Constraint Programming

Transcription

1 1 / 28 Constraint Programming Lecture 4. Global Constraints. Ruslan Sadykov INRIA Bordeaux Sud-Ouest 5 February 2018

2 2 / 28 Importance of global constraints A global constraint is a union of simple constraints. Use of global constraints facilitates the modeling (smaller number of constraints, libraries of constraints) ; accelerates the solving (specialised, and thus efficient, algorithms for propagation). Important Global constraints contribute a lot to the succes of Constraint Programming in practice. Catalogue of global constraints : http ://sofdem.github.io/gccat/

3 2 / 28 Importance of global constraints A global constraint is a union of simple constraints. Use of global constraints facilitates the modeling (smaller number of constraints, libraries of constraints) ; accelerates the solving (specialised, and thus efficient, algorithms for propagation). Important Global constraints contribute a lot to the succes of Constraint Programming in practice. Catalogue of global constraints : http ://sofdem.github.io/gccat/

4 2 / 28 Importance of global constraints A global constraint is a union of simple constraints. Use of global constraints facilitates the modeling (smaller number of constraints, libraries of constraints) ; accelerates the solving (specialised, and thus efficient, algorithms for propagation). Important Global constraints contribute a lot to the succes of Constraint Programming in practice. Catalogue of global constraints : http ://sofdem.github.io/gccat/

5 2 / 28 Importance of global constraints A global constraint is a union of simple constraints. Use of global constraints facilitates the modeling (smaller number of constraints, libraries of constraints) ; accelerates the solving (specialised, and thus efficient, algorithms for propagation). Important Global constraints contribute a lot to the succes of Constraint Programming in practice. Catalogue of global constraints : http ://sofdem.github.io/gccat/

6 2 / 28 Importance of global constraints A global constraint is a union of simple constraints. Use of global constraints facilitates the modeling (smaller number of constraints, libraries of constraints) ; accelerates the solving (specialised, and thus efficient, algorithms for propagation). Important Global constraints contribute a lot to the succes of Constraint Programming in practice. Catalogue of global constraints : http ://sofdem.github.io/gccat/

7 3 / 28 Lignes directrices «Simple» constraints All-diff GCC Constraints for scheduling

8 4 / 28 Global constraint ScalProd scal_prod(x 1,..., X n, c 1,..., c n, v) Equivalent to n c i X i = v. i=1 Rules for achieving arc-b-consistency ( D Xi = [x i, x i ], c i > 0 ) x i max x i, v 1 j n: j i c i } max {c j x j, c j x j x i min x i, v 1 j n: j i c i } min {c j x j, c j x j

9 4 / 28 Global constraint ScalProd scal_prod(x 1,..., X n, c 1,..., c n, v) Equivalent to n c i X i = v. i=1 Rules for achieving arc-b-consistency ( D Xi = [x i, x i ], c i > 0 ) x i max x i, v 1 j n: j i c i } max {c j x j, c j x j x i min x i, v 1 j n: j i c i } min {c j x j, c j x j

10 5 / 28 Global constraint Element element(x, v 1,..., v n, Y ) Equivalent to X = v Y. We should have D Y {1,..., n}. This constraint allows one to use variables as indices. Can be represented as the following explicit constraint : (X, Y ) {(v i, i)} 1 i n. So, the arc-consistency can be achieved using classic algorithms (AC-3, AC-4, etc).

11 5 / 28 Global constraint Element element(x, v 1,..., v n, Y ) Equivalent to X = v Y. We should have D Y {1,..., n}. This constraint allows one to use variables as indices. Can be represented as the following explicit constraint : (X, Y ) {(v i, i)} 1 i n. So, the arc-consistency can be achieved using classic algorithms (AC-3, AC-4, etc).

12 5 / 28 Global constraint Element element(x, v 1,..., v n, Y ) Equivalent to X = v Y. We should have D Y {1,..., n}. This constraint allows one to use variables as indices. Can be represented as the following explicit constraint : (X, Y ) {(v i, i)} 1 i n. So, the arc-consistency can be achieved using classic algorithms (AC-3, AC-4, etc).

