Combining Constraint Programming and Integer Programming

Transcription

1 Combnng Constrant Programmng and Integer Programmng GLOBAL CONSTRAINT OPTIMIZATION COMPONENT Specal Purpose Algorthm mn c T x +(x- 0 ) x( + ()) =1 x( - ()) =1 FILTERING ALGORITHM COST-BASED FILTERING ALGORITHM CUT GENERATOR IP- Model mn Z = c x = 1 V x = 1 V S V V V V x x 1 SV S V\S x 0 and nteger 1

2 Constrant Programmng (CP) Prelmnares CP on Fnte Domans CP(FD): a programmng paradgm explotng Constrant Satsfacton technques A Constrant Satsfacton Problem (CSP) conssts of: a set of varables (V 1, V 2,,V n ) a dscrete doman (D 1,D 2,,D n ) for each varable a set of constrants on te varables: relatons among varables whch represent a subset of the Cartesan product of the domans D 1 x D 2 x... x D n Soluton of a CSP: an assgnment of values to varables consstent wth the constrants E. Tsang: : Foundatons of Constrant Satsfacton Academc Press,

3 Each varable s nvolved n many constrants Any change n the doman of a sngle varable may propagate to other varables. Constrants agents vew: durng ther lfetme, the constrants alternate ther status between suspended and woken states (trggered by events). Example: CP Prelmnares nteracton among constrants X = Y + 1 X::[1..5], Y::[1..5], Z::[1..5] Y = Z + 1 X = Z - 1 3

4 X = Y + 1 X::[1..5], Y::[1..5], Z::[1..5] Y = Z + 1 X = Z - 1 Frst propagaton of X = Y + 1 yelds X::[2..5], Y::[1..4], Z::[1..5] X = Y + 1 s then suspended 4

5 Second propagaton of Y = Z + 1 yelds X::[2..5], Y::[2..4], Z::[1..3] Y = Z + 1 s then suspended The doman of Y has changed, X = Y + 1 s then awakened X::[3..5], Y::[2..4], Z::[1..3] X = Y + 1 s then suspended 5

6 Thrd propagaton of X = Z - 1 yelds X::[] Y::[2..4] Z::[1..3] FAILURE detected 6

7 CP Prelmnares search Prunng all the nfeasble values from varable domans often has the same complexty as solvng the orgnal problem Propagaton algorthms are then ncomplete,.e., when propagaton s stopped, stll some values left n the varable domans can be nconsstent a SEARCH step s executed. P Branchng strateges defne the way of parttonng the problem P nto easer subproblems P 1, P 2,, P n. To each subproblem: apply agan propagaton. P 1 P 2 P 3 P 4 P n New branches can be pruned because of the new nformaton derved from the branchng 7

8 The search scheme varable selecton contnue on success value selecton constrant propagaton backtrack on falure const. 1 const 2 const 3 8

9 CP Prelmnares: dealng wth an obectve functon What about the obectve functon? As soon as a feasble soluton of value Z * (say) s found, a new boundng constrant s added so as to mpose that further solutons must have a better value: Z < Z * Hence, CP solves a sequence of feasblty problems, constraned to mprove the obectve functon value. However: Z s n general the sum of the values assumed by several varables the propagaton of the boundng constrant s typcally very weak 9

10 Example: (ATSP) Asymmetrc Travelng SalesmanProblem Gven a drected graph G=(V,A) wth nodes 0,1,, n-1 arc costs c Fnd a Hamltonan tour (= closed crcut passng exactly once through all nodes) of mnmum total cost 0 c 10

11 ATSP: a CP model Gven the drected graph G=(V,A), assocate to each node a varable X whose doman contans the next possble nodes n a smple path. CP standard modules nclude the followng path constrant: X 0 ::D 0, X 1 ::D 1,..., X k ::D k path([x 0,X 1,,X k ]) sayng that the assgnment of varables X 0,X 1,,X k has to defne a smple path nvolvng all nodes 0,,k. 11

12 The Asymmetrc Travellng Salesman Problem (ATSP) can then be formulated through the CP path constrant as follows: one of the nodes, say node 0, s duplcated generatng node n the constrant path([x 0,X 1,,X n ]) s mposed 0 n c 12

13 13 ATSP: an IP model A x S V S x V x V x To Subect x c Mnmze S S V V A >= >= = = = = ), ( nteger 0 (no subtours) 0 : 1 1) (outdegree 1 1) (ndegree 1 : : ), ( thesoluton n ff x ), ( = 1 0 {0,1} x

14 Comparng the CP and IP models CP- Model: X ::[v 1,v 2,,v n ] =0..n-1 path([x 0,X 1,,X n ]) C ::[c 1,c 2,,c n ]=0..n-1 C n = 0; X n = 0; C 0 + +C n-1 = Z mnmze(z) X = v path ([X 0, X 1,.., X n ]) C =c Mappng IP- Model mn Z = c x = 1 V x = 1 V S V V V V x x 1 SV S V\S x 0 and nteger x =1 lnear constr.s c ff x =1 14

15 Pure IP vs Pure CP [red,blue,green,yellow,pnk] [red,blue,green,yellow,pnk] V 1 V 2 V 3 V 4 V 5 [red,blue,green,yellow,pnk] [red,blue,green,yellow,pnk] Tght LP relaxaton, easy feasblty: IP wns Almost useless LP relaxaton, hard feasblty: CP wns 15

16 CP+IP: prelmnary results Although CP s not compettve for problems lke TSP and ATSP, the addton of an IP optmzaton component wthn the CP framework allows for the soluton of nstances of sgnfcantly larger sze (Lod et al., 2003) Pure CP gets stuck on problems wth nodes. TSP and ATSP nstances Instance pure AP AP + Lagrangean relaxaton of cuts Opt Tme Fals Opt Tme Fals Gr Fr Bays Dantzg > RY48P 14854* > K 16