Upload
brendan-austin
View
38
Download
0
Embed Size (px)
DESCRIPTION
These slides are provided as a teaching support for the community. They can be freely modified and used as far as the original authors (T. Schiex and J. Larrosa) contribution is clearly mentionned and visible and that any modification is acknowledged by the author of the modification. - PowerPoint PPT Presentation
Citation preview
Soft constraint processingThomas SchiexINRA ToulouseFranceJavier LarrosaUPC BarcelonaSpainThese slides are provided as a teaching support for the community. They can be freely modified and used as far as the original authors (T. Schiex and J. Larrosa) contribution is clearly mentionned and visible and that any modification is acknowledged by the author of the modification.
CP06
OverviewFrameworks Generic and specificAlgorithmsSearch: complete and incompleteInference: complete and incompleteIntegration with CPSoft as hardSoft as global constraint
CP06
Parallel mini-tutorialCSP SAT strong relationAlong the presentation, we will highlight the connections with SAT
Multimedia trick:SAT slides in yellow background
CP06
Why soft constraints?CSP framework: natural for decision problemsSAT framework: natural for decision problems with boolean variables
Many problems are constrained optimization problems and the difficulty is in the optimization part
CP06
Why soft constraints?Given a set of requested pictures (of different importance) select the best subset of compatible pictures subject to available resources:3 on-board camerasData-bus bandwith, setup-times, orbiting
Best = maximize sum of importance Earth Observation Satellite Scheduling
CP06
Why soft constraints?Frequency assignment
Given a telecommunication networkfind the best frequency for each communication link avoiding interferences
Best can be:Minimize the maximum frequency (max)Minimize the global interference (sum)
CP06
Why soft constraints?Combinatorial auctionsGiven a set G of goods and a set B of bidsBid (bi,vi), bi requested goods, vi value find the best subset of compatible bidsBest = maximize revenue (sum)
CP06
Why soft constraints?Probabilistic inference (bayesian nets)
Given a probability distribution defined by a DAG of conditional probability tablesand some evidence find the most probable explanation for the evidence (product)
CP06
Why soft constraints?Even in decision problems:users may have preferences among solutions
Experiment: give users a few solutions and they will find reasons to prefer some of them.
CP06
ObservationOptimization problems are harder than satisfaction problems
CSP vs. Max-CSP
CP06
Why is it so hard ?Problem P(alpha): is there an assignment of cost lower than alpha ? Proof of inconsistency Proof of optimalityHarder than finding an optimum
CP06
NotationX={x1,..., xn} variables (n variables)D={D1,..., Dn} finite domains (max size d)
ZYX, tY is a tuple on YtY[Z] is its projection on ZtY[-x] = tY[Y-{x}]is projecting out variable x fY: xiY Di Eis a cost function on Y
CP06
Generic and specific frameworksValued CNweighted CNSemiring CNfuzzy CN
CP06
Costs (preferences)E costs (preferences) setordered by if a b then a is better than bCosts are associated to tuplesCombined with a dedicated operatormax: priorities+: additive costs*: factorized probabilities
Fuzzy/possibilistic CNWeighted CNProbabilistic CN, BN
CP06
Soft constraint network (CN)(X,D,C)X={x1,..., xn} variablesD={D1,..., Dn} finite domainsC={f,...} cost functionsfS, fij, fi fscope S,{xi,xj},{xi}, fS(t): E (ordered by , T)
Obj. Function: F(X)= fS (X[S]) Solution: F(t) T Task: find optimal solutionidentity commutative associative monotonicanihilator
CP06
Specific frameworks
CP06
Weighted Clauses(C,w)weighted clauseCdisjunction of literalswcost of violation w E (ordered by , T) combinator of costsCost functions = weighted clauses(xi xj, 6), (xi xj, 2),(xi xj, 3)
xixjf(xi,xj)006010102113
CP06
Specific weighted prop. logics
CP06
CSP example (3-coloring)x3x2x5x1x4For each edge:(hard constr.)
