1. ALIMENTATION AGRICULTURE ENVIRONNEMENT Max-CSP solver competition 2008 toulbar2 Marti Sanchez 1, Sylvain Bouveret 2, Simon de Givry 1, Federico Heras 3, Philippe Jegou 4, Javier Larrosa 3, Samba Ndiaye 4, Emma Rollon 3, Thomas Schiex 1, Cyril Terrioux 4, Gerard Verfaillie 2, Matthias Zytnicki 1 1. INRA, Toulouse, France 2. ONERA, Toulouse, France 3. UPC, Barcelona, Spain 4. LSIS, Marseilles, France

2. ALIMENTATION AGRICULTURE ENVIRONNEMENT Framework: Weighted CSP m (X,D,W,m) n X={X 1,..., X n } n variables nd D={D 1,..., D n } n finite domains of maximum size d e W, a set of e cost functions W S, W i, W ∅ with scopes S, {X i }, ∅ m W S associates a cost in [0,m] to any assignment of S m  [1,+  ] m  [1,+  ] is associated to completely forbidden assignments A Find a complete assignment A minimizing ∑ S  W W S ( A[S] ) NP-hard problem

3. ALIMENTATION AGRICULTURE ENVIRONNEMENT Depth-First Branch and Bound (DFBB) (LB) Lower Bound (UB) Upper Bound If  UB then prune Variables under estimation of the best solution in the sub-tree = best solution so far Each node is a soft constraint subproblem LB WWWW W  = W  m = m O(d n ) Time complexity: O(d n ) O(n) Space complexity: O(n) m Obtained by enforcing soft local consistency Binary branching scheme instead of value enumeration Dynamic variable and value ordering heuristics No initial upper bound

4. Enforcing local consistency by Equivalence Preserving Transformations 1 0 1 0 X1X2 W  =0 1 1 Cost extension from a unary to a binary cost function 1 0 0 0 X1X2 2 1 1 W  =0 Cost projection from a binary to a unary cost function 1 1 0 0 X1X2 2 W  =0 0 0 0 0 X1X2 W=1W=1 2 Cost projection from a unary cost function to W to W 

5. ALIMENTATION AGRICULTURE ENVIRONNEMENT Hierarchy NC * O(nd) AC * O(n 2 d 3 )DAC * O(ed 2 ) FDAC * O(end 3 ) AC NC DAC Special case: CSP (m=1) EDAC * O(ed 2 max{nd,m}) VAC O(ed 2 m/  )  [0,1] OSAC O(poly(ed+n)) Solve tree-like constraint graphs Solve submodular cost functions Non-binary constraints GAC Ternary EDAC O(ed 3 max{nd,m})

6. ALIMENTATION AGRICULTURE ENVIRONNEMENT Last-conflict variable heuristic (Lecoutre et al, 2006) Basic form of Conflict Back-Jumping

7. ALIMENTATION AGRICULTURE ENVIRONNEMENT Boosting search with variable elimination (Larrosa and Dechter, 2003) At each search node Eliminate all unassigned variables with degree ≤ p Select an unassigned variable x Branch on the values of x Properties BB-VE(-1) is Depth-First Branch and Bound BB-VE(w) is Variable Elimination BB-VE(1) is similar to Cycle-Cutset BB-VE(2) is well suited with soft local consistencies (add binary constraints only, independent of the elimination order)

8. ALIMENTATION AGRICULTURE ENVIRONNEMENT Depth-First Branch and Bound exploiting a Tree Decomposition (Terrioux et al, ECAI 2004) (Givry et al, AAAI 2006) Toulbar2/BTD MCS tree decomposition heuristic (Marseilles’ toolkit) in preprocessing Root selection maximizing cluster size Same search as DFBB inside clusters (DVO, CBJ, VE(2)) Limited caching (keep only small separators, s=4) Soft-08 workshop Russian Doll Search pruning scheme (Soft-08 workshop) ts t = largest cluster size, s = largest separator size Tree search with Restricted variable ordering Restricted variable ordering Graph-based backjumping Graph-based backjumping Graph-based learning Graph-based learning Time: O(d t ) Time: O(d t ) Space: O(d s ) Space: O(d s )

9. ALIMENTATION AGRICULTURE ENVIRONNEMENT toulbar2 Open source WCSP solver in C++ (release 0.7) http://mulcyber.toulouse.inra.fr/gf/project/toulbar2 Large set of benchmarks http://carlit.toulouse.inra.fr/cgi-bin/awki.cgi/SoftCSP Frequency assignment (Cabon et al., 1999) n≤458, d=44, e(2)≤5000 Satellite management (Bensana et al., 1999) n≤364, d=4, e(2-3)≤10108 Mendelian error detection (Sanchez et al, 2008) n≤20000, d≤66, e(3)≤30000 RNA gene finding (Zytnicki et al, 2008) n≈20, d>100 million!, e(4)≈10 Tag SNP selection (Sanchez et al, 2008) n≤251, d≤175, e(2)≤13649 … Link to ILOG Solver Encapsulates a WCSP as a global constraint