13 6 / 28 Lignes directrices «Simple» constraints All-diff GCC Constraints for scheduling

14 7 / 28 Global constraint all-different all-different(x 1,..., X n ) Replaces n2 2 binary constraints X i X j, 1 i < j n. The most used constraint in practice : assignment, permutation,... We can achieve arc-consistency in time O( nnd) using graph theory (Régin, 94) tools (to compare with complexity O(n 2 d 2 ) if we take constraints one by one).

15 7 / 28 Global constraint all-different all-different(x 1,..., X n ) Replaces n2 2 binary constraints X i X j, 1 i < j n. The most used constraint in practice : assignment, permutation,... We can achieve arc-consistency in time O( nnd) using graph theory (Régin, 94) tools (to compare with complexity O(n 2 d 2 ) if we take constraints one by one).

16 7 / 28 Global constraint all-different all-different(x 1,..., X n ) Replaces n2 2 binary constraints X i X j, 1 i < j n. The most used constraint in practice : assignment, permutation,... We can achieve arc-consistency in time O( nnd) using graph theory (Régin, 94) tools (to compare with complexity O(n 2 d 2 ) if we take constraints one by one).

17 8 / 28 Matching problem in a graph Bipartite graph is a graph vertices of which can be partitioned in two subsets U and V such that each edge incident to one vertex in U and to one in V. Matching in a graph is a set of disjoint edges (which do have common incident vertices) Maximum matching in a graph is a maximum size matching. Alternating path (given a matching) is a path in which the edges belong alternatively to the matching and not to the matching.

18 9 / 28 All-different constraint : propagation Algorithm : 1. We construct a bipartite «value» graph. 2. We search for a maximum matching (if its size is < n, then there is no solution). 3. Given this matching, we establish edges which do not belong to an alternating circuit, an alternating path such that one of its extremities is a free vertex, the matching found. 4. We remove values which correspond to these edges. Exemple : D x1 = {1, 2}, D x2 = {2, 3}, D x3 = {1, 3}, D x4 = {3, 4}, D x5 = {2, 4, 5, 6}, D x6 = {5, 6, 7} all-different(x 1,..., x 6 ) x 1 x 2 x 3 x 4 x 5 x

19 9 / 28 All-different constraint : propagation Algorithm : 1. We construct a bipartite «value» graph. 2. We search for a maximum matching (if its size is < n, then there is no solution). 3. Given this matching, we establish edges which do not belong to an alternating circuit, an alternating path such that one of its extremities is a free vertex, the matching found. 4. We remove values which correspond to these edges. Exemple : D x1 = {1, 2}, D x2 = {2, 3}, D x3 = {1, 3}, D x4 = {3, 4}, D x5 = {2, 4, 5, 6}, D x6 = {5, 6, 7} all-different(x 1,..., x 6 ) x 1 x 2 x 3 x 4 x 5 x

20 9 / 28 All-different constraint : propagation Algorithm : 1. We construct a bipartite «value» graph. 2. We search for a maximum matching (if its size is < n, then there is no solution). 3. Given this matching, we establish edges which do not belong to an alternating circuit, an alternating path such that one of its extremities is a free vertex, the matching found. 4. We remove values which correspond to these edges. Exemple : D x1 = {1, 2}, D x2 = {2, 3}, D x3 = {1, 3}, D x4 = {3, 4}, D x5 = {2, 4, 5, 6}, D x6 = {5, 6, 7} all-different(x 1,..., x 6 ) x 1 x 2 x 3 x 4 x 5 x

21 9 / 28 All-different constraint : propagation Algorithm : 1. We construct a bipartite «value» graph. 2. We search for a maximum matching (if its size is < n, then there is no solution). 3. Given this matching, we establish edges which do not belong to an alternating circuit, an alternating path such that one of its extremities is a free vertex, the matching found. 4. We remove values which correspond to these edges. Exemple : D x1 = {1, 2}, D x2 = {2, 3}, D x3 = {1, 3}, D x4 = { 3, 4}, D x5 = { 2, 4, 5, 6}, D x6 = {5, 6, 7} all-different(x 1,..., x 6 ) x 1 x 2 x 3 x 4 x 5 x

