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IJCAI 20091 Russian Doll Search with Tree Decomposition Martí Sànchez, David Allouche, Simon de Givry, and Thomas Schiex INRA Toulouse, FRANCE

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IJCAI 20092 Real applications with a structure SPOT5 #509 (Constraints 4(3), 1999) langladeM7 sheep pedigree (Constraints 13(1), 2008) CELAR SCEN-07r (Constraints 4(1), 1999) Earth Observation Satellite Management Radio Link Frequency Assignment Mendelian Error Detection Tag SNP Selection HapMap chr01 r 2 ≥0.8 #14481

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IJCAI 20093 Framework: Weighted CSP k(X,D,F,k) n –X={X 1,..., X n } n variables nd –D={D 1,..., D n } n finite domains of maximum size d m –F={f 1,…,f m }, a set of m cost functions f S with scope S, assigns a positive cost to any assignment of S f ∅ f ∅ problem lower bound (obtained by enforcing local consistency by Equivalence Preserving Transformations) –k –k, maximum cost and global upper bound Find an assignment A of all the variables which minimizes ∑ f S F f S ( A[S] ) NP-hard problem Improve bounds computation compared to BTD (Jégou&Terrioux, ECAI 2004; de Givry et al., AAAI 2006)

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IJCAI 20094 Soft 2-Coloring example A D BC E F G J I H K XiXi XjXj f ij 1 0 0 1

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IJCAI 20095 Soft 2-Coloring example A D BC E F G J I H K 5 Optimal solution of cost 5

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IJCAI 20096 C5 C4 C3 C2 C1 Tree Decomposition C3 C1 C4 C2 C5 A D BC E F G J I H K {D} {E,F} {D,I,J} The set of clusters covers the set of variables and the set of cost functions Separator = intersection between two connected clusters

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IJCAI 20097 C4 C3 C2 C1 Search with Tree Decomposition C5 A D BC E F G J I H K The assignment of a separator disconnects the problem into two independent subproblems C3 C1 C4 C2 C5 {D} {E,F} {D,I,J}

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IJCAI 20098 C4 C5 C3 C2 01 0101 0101010101010101 01010101 0 1 01010101 0101 E G F D 01 A D BC E F G J I H K Search with Tree Decomposition K J I H P2 / {(D=red)} C3 C1 C4 C2 C5 P4 / {(D=red)} P2 / {(D=green)} P4 / {(D=green)} {D} {E,F} {D,I,J}

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IJCAI 20099 Search with Tree Decomposition 0101 01010101 01 F G E D 0 P2 / {(D=red)} C4 C5 C3 C2 A D BC E F G J I H K

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IJCAI 200910 Search with Tree Decomposition 0101 01010101 01 F G E D 0 Record the optimum of P2 / {(D=red)} LB P2 / {(D=red)} = 1 OPT P2 / {(D=red)} = true P2 / {(D=red)} k Assume a global upper bound k = 5. It may be useless to compute the optimum of P2 / {(D=red)}, only a lower bound is needed! C4 C5 C3 C2 A D BC E F G J I H K

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IJCAI 200911 Backtrack bounded by Tree Decomposition 0101 01010101 01 F G E D 0 P2 / {(D=red)} k Assume a global upper bound k = 5. It may be useless to compute the optimum of P2, only a lower bound is needed! Add a local upper bound: k UB P2 / {(D=red)} = k – 3 – LB P4 / {(D=red)} k UB P2 / {(D=red)} = k – 3 – max ( f C4 + f C5, LB P4 / {(D=red)} ) Maintaining local consistencyRecorded during search C3 C1 C4 C2 C5 BTD C4 C5 C3 C2 A D BC E F G J I H K (Jégou & Terrioux, ECAI 2004) (de Givry et al., AAAI 2006)

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IJCAI 200912 Russian Doll Search with Tree Decomposition k UB P2 / {(D=red)} = k – 3 – max ( f C4 + f C5, LB P4 / {(D=red)}, LB P4 RDS ) Found during searchObtained in preprocessing RDS-BTD lower bound is a maximum between Local Consistency Local Consistency Recorded lower bounds Recorded lower bounds RDS lower bounds RDS lower bounds –Pi RDS is a relaxation of Pi / A by removing separator variables and cost functions on them –For all i, solves Pi RDS in a bottom-up order using BTD exploiting RDS lb C3 C1 C4 C2 C5 The optimum of Pi RDS is a lower bound of Pi / A for any separator assignment A