xixjf(xi,xj)bbTbgbrgbggTgrrbrgrrT
CP06
Weighted CSP example ( = +)x3x2x5x1x4F(X): number of non blue verticesFor each vertex
CP06
Some important detailsT = maximum acceptable violation. Empty scope soft constraint f (a constant)Gives an obvious lower bound on the optimumIf you do not like it: f =
Additional expression power
CP06
Weighted CSP example ( = +)x3x2x5x1x4F(X): number of non blue verticesFor each vertexFor each edge:T=6f = 0T=3Optimal coloration with less than 3 non-blue
xixjf(xi,xj)bbTbg0br0gb0ggTgr0rb0rg0rrT
CP06
General frameworks and cost structures hard{,T}totallyorderedSemiring CSP
Valued CSPidempotentfairmulticriterialatticeorderedmultiple
CP06
Idempotencya a = a (for any a)
For any fS implied by (X,D,C)(X,D,C) (X,D,C{fS})Classic CN: = andPossibilistic CN: = maxFuzzy CN: = max
CP06
FairnessAbility to compensate for cost increases by subtraction using a pseudo-difference:
For b a, (a b) b = aClassic CN: ab = or (max)Fuzzy CN: ab = maxWeighted CN: ab = a-b (aT) else T Bayes nets:ab = /
CP06
Processing Soft constraintsSearchcomplete (systematic)incomplete (local)Inferencecomplete (variable elimination)incomplete (local consistency)
CP06
Systematic searchBranch and bound(s)
CP06
I - Assignment (conditioning)f[xi=b]g(xj)g[xj=r]h
xixjf(xi,xj)bbTbg0br3gb0ggTgr0rb0rg0rrT
xjbTg0r3
03
CP06
I - Assignment (conditioning){(xyz,3), (xy,2)}
x=true(y,2)(,2)y=false empty clause. It cannot be satisfied,2 is necessary cost
CP06
Systematic search(LB) Lower Bound (UB) Upper BoundIf then prunevariablesunder estimation of the best solution in the sub-tree= best solution so farEach node is a soft constraint subproblemLBf= f= TUBT
CP06
Improving the lower bound (WCSP)Sum up costs that will necessarily occur (no matter what values are assigned to the variables)
PFC-DAC (Wallace et al. 1994)PFC-MRDAC (Larrosa et al. 1999)Russian Doll Search (Verfaillie et al. 1996)Mini-buckets (Dechter et al. 1998)
CP06
Improving the lower bound (Max-SAT)Detect independent subsets of mutually inconsistent clauses
LB4a (Shen and Zhang, 2004)UP (Li et al, 2005)Max Solver (Xing and Zhang, 2005)MaxSatz (Li et al, 2006)
CP06
Local searchNothing really specific
CP06
Local searchBased on perturbation of solutions in a local neighborhood
Simulated annealingTabu searchVariable neighborhood searchGreedy rand. adapt. search (GRASP)Evolutionary computation (GA)Ant colony optimization
See: Blum & Roli, ACM comp. surveys, 35(3), 2003For booleanvariables: GSAT
CP06
Boosting Systematic Search with Local SearchDo local search prior systematic search Use best cost found as initial T If optimal, we just prove optimality In all cases, we may improve pruningLocal search(X,D,C)time limitSub-optimalsolution
CP06
Boosting Systematic Search with Local Search Ex: Frequency assignment problemInstance: CELAR6-sub4#var: 22 , #val: 44 , Optimum: 3230Solver: toolbar 2.2 with default optionsT initialized to 100000 3 hoursT initialized to 3230 1 hourOptimized local search can find the optimum in a less than 30 (incop)
CP06
Complete inference Variable (bucket) eliminationGraph structural parameters
CP06
II - Combination (join with , + here)= 0 6
xixjf(xi,xj)bb6bg0gb0gg6
xjxkg(xj,xk)bb6bg0gb0gg6
xixjxkh(xi,xj,xk)bbb12bbg6bgb0bgg6gbb6gbg0ggb6ggg12
CP06
III - Projection (elimination)f[xi]002g[]0Min
xixjf(xi,xj)bb4bg6br0gb2gg6gr3rb1rg0rr6
xig(xi)bgr
h
CP06
Properties
Replacing two functions by their combination preserves the problem
If f is the only function involving variable x, replacing f by f[-x] preserves the optimum
CP06
Variable eliminationSelect a variableSum all functions that mention itProject the variable out
Complexity Time: (exp(deg+1)) Space: (exp(deg))
CP06
Variable elimination (aka bucket elimination)Eliminate Variables one by one.When all variables have been eliminated, the problem is solvedOptimal solutions of the original problem can be recomputed Complexity: exponential in the induced width
CP06
Elimination order influence{f(x,r), f(x,z), , f(x,y)}Order: r, z, , y, x
xrzy
CP06
Elimination order influence{f(x,r), f(x,z), , f(x,y)}Order: r, z, , y, x
xrzy
CP06
Elimination order influence{f(x), f(x,z), , f(x,y)}Order: z, , y, x
xzy
CP06
Elimination order influence{f(x), f(x,z), , f(x,y)}Order: z, , y, x
xzy
CP06
Elimination order influence{f(x), f(x), f(x,y)}Order: y, x
xy
CP06
Elimination order influence{f(x), f(x), f(x,y)}Order: y, x
xy
CP06
Elimination order influence{f(x), f(x), f(x)}Order: x
x
CP06
Elimination order influence{f(x), f(x), f(x)}Order: x
x
CP06
Elimination order influence{f()}Order:
CP06
Elimination order influence{f(x,r), f(x,z), , f(x,y)}Order: x, y, z, , r
xrzy
CP06
Elimination order influence{f(x,r), f(x,z), , f(x,y)}Order: x, y, z, , r
xrzy
CP06
Elimination order influence{f(r,z,,y)}Order: y, z, r
rzyCLIQUE
CP06
Induced widthFor G=(V,E) and a given elimination (vertex) ordering, the largest degree encountered is the induced width of the ordered graph
Minimizing induced width is NP-hard.