22 10 / 28 Global constraint assignment assignment(x 1,..., X n, Y 1,..., Y n ) It is the symmetric all-different constraint : X i = j Y j = i, 1 i, j n. We should have D X {1,..., n}, D Y {1,..., n}. Used for mutual assignment. We can achieve arc-consistency using the same algorithm as for the all-different constraint. X 1 X 2 X 3 X 4 X 5 Y 1 Y 2 Y 3 Y 4 Y 5

23 10 / 28 Global constraint assignment assignment(x 1,..., X n, Y 1,..., Y n ) It is the symmetric all-different constraint : X i = j Y j = i, 1 i, j n. We should have D X {1,..., n}, D Y {1,..., n}. Used for mutual assignment. We can achieve arc-consistency using the same algorithm as for the all-different constraint. X 1 X 2 X 3 X 4 X 5 Y 1 Y 2 Y 3 Y 4 Y 5

24 10 / 28 Global constraint assignment assignment(x 1,..., X n, Y 1,..., Y n ) It is the symmetric all-different constraint : X i = j Y j = i, 1 i, j n. We should have D X {1,..., n}, D Y {1,..., n}. Used for mutual assignment. We can achieve arc-consistency using the same algorithm as for the all-different constraint. X 1 X 2 X 3 X 4 X 5 Y 1 Y 2 Y 3 Y 4 Y 5

25 10 / 28 Global constraint assignment assignment(x 1,..., X n, Y 1,..., Y n ) It is the symmetric all-different constraint : X i = j Y j = i, 1 i, j n. We should have D X {1,..., n}, D Y {1,..., n}. Used for mutual assignment. We can achieve arc-consistency using the same algorithm as for the all-different constraint. X 1 X 2 X 3 X 4 X 5 Y 1 Y 2 Y 3 Y 4 Y 5

26 11 / 28 Lignes directrices «Simple» constraints All-diff GCC Constraints for scheduling

27 12 / 28 Global constraint GCC GCC(X 1,..., X n, v 1,..., v k, l 1,..., l k, u 1,..., u k ) This global cardinality constraint is a generalisation of all-different : the number of times each value v j is taken should be inside interval [l j, u j ] (for all-different, l j = 0, u j = 1, j). Often used in practice : complicated assignment, distribution,... We can achieve the arc-consistency for constraint GCC in time O(n 2 d) by using the maximum flow algorithm (Régin, 99).

28 12 / 28 Global constraint GCC GCC(X 1,..., X n, v 1,..., v k, l 1,..., l k, u 1,..., u k ) This global cardinality constraint is a generalisation of all-different : the number of times each value v j is taken should be inside interval [l j, u j ] (for all-different, l j = 0, u j = 1, j). Often used in practice : complicated assignment, distribution,... We can achieve the arc-consistency for constraint GCC in time O(n 2 d) by using the maximum flow algorithm (Régin, 99).

29 12 / 28 Global constraint GCC GCC(X 1,..., X n, v 1,..., v k, l 1,..., l k, u 1,..., u k ) This global cardinality constraint is a generalisation of all-different : the number of times each value v j is taken should be inside interval [l j, u j ] (for all-different, l j = 0, u j = 1, j). Often used in practice : complicated assignment, distribution,... We can achieve the arc-consistency for constraint GCC in time O(n 2 d) by using the maximum flow algorithm (Régin, 99).

30 13 / 28 Maximum flow in a graph Notations Let D = (V, A) be an directed graph in which to each arc (i, j) A we associate a capacity u ij. Flow definition A flow in graph D is a function f defined on arcs of D : 0 f ij u ij, (i, j) A, f ij = f ji, j V \ {s, d}. i: (i,j) A i: (j,i) A Problem Find the maximum flow which can be sent from s to t.

31 14 / 28 Construction of maximum flow Algorithm : 1. We start with a feasible flow f (for example, zero flow). 2. We construct the residual directed graph R = (V, A ) : f ij > 0 (j, i) A ; fij < u ij (i, j) A. 3. If there exists a path from s to t in the residual graph, we increase the flow along this path as much as possible. 4. If such path does not exists, the flow is maximum. Example : 3,4 3 5,5 s = 1 2,3 t = 4 2,2 2 0,1

