Local consistency in soft constraint networks

1
Local consistency in soft constraint networks

• Thomas Schiex
• Matthias Zytnicki
• INRA Toulouse
• France

Special thanks to Javier Larrosa UPC
Barcelona Spain
2
Overview

• Introduction and definitions
• Why soft constraints?
• Weighted CSP
• Existing approaches
• Soft as hard global soft constraints
• Soft as soft AC, DAC and FDAC
• Putting the 2 together (and more)
• Global soft constraints
• Bound consistency

3
Why soft constraints?

• CSP framework for decision problems
• Many problems are optimization problems
• Economics (combinatorial auctions)
• Given a set G of goods and a set B of bids
• Bid (Bi , Vi ), Bi requested goods, Vi value
• find the best subset of compatible bids
• Best maximize revenue (sum)

4
Why soft constraints?

• Satellite scheduling
• Spot 5 is an earth observation satellite
• It has 3 on-board cameras
• Given a set of requested pictures (of different
  importance)
• Resources, data-bus bandwidth, setup-times,
  orbiting
• select a subset of compatible pictures with max.
  importance (sum)

5
Why soft constraints

• Even in decision problems, the user may have
  preferences among solutions.
• It happens in most real problems.
• Experiment give users a few solutions and they
  will find reasons to prefer some of them.

6
Soft constraint network
Schiex, Fargier, Verfaillie 1995Bistarelli,
Rossi 95

• (X,D,C)
• Xx1,..., xn variables
• DD1,..., Dn finite domains
• Cc1,..., ce cost functions
• var(ci) scope
• ci(t) ? E (ordered cost domain, T, ?)
• Obj. Function F(X) ?ci (X)
• Solution F(t) ? T
• Soft CN find minimal cost solution

annihilator
identity

• commutative
• associative
• monotonic

7
Weighted CSP example (? )
Shapiro 81Freuder 92
For each vertex
x3
x2
x1
x4
For each edge
x5
F(X) number of non blue vertices
8
1st approach Soft as Hard
Petit, Regin, Bessiere 2000

• Soft constraint c reified in c with extra cost
  xc variable.
• (t,a) ? c iff a c(t)
• Define cost SUM(xc) (and more)
• This is it Now optimize cost.
• But how will propagation occur ?

9
Propagation a key issue
Larrosa, Meseguer, Schiex 1999Petit, Regin,
Bessiere 2001

• Use the PFC-MRDAC principles
• Associate each constraint to 1 of its var C(x)
• Compute inc(x,a) cost payed on C(x) if xa
• Sum(Min(inc(x,a))) is a lower bound on a cost
  variable associated with all soft constraints.
• Requires one artificial soft global constraint
  reifying all soft constraints with a single
  associated cost variable

10
Global soft constraints
Petit et al 2001van Hoeve et al.
2004Beldiceanu, Petit 2004

• All-diff, GCC can also be reified
• Enforcing performs domain consistency
• Removes values that have no supporting tuple in
  the constraint
• Value removal in xc and original variables

11
The CSP approach
Larrosa 2002

• T maximum violation.
• Can be set to a bounded max. violation k
• Solution F(t) lt k Top
• Empty scope soft constraint c? (a constant)
• Gives an obvious lower bound on the optimum
• If you do not like it c? ?
• Similar to xc of the soft global constraint.

12
A new operation on weighted networks
Schiex 2000

• Projection of cij on Xi with compensation
• Equivalence preserving transformation
• Can be reversed

v
v
1
0
0
0
w
w
0
1
i
j
13
Node Consistency (NC)
Larrosa 2002

• For all variable i
• ?a, C? Ci (a)ltT
• ? a, Ci (a) 0
• Complexity
• O(nd)

T
4
C?
0
1
x
v
3
2
z
w
0
v
2
0
1
1
0
w
1
v
1
0
1
w
1
y
14
Arc Consistency (AC)
Schiex 2000Larrosa 2002Larrosa, Schiex
2003Copper 2003Cooper, Schiex 2004Larrosa,
Schiex 2003Larrosa, Schiex 2004

• NC
• For all Cij
• ?a ? b
• Cij(a,b) 0
• b is a support
• complexity
• O(n 2d 3)

T4
C?
1
2
x
z
w
0
v
2
0
1
w
0
v
1
0
1
0
1
0
w
1
y
15
PFC-MRDAC/DC on reifieddominated by AC
16
Directional AC (DAC)
Schiex 2000Copper 2003Cooper, Schiex
2004Larrosa, Schiex 2003
xltyltz

• NC
• For all Cij (iltj)
• ?a ? b
• Cij(a,b) Cj(b) 0
• b is a full-support
• complexity
• O(ed 2)

T4
C?
1
2
x
v
2
2
z
w
0
v
2
0
1
1
w
0
1
v
1
0
1
1
1
w
0
y
17
Full AC (FAC)
Schiex 2000

• NC
• For all Cij
• ?a ? b
• Cij(a,b) Cj(b) 0
• (full support)

T4
C? 0
x
1
v
0
1
z
w
0
1
v
0
1
0
w
1
Thats our starting point! No termination !!!
18
Full DAC (FDAC)
Copper 2003Cooper, Schiex 2004Larrosa,
Schiex 2003
xltyltz