CP06
History / terminologySAT: Directed Resolution (Davis and Putnam, 60)Operations Research: Non serial dynamic programming (Bertel Brioschi, 72) Databases: Acyclic DB (Beeri et al 1983) Bayesian nets: Join-tree (Pearl 88, Lauritzen et Spiegelhalter 88)Constraint nets: Adaptive Consistency (Dechter and Pearl 88)
CP06
Boosting search with variable elimination: BB-VE(k)At each nodeSelect an unassigned variable xiIf degi k then eliminate xiElse branch on the values of xi
PropertiesBE-VE(-1) is BBBE-VE(w*) is VEBE-VE(1) is similar to cycle-cutset
CP06
Boosting search with variable elimination Ex: still-life (academic problem)Instance: n=14#var:196 , #val:2Ilog Solver 5 daysVariable Elimination 1 dayBB-VE(18) 2 seconds
CP06
Memoization fights thrashing=Different nodes, Same subproblemstoreretrieveDetecting subproblems equivalence is hardVVtPtPPtt
CP06
Context-based memoizationP=P, if|t|=|t| andsame assign. to partially assigned cost functionsttPP
CP06
MemoizationDepth-first B&B with,context-based memoizationindependent sub-problem detection is essentialy equivalent to VETherefore space expensiveFresh approach: Easier to incorporate typical tricks such as propagation, symmetry breaking,Algorithms:Recursive Cond. (Darwiche 2001)BTD (Jgou and Terrioux 2003)AND/OR (Dechter et al, 2004)
Adaptive memoization: time/space tradeoff
CP06
SAT inferenceIn SAT, inference = resolution xAxB------------ ABEffect: transforms explicit knowledge into implicitComplete inference: Resolve until quiescenceSmart policy: variable by variable (Davis & Putnam, 60). Exponential on the induced width.
