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Local consistency in soft constraint networks. Thomas Schiex Matthias Zytnicki INRA – Toulouse France. Special thanks to: Javier Larrosa UPC – Barcelona Spain. Overview. Introduction and definitions Why soft constraints? Weighted CSP Existing approaches - PowerPoint PPT Presentation
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Local consistency in soft constraint networks
Thomas SchiexMatthias ZytnickiINRA – ToulouseFrance
Special thanks to:Javier LarrosaUPC – BarcelonaSpain
Tokyo 2004 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
Tokyo 2004 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)
Tokyo 2004 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)
Tokyo 2004 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.
Tokyo 2004 6
Soft constraint network
(X,D,C) X={x1,..., xn} variables D={D1,..., Dn} finite domains C={c1,..., 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
identityannihilator
• commutative• associative• monotonic
[Schiex, Fargier, Verfaillie 1995][Bistarelli, Rossi 95]
Tokyo 2004 7
Weighted CSP example ( = +)
x3
x2
x5
x1 x4
F(X): number of non blue vertices
For each vertex
For each edge:
xi c(xi
)
b 0
g 1
r 1
xi xj c(xi,x
j)
b b T
b g 0
b r 0
g b 0
g g T
g r 0
r b 0
r g 0
r r T
[Shapiro 81][Freuder 92]
Tokyo 2004 8
1st approach: Soft as Hard
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 ?
[Petit, Regin, Bessiere 2000]
Tokyo 2004 9
Propagation: a key issue Use the PFC-M[R]DAC principles
Associate each constraint to 1 of its var: C(x)
Compute #inc(x,a): cost payed on C(x) if x=a
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
[Larrosa, Meseguer, Schiex 1999][Petit, Regin, Bessiere 2001]
Tokyo 2004 10
Global soft constraints
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
[Petit et al 2001][van Hoeve et al. 2004][Beldiceanu, Petit 2004]…
Tokyo 2004 11
The CSP approach
T = maximum violation. Can be set to a bounded max. violation k Solution: F(t) < 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.
[Larrosa 2002]
Tokyo 2004 12
Projection of cij on Xi with compensation
Equivalence preserving transformation Can be reversed
1
A new operation on weighted networks
w
v v
w
0
0
0
i j1
0
[Schiex 2000]
Tokyo 2004 13
Node Consistency (NC*)
For all variable i a, C + Ci (a)<T a, Ci (a)= 0
Complexity:
O(nd)w
v
v
v
w
w
C =T =
32
2
11
1
1
1
0
0
1
x
y
z
0
0
014
[Larrosa 2002]
Tokyo 2004 14
0
Arc Consistency (AC*)
NC*
For all Cij a b
Cij(a,b)= 0
b is a support complexity:
O(n 2d 3)
wv
v
w
w
C =T=4
2
11
1
0
0
0
0
1
x
y
z
1
12
0
[Schiex 2000][Larrosa 2002][Larrosa, Schiex 2003][Copper 2003][Cooper, Schiex 2004][Larrosa, Schiex 2003][Larrosa, Schiex 2004]
Tokyo 2004 15
PFC-MRDAC/DC on reifieddominated by AC*
Tokyo 2004 16
1
1
Directional AC (DAC*)
NC*
For all Cij (i<j) a b
Cij(a,b) + Cj(b) = 0
b is a full-support complexity:
O(ed 2)
w
v
v
v
w
w
C =T=4
22
2
1
1
1
0
0
0
0
x
y
z
x<y<z
0
1
1
12
[Schiex 2000][Copper 2003][Cooper, Schiex 2004][Larrosa, Schiex 2003]
Tokyo 2004 17
0
1
Full AC (FAC*)
NC*
For all Cij a b
Cij(a,b) + Cj(b) = 0
(full support)
w
v
v
w
C =0 T=4
01
0
1
0
x
z1
1
That’s our starting point!No termination !!!
[Schiex 2000]
Tokyo 2004 18
0
0
1
1
Full DAC (FDAC*)
NC*
For all Cij (i<j) a b
Cij(a,b) + Cj(b) = 0(full support)
For all Cij (i>j) a b, Cij(a,b) = 0(support)
Complexity: O(end3)
w
v
v
v
w
w
C =T=4
22
2
11
1
0
0
0
x
y
z
x<y<z
21
12
[Copper 2003][Cooper, Schiex 2004][Larrosa, Schiex 2003]
Tokyo 2004 19
Hierarchy
NC* O(nd)
AC* O(n 2d 3) DAC* O(ed 2)
FDAC* O(end 3) AC
NC
DAC
Special case: CSP (Top=1)
Tokyo 2004 20
Maintaining LC
BT(X,D,C)if (X=) then Top :=c
else xj := selectVar(X)
forall aDj do
cC s.t. xj var(c) c:=Assign(c, xj ,a)
c:= cC s.t. var(c)= c
if (LC) then BT(X-{xj},D-{Dj},C)
[Larrosa, Schiex 2003]
Tokyo 2004 21
BT
MNC
MAC/MDAC
MFDAC
[Larrosa, Schiex 2003]
Tokyo 2004 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 PFC{M}RDAC
http://carlit.toulouse.inra.fr/cgi-bin/awki.cgi/ToolBarIntro
Tokyo 2004 23
CPU time
n. of variables
Tokyo 2004 24
Soft global constraints
A network is -inverse consistent iff For all c, there is a t s.t. c(t)=0
-inverse+NC* 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(t[i])=0 Stronger and terminates: new definition
of soft global constraints algorithms ?
Tokyo 2004 25
Large domains and soft constraints
Bound-NC*: for all variable i [lbi,ubi] C + Ci (lbi)<T, C + Ci (ubi)<T
Bound-AC*: for all variable i [lbi,ubi] tl,tu s.t. tl[i]=lbi, tu[i]=ubi, c(tl)<T, c(tu)<T 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?
Tokyo 2004 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, 5955-7’30”)
Seems better to lift classical to soft rather than plunging soft into classical
(but for the need for a complete solver rewriting…)
Tokyo 2004 27
References (Send more)
[Baptiste, Le Pape, Peridy 1998] Global constraints for Partial CSPs: A case study of resource and due-date constraints. CP’98.
[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. Soft’03 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 CP’99) [Régin, Petit, Bessière, Puget 2001] New lower bounds of constraint violations for over constrained
problems. CP’2001. [Schiex 2000] Arc consistency for Soft constraints [Schiex, Fargier, Verfaillie 1995] Valued CSP: hard and easy problems. IJCAI. 1995. [van Hoeve, Pesant, Martin-Rousseau 2004] On Global Warming (Softening global constraints).
Soft’04.