• NC
• For all Cij (iltj)
• ?a ? b
• Cij(a,b) Cj(b) 0
• (full support)
• For all Cij (igtj)
• ?a ? b, Cij(a,b) 0
• (support)
• Complexity O(end3)

T4
C?
1
2
x
v
2
2
z
w
0
v
2
2
1
0
1
w
0
v
0
1
1
1
0
1
w
y
19
Hierarchy
Special case CSP (Top1)
NC
NC O(nd)
DAC
AC O(n 2d 3)
DAC O(ed 2)
AC
FDAC O(end 3)
20
Maintaining LC
Larrosa, Schiex 2003

• BT(X,D,C)
• if (X?) then Top c?
• else
• xj selectVar(X)
• forall a?Dj do
• ?c?C s.t. xj ?var(c) cAssign(c, xj
  ,a)
• c? ?c?C s.t. var(c) ? c
• if (LC) then BT(X-xj,D-Dj,C)

21
Larrosa, Schiex 2003
MFDAC
MAC/MDAC
MNC
BT
22
Maintaining local consistency

• Ex Frequency assignment problem
• Instance CELAR6-sub4 (Proof of optimality)
• var 22 , val 44 , Optimum 3230
• Solver toolbar PIII 800MHz (Linux/gcc 3.3)
• MNC? 1 year (estimated)
• MFDAC ? 1 hour
• Typ. much better than PFCMRDAC

http//carlit.toulouse.inra.fr/cgi-bin/awki.cgi/To
olBarIntro
23
CPU time
n. of variables
24
Soft global constraints

• A network is ?-inverse consistent iff
• For all c, there is a t s.t. c(t)0
• ?-inverseNC ? domain consistency on global
  reified constraints
• Weaker than AC, DAC or FDAC
• Full ?-inverse consistency
• For all c, there is a t s.t. c(t)?ci(ti)0
• Stronger and terminates new definition of soft
  global constraints algorithms ?

25
Large domains and soft constraints

• Bound-NC for all variable i ? lbi,ubi
• C? Ci (lbi)ltT, C? Ci (ubi)ltT
• Bound-AC for all variable i ? lbi,ubi
• ? tl,tu s.t. tlilbi, tuiubi, c(tl)ltT,
  c(tu)ltT
• ? t s.t. c(t)0 (?-inverse consistency)
• Requires only two ci per variable
• If complete ci are available full Bound-AC
• Same good properties as 2b-consistency?

26
Conclusion

• AC,DAC,FDAC stronger than PFC-MRDAC
• Nice integration and possible strengthening of
  soft global constraints enforcing
• Extension of bound-consistency
• Offers additional crucial heuristics info
• ci(a) (CELAR6-SUB4 w/o 5955- 1 hour, 5955-730)
• Seems better to lift classical to soft rather
  than plunging soft into classical
• (but for the need for a complete solver
  rewriting)

27
References (Send more)

• Baptiste, Le Pape, Peridy 1998 Global
  constraints for Partial CSPs A case study of
  resource and due-date constraints. CP98.
• Beldiceanu, Petit 2004 Cost evaluation of soft
  global constraints. CPAIOR 2004.
• Bistarelli, Rossi 1995 Semiring CSP. IJCAI 1995
  (see also JACM 1995).
• Brown 2003 Soft consistencies for Weighted
  CSPs. Soft03 workshop (CP 2003)
• Cooper, Schiex 2004 Arc consistency for Soft
  Constraints, AIJ, 2004.
• Cooper 2003 Reduction operations in fuzzy or
  valued constraint satisfaction, Fuzzy sets and
  systems, 2003
• de Givry, Larrosa, Meseguer, Schiex 2003
  Solving Max-SAT as weighted CSP. CP 2003.
• Freuder, Wallace 1992 Partial Constraint
  Satisfaction. AIJ. 1992.
• Larrosa, Meseguer, Schiex 1999 Maintaining
  reversible DAC for Max-CSP. AIJ. 1999.
• Larrosa 2002 Node and Arc consistency in
  weighted CSP
• Larrosa, Schiex 2004 Solving weighted CSP by
  maintaining arc consistency, AIJ, 2004.
• Larrosa, Schiex 2003 In the quest of the best
  form of local consistency for Weighted CSP.
  IJCAI. 2003
• Petit, Régin, Bessière 2000 Meta-constraints on
  violations for over constrained problems. ICTAI
  2000.
• Petit, Régin, Bessière 2001 Specific filtering
  algorithms for over-constrained problems. CP2001.
  
• Régin, Puget, Petit 2002 Representation of hard
  constraints by soft constraints. JFPLC, 2002.
• Régin 2002 Cost-based AC for the global
  cardinality constraint, Constraints 2002 (see
  also CP99)
• Régin, Petit, Bessière, Puget 2001 New lower
  bounds of constraint violations for over
  constrained problems. CP2001.
• Schiex 2000 Arc consistency for Soft
  constraints
• Schiex, Fargier, Verfaillie 1995 Valued CSP
  hard and easy problems. IJCAI. 1995.