CP06
Fair SAT Inference(x A,u), (x B,w) (A B,m), (x A,um),(x B, wm),(x A B,m),(x A B,m)where: m=min{u,w} Effect: moves knowledge
CP06
Example: Max-SAT (=+, =-)(xy,3), (xz,3)=xyyzxxyzyzx33z333(yz,3),(xy,3-3),(xz,3-3),(xyz,3),(xyz,3)
CP06
Properties (Max-SAT)In SAT, collapses to classical resolutionSound and completeVariable elimination:Select a variable xResolve on x until quiescenceRemove all clauses mentioning x
Time and space complexity: exponential on the induced width
CP06
Change
CP06
Incomplete inferenceLocal consistencyRestricted resolution
CP06
Incomplete inference
Tries to trade completeness for space/timeProduces only specific classes of cost functions Usually in polynomial time/space
Local consistency: node, arcEquivalent problemCompositional: transparent useProvides a lb on consistencyoptimal cost
CP06
Classical arc consistency A CSP is AC iff for any xi and cijci= ci (cij cj)[xi] namely, (cij cj)[xi] brings no new information on xiwvvw0000ijTTcij cj(cij cj)[xi]
xixjcijvv0vw0wvTwwT
xic(xi)v0wT
CP06
Enforcing AC for any xi and cij ci:= ci (cij cj)[xi] until fixpoint (unique)wvvw0000ijTTcij cjT(cij cj)[xi]
xixjcijvv0vw0wvTwwT
xic(xi)v0wT
CP06
Arc consistency and soft constraints for any xi and fijf=(fij fj)[xi] brings no new information on xiwvvw0000ij21fij fj1Always equivalent iff idempotent(fij fj)[xi]
xixjfijvv0vw0wv2ww1
Xif(xi)v0w1
CP06
Idempotent soft CNThe previous operational extension works on any idempotent semiring CNChaotic iteration of local enforcing rules until fixpointTerminates and yields an equivalent problemExtends to generalized k-consistency
Total order: idempotent ( = max)
CP06
Non idempotent: weighted CN for any xi and fijf=(fij fj)[xi] brings no new information on xiwvvw0000ij21fij fj1EQUIVALENCE LOSTfij fj [xi]
xixjfijvv0vw0wv2ww1
xif(xi)v0w1
CP06
IV - Subtraction of cost functions (fair)Combination+Subtraction: equivalence preserving transformation2wvvw000ij1011
CP06
(K,Y) equivalence preserving inferenceFor a set K of cost functions and a scope YReplace K by (K)Add (K)[Y] to the CN (implied by K)Subtract (K)[Y] from (K)
Yields an equivalent networkAll implicit information on Y in K is explicit
Repeat for a class of (K,Y) until fixpoint
CP06
Full AC (FAC*): ({fij,fj},{xi}) EPINC*For all fij a b fij(a,b) + fj(b) = 0(full support)
01wvvwf =0 T=401010xz11Thats our starting point!No termination !!!
CP06
Arc Consistency (AC*): ({fij},{xi}) EPINC*For all fij a b fij(a,b)= 0b is a supportcomplexity:O(n 2d 3)
0wvvwwf =T=4211100001xyz1120
CP06
Neighborhood Resolution
if |A|=0, enforces node consistency
if |A|=1, enforces arc consistency(x A,u), (x A,w) (A,m), (x A,um),(x A, wm),(x A A,m),(x A A,m)
CP06
Confluence is lostw01yxvw00v1f = 0 1
CP06
Confluence is lostFinding an AC closure that maximizes the lb is an NP-hard problem (Cooper & Schiex 2004).Well one can do better in pol. time (OSAC, IJCAI 2007)w01yxvw00vf = 0 1
CP06
HierarchyNC* O(nd)AC* O(n 2d 3)DAC* O(ed 2)FDAC* O(end 3)ACNCDACSpecial case: CSP (Top=1)EDAC* O(ed2 max{nd,T})
CP06
Boosting search with LCBT(X,D,C)if (X=) then Top :=f else xj := selectVar(X) forall aDj do fSC s.t. xj S fS := fS [xj =a] if (LC) then BT(X-{xj},D-{Dj},C)
CP06
BTMNCMAC/MDACMFDACMEDAC
CP06
Boosting Systematic Search with Local consistency Frequency assignment problem CELAR6-sub4 (22 var, 44 val, 477 cost func):
MNC*1 yearMFDAC* 1 hour
CELAR6 (100 var, 44 val, 1322 cost func):
MEDAC+memoization 3 hours (toolbar-BTD)
CP06
Beyond Arc ConsistencyPath inverse consistency PIC (Debryune & Bessire)xyz(x,a) can be pruned because there are two other variables y,z such that (x,a) cannot be extended to any of their values.({fy, fz, fxy, fxz, fyz},{x}) EPIabc
CP06
Beyond Arc ConsistencySoft Path inverse consistency PIC*223({fy, fz, fxy, fxz, fyz},x) EPIfy fz fxy fxz fyz(fyfzfxyfxzfyz)[x]xyz122
xyzaaa2aab5aba2abb3baa0bab0bba2bbb0
xa2b0
xyzaaa0aab3aba0abb1baa0bab0bba2bbb0
CP06
Hyper-resolution (2 steps)(lhA,u),(lqA,v),(hqA,u)(hqA,m),(lhA,u-m),(lqA,v-m),(lhqA,m),(lqhA,m),(hqA,u)(qA,m),(hqA,m-m),(hqA,u-m),(lhA,u-m),(lqA,v-m),(lhqA,m),(lqhA,m)if |A|=0, equal to soft PICImpressive empirical speed-ups==
CP06
Complexity & Polynomial classesTree = induced width 1Idempotent or not
CP06
Polynomial classesIdempotent VCSP: min-max CN
Can use -cuts for lifting CSP classesSufficient condition: the polynomial class is conserved by -cuts
Simple TCSP are TCSP where all constraints use 1 interval: xi-xj[aij,bij]
Fuzzy STCN: any slice of a cost function is an interval (semi-convex function) (Rossi et al.)