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IJCAI 200913 Russian Doll Search with Tree Decomposition A D BC E F G J I H K P3 RDS A D BC E F G J I H K P2 RDS A D BC E F G J I H K P5 RDS A D BC E F G J I H K P4 RDS P1=P1 RDS A D BC E F G J I H K Same complexities as BTD: Time: O(exp(w+1)) Space: O(exp(s)) s w = tree-width = largest cluster size –1, s = largest separator size LB P3 RDS =0 LB P2 RDS =1 LB P5 RDS =0 LB P4 RDS =1 LB P1 RDS =5 LB P3 / {(E=red),(F=green)} =1, …

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IJCAI 200914 Related Work A D BC E F G J H K I C6 C2 C3 C5 C4 C11 C7 C8 C10 C9 C1 RDS (Verfaillie et al, AAAI 1996) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

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IJCAI 200915 Related Work A D BC E F G J H K I Pseudo-Tree RDS (Larrosa et al, ECAI 2002) AND/OR graph search (Marinescu & Dechter, AAAI 2006) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C6 C2 C3 C5 C4 C11 C7 C8 C10 C9 C1

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IJCAI 200916 Exploiting small separators only Merge clusters i and father(i) p if |sep i,father(i) | > p Gives more freedom for the dynamic variable ordering heuristic C5 C4 C3 C2 C1 A D BC E F G J I H K C3 C1 C4 C2 C5 {D} {E,F} {D,I,J} p p = 2 C3 C1 C4 U C5 C2 {D} {E,F} C4 U C5 (Jégou et al., CP 2007)

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IJCAI 200917 Exploiting small separators only A D BC E F G J H K I DFBB C1 p p = 0

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IJCAI 200918 Tag SNP Selection Set covering problem with additional coverage quadratic criteria

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IJCAI 200919 Tag SNP Selection ∞ 4 DFBB BTD RDS-BTD 25 largest instances from HapMap human chr01 (r 2 ≥ 0.5) 171 ≤ n/2 ≤ 777 30 ≤ d ≤ 266 6% ≤ m ≤ 37% 14% ≤ w ≤ 23% Solved by toulbar2 toulbar2 v.0.8 with initial upper bound found by FESTA and with MCS tree decomposition Time limit: 2 hours p

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IJCAI 200920 Radio Link Frequency Assignment CELAR SCEN-07r (Constraints 4(1), 1999)

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IJCAI 200921 Radio Link Frequency Assignment Maximum Cardinality Search tree decomposition heuristic Root selection: largest (SCEN-06) / most costly (SCEN-07r) cluster Last-conflict variable ordering (Lecoutre et al., 20006) and dichotomic binary branching Closing 1 open problem by exploiting tree decompostion CELARndmkpwDFBBBTD RDS- BTD SCEN-0610044350 ∞∞ 112588 sec.221 sec. 316 sec. SCEN-07r16244764354008353- (> 50 days) 6 days 4.5 days RECORDS : SCEN-07r p = 3 BTD 90528 recorded LB (20 %) RDS-BTD 29453 recorded LB (6.4 %) RECORDS : SCEN-06 p = ∞ BTD 5633 recorded LB RDS-BTD 7145 recorded LB

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IJCAI 200922 Conclusion and perspectives RDS-BTDRDS-BTD generalizes DFBB, RDS, PT-RDS, BTD,… exploiting better bounds Needs only linear extra space But there are not obtained in polynomial time small separatorsRestriction to small separators gives more robustness Give more freedom to dynamic variable ordering inside clusters Perspectives Produce any-time lower bounds (Iterative Deepening) p Dedicated heuristics to find small separators, p auto-tuning toulbar2 toulbar2 C++ solver version 0.8 ( http://mulcyber.toulouse.inra.fr/projects/toulbar2 )

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IJCAI 200923 Enforcing local consistency by Equivalence Preserving Transformations 1 0 1 0 AB f =0 1 1 Cost extension from a unary to a binary cost function 1 0 0 0 AB 2 1 1 f =0 Cost projection from a binary to a unary cost function 1 1 0 0 AB 2 f =0 0 0 0 0 AB f =1 2 Cost projection from a unary cost function to f to f

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IJCAI 200924 A1 A2 B1 B2 T|F a c d b b a D C B AE Weighted CSP model 1.0 0.8 0.9 1.0 0.8 B2Representativity b0 a10 c100 d0 A1B2Constraint2 False a .. 0 B1Tagsnp TrueM False0 A1A2Constraint1 Falsea Truea0 Falseb0.. 0 A1B1Dispersion True 90 TrueFalse0 True0 False 0 a a

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