constraint relaxations in Bool(P)

Maintaining Bool(P) by dynamic AC (AC/DC2 − [FLAIRS 2005] )

Relaxation proceduce:

1

Restoring restorable values (see above)

2

Propagating restored values to neighborhood (new support)

3

Rechecking the restored values for AC

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 8 / 13

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SLIDE 23

Avantages of DynVAC

. Bool(P) Saving work by keeping viable values and some deleted values from the previous iterations

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 9 / 13

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SLIDE 24

Maintaining VAC in search

Going down

x = a, x ≤ a, x ≥ a: domain restriction, done by AC x = a: if (x, a) has a positive cost

◮ this cost goes directly to w∅ ◮ relaxation of Bool(P): done by relaxation proceduce

Going up

rebuilding all domains in Bool(P) thanks to the ”killer” data-structure (backtrackable)

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 10 / 13

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SLIDE 25

Experimentation: pre-processing

Cost Function Library DynVAC is faster than VAC for celar (1.6x), tagsnp (3x), warehouse (5x), but significantly slower for maxclique problems. A domain based heuristic handles those pathological cases.

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 11 / 13

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SLIDE 26

VAC DynVAC ratio time(sec) time(sec) time nodes Results in search CELAR cl6-2 29 46 1,59 1.17 gr11 470 366 0,78 0,65 gr13 1.431 1.144 0,8 0,51 sc06-18 1.511 736 0,49 0,77 sc06-20 765 508 0,66 1,26 sc06 5.227 2.454 0,47 0,79 ... WAREHOUSE capa 2.462 1.013 0,41 1,00 capb 3.019 6.168 2,04 1,35 capc 2.027 1.228 0,61 0,75 capmo5 75 14 0,19 0.74 capmq1 5.111 2.374 0,46 0,68 capmq2 6.520 3.209 0,49 1,05 ... average (47 prob) 1831 1117 0,61 0,83

Only some worse and best cases are presented.

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 12 / 13

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SLIDE 27

Conclusion

DynVAC

incremental algorithm for enforcing VAC in WCSPs faster than VAC for large costs and large domains problems a heuristic that gets rid of pathological cases

Perspective

extension to non-binary cost functions

  • H. Nguyen et al. (INRA and LIRMM)

Dynamic Virtual Arc Consistency JFPC’2013 13 / 13