CP06
Hardness in the additive case(weighted/boolean)MaxSat is MAXSNP complete (no PTAS)Weighted MaxSAT is FPNP-completeMaxSAT is FPNP[O(log(n))] complete: weights !MaxSAT tractable langages fully characterized (Creignou 2001)
MaxCSP langage: feq(x,y) : (x = y) ? 0 : 1 is NP-hard.Submodular cost function lang. is polynomial.(u x, v y f(u,v)+f(x,y) f(u,y)+f(x,v)) (Cohen et al.)
CP06
Integration of soft constraints into classical constraint programmingSoft as hardSoft local consistency as a global constraint
CP06
Soft constraints as hard constraintsone extra variable xs per cost function fS all with domain E fS cS{xS} allowing (t,fS(t)) for all t(S)one variable xC = xs (global constraint)
CP06
Soft as Hard (SaH)Criterion represented as a variableMultiple criteria = multiple variablesConstraints on/between criteria
Weaknesses:Extra variables (domains), increased aritiesSaH constraints give weak GAC propagationProblem structure changed/hidden
CP06
Soft AC stronger than SasH GAC
Take a WCSPEnforce Soft AC on itEach cost function contains at least one tuple with a 0 cost (definition)Soft as Hard: the cost variable xC will have a lb of 0
The lower bound cannot improve by GAC
CP06
> Soft AC stronger than SasH GAC111111f=1abx1x3x2x1x3x2101010x12x23xcxc=x12+x232
CP06
Soft local Consistency as a Global constraint (=+)Global constraint: Soft(X,F,C)XvariablesFcost functionsCinterval cost variable (ub = T)
Semantics: X U{C} satisfy Soft(X,F,C) ifff(X)=CEnforcing GAC on Soft is NP-hardSoft consistency: filtering algorithm (lbf)
CP06
Example: soft quasi-group (motivated by sports scheduling)
Alldiff(xi1,,xin) i=1..mAlldiff(x1j,,xmj) j=1..nSoft(X,{fij},[0..k],+)Minimize #neighbors of different parityCost 1
34
CP06
Global soft constraints
CP06
Global soft constraintsIdea: define a library of useful but non-standard objective functions along with efficient filtering algorithms
AllDiff (2 semantics: Petit et al 2001, van Hoeve 2004)Soft global cardinality (van Hoeve et al. 2004)Soft regular (van Hoeve et al. 2004) all enforce reified GAC
CP06
ConclusionA large subset of classic CN body of knowledge has been extended to soft CN, efficient solving tools exist.Much remains to be done:Extension: to other problems than optimization (counting, quantification)Techniques: symmetries, learning, knowledge compilation Algorithmic: still better lb, other local consistencies or dominance. Global (SoftAsSoft). Exploiting problem structure.Implementation: better integration with classic CN solver (Choco, Solver, Minion)Applications: problem modelling, solving, heuristic guidance, partial solving.
CP06
30 of publicity
CP06
Open source libraries Toolbar and Toulbar2
Accessible from the Soft wiki site:carlit.toulouse.inra.fr/cgi-bin/awki.cgi/SoftCSPAlg: BE-VE,MNC,MAC,MDAC,MFDAC,MEDAC,MPIC,BTDILOG connection, large domains/problemsRead MaxCSP/SAT (weighted or not) and ERGO formatThousands of benchmarks in standardized formatPointers to other solvers (MaxSAT/CSP)Forge mulcyber.toulouse.inra.fr/projects/toolbar (toulbar2)
Pwd: bia31
CP06
Thank you for your attentionThis is it !S. Bistarelli, U. Montanari and F. Rossi, Semiring-based Constraint Satisfaction and Optimization, Journal of ACM, vol.44, n.2, pp. 201-236, March 1997.S. Bistarelli, H. Fargier, U. Montanari, F. Rossi, T. Schiex, G. Verfaillie. Semiring-Based CSPs and Valued CSPs: Frameworks, Properties, and Comparison. CONSTRAINTS, Vol.4, N.3, September 1999.S. Bistarelli, R. Gennari, F. Rossi. Constraint Propagation for Soft Constraint Satisfaction Problems: Generalization and Termination Conditions , in Proc. CP 2000C.Blum and A.Roli. Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM Computing Surveys, 35(3):268-308, 2003. T. Schiex, Arc consistency for soft constraints, in Proc. CP2000.M. Cooper, T. Schiex. Arc consistency for soft constraints, Artificial Intelligence, Volume 154 (1-2), 199-227 2004.M. Cooper. Reduction Operations in fuzzy or valued constraint satisfaction problems. Fuzzy Sets and Systems 134 (3) 2003.A. Darwiche. Recursive Conditioning. Artificial Intelligence. Vol 125, No 1-2, pages 5-41.R. Dechter. Bucket Elimination: A unifying framework for Reasoning. Artificial Intelligence, October, 1999. R. Dechter, Mini-Buckets: A General Scheme For Generating Approximations In Automated Reasoning In Proc. Of IJCAI97
CP06
References
S. de Givry, F. Heras, J. Larrosa & M. Zytnicki. Existential arc consistency: getting closer to full arc consistency in weighted CSPs. In IJCAI 2005.W.-J. van Hoeve, G. Pesant and L.-M. Rousseau. On Global Warming: Flow-Based Soft Global Constraints. Journal of Heuristics 12(4-5), pp. 347-373, 2006. P. Jegou & C. Terrioux. Hybrid backtracking bounded by tree-decomposition of constraint networks. Artif. Intell. 146(1): 43-75 (2003) J. Larrosa & T. Schiex. Solving Weighted CSP by Maintaining Arc Consistency. Artificial Intelligence. 159 (1-2): 1-26, 2004. J. Larrosa and T. Schiex. In the quest of the best form of local consistency for Weighted CSP, Proc. of IJCAI'03J. Larrosa, P. Meseguer, T. Schiex Maintaining Reversible DAC for MAX-CSP. Artificial Intelligence.107(1), pp. 149-163. R. Marinescu and R. Dechter. AND/OR Branch-and-Bound for Graphical Models. In proceedings of IJCAI'2005. J.C. Regin, T. Petit, C. Bessiere and J.F. Puget. An original constraint based approach for solving over constrained problems. In Proc. CP'2000. T. Schiex, H. Fargier et G. Verfaillie. Valued Constraint Satisfaction Problems: hard and easy problems In Proc. of IJCAI 95. G. Verfaillie, M. Lemaitre et T. Schiex. Russian Doll Search Proc. of AAAI'96.
CP06
References
M. Bonet, J. Levy and F. Manya. A complete calculus for max-sat. In SAT 2006.M. Davis & H. Putnam. A computation procedure for quantification theory. In JACM 3 (7) 1960.I. Rish and R. Dechter. Resolution versus Search: Two Strategies for SAT. In Journal of Automated Reasoning, 24 (1-2), 2000.F. Heras & J. Larrosa. New Inference Rules for Efficient Max-SAT Solving. In AAAI 2006. J. Larrosa, F. Heras. Resolution in Max-SAT and its relation to local consistency in weighted CSPs. In IJCAI 2005.C.M. Li, F. Manya and J. Planes. Improved branch and bound algorithms for max-sat. In AAAI 2006.H. Shen and H. Zhang. Study of lower bounds for max-2-sat. In proc. of AAAI 2004.Z. Xing and W. Zhang. MaxSolver: An efficient exact algorithm for (weighted) maximum satisfiability. Artificial Intelligence 164 (1-2) 2005.
CP06
SoftasHard GAC vs. EDAC 25 variables, 2 values binary MaxCSPToolbar MEDACopt=34220 nodescpu-time = 0
GAC on SoftasHard, ILOG Solver 6.0, solveopt = 34339136 choice pointscpu-time: 29.1Uses table constraints
CP06
Other hints on SoftasHard GACMaxSAT as Pseudo Boolean SoftAsHardFor each clause:c = (xz,pc) cSAH = (xzrc) Extra cardinality constraint: pc.rc k
Used by SAT4JMaxSat (MaxSAT competition).
CP06
MaxSAT competition (SAT 2006)Unweighted MaxSAT
CP06
MaxSAT competition (SAT 2006)Weighted
CP06