32 14 / 28 Construction of maximum flow Algorithm : 1. We start with a feasible flow f (for example, zero flow). 2. We construct the residual directed graph R = (V, A ) : f ij > 0 (j, i) A ; fij < u ij (i, j) A. 3. If there exists a path from s to t in the residual graph, we increase the flow along this path as much as possible. 4. If such path does not exists, the flow is maximum. Example : 3,4 3 5,5 s = 1 2,3 t = 4 2,2 2 0,1 3 s = 1 t = 4 2

33 14 / 28 Construction of maximum flow Algorithm : 1. We start with a feasible flow f (for example, zero flow). 2. We construct the residual directed graph R = (V, A ) : f ij > 0 (j, i) A ; f ij < u ij (i, j) A. 3. If there exists a path from s to t in the residual graph, we increase the flow along this path as much as possible. 4. If such path does not exists, the flow is maximum. Example : 3,4 3 5,5 s = 1 2,3 t = 4 2,2 2 0,1 3 s = 1 t = 4 2 4,4 3 5,5 s = 1 1,3 t = 4 2,2 2 1,1

34 14 / 28 Construction of maximum flow Algorithm : 1. We start with a feasible flow f (for example, zero flow). 2. We construct the residual directed graph R = (V, A ) : f ij > 0 (j, i) A ; f ij < u ij (i, j) A. 3. If there exists a path from s to t in the residual graph, we increase the flow along this path as much as possible. 4. If such path does not exists, the flow is maximum. Example : 3,4 3 5,5 s = 1 2,3 t = 4 2,2 2 0,1 3 s = 1 t = 4 2 4,4 3 5,5 s = 1 1,3 t = 4 2,2 2 1,1

35 15 / 28 Propagation of constraint GCC I We construct the «value» graph : Peter M(1,2) Paul Mary D(1,2) s John N(1,1) t Bob Mike B(0,2) Julia O(0,2)

36 16 / 28 Propagation of constraint GCC II We find the maximum flow : Peter M(1,2) Paul Mary D(1,2) 2 2 s John N(1,1) 1 1 t Bob Mike B(0,2) 1 Julia O(0,2)

37 17 / 28 Propagation of constraint GCC III We construct the residual graph induces by the maximum flow : Peter M(1,2) Paul Mary D(1,2) s John N(1,1) t Bob Mike B(0,2) Julia O(0,2)

38 18 / 28 Propagation of constraint GCC IV We establish non-saturated arcs which between variables and values which do not belong to any circuit in the residual graph (decomposition in strongly connected components) : Peter M(1,2) Paul Mary D(1,2) s John N(1,1) t Bob Mike B(0,2) Julia O(0,2)

39 19 / 28 Propagation of constraint GCC V We remove values which correspond to these edges : Peter M(1,2) Paul Mary John D(1,2) N(1,1) Bob Mike Julia B(0,2) O(0,2)

40 20 / 28 Lignes directrices «Simple» constraints All-diff GCC Constraints for scheduling

41 21 / 28 Global constraint disjunctive disjunctive(x 1,..., X n, p 1,..., p n ) Replaces n2 2 logical binary constraints : X i + p i X j Xi X j + p j, i, j : i j. Often used for scheduling problems (often with D xi = [r i, d i p i ])

42 22 / 28 Global constraint disjunctive : propagation I Achieving arc-b-consistency for this constraint is NP-hard. Weaker («Edge-Finding») propagation is used : max d i min r i < p i Ω précède k i Ω i Ω {k} r k max Ω Ω Ω {k} { min i Ω r i + i Ω p i } k = 1 Ω = {2, 3} r k 8

43 22 / 28 Global constraint disjunctive : propagation I Achieving arc-b-consistency for this constraint is NP-hard. Weaker («Edge-Finding») propagation is used : max d i min r i < p i Ω précède k i Ω i Ω {k} r k max Ω Ω Ω {k} { min i Ω r i + i Ω p i } k = 1 Ω = {2, 3} r k 8

44 23 / 28 Global constraint disjunctive : propagation II «Edge-Finding» detects if job k should be executed before (after) all jobs in set Ω. We can verify all possible pairs (Ω, k) in time O(n log n). (Carlier & Pinson, 94). «Not-First/Not-Last» detects if job k should be execute before (after) at least one job in set Ω. We can verify all possible pairs (Ω, k) in time O(n log n). (Vilím, 04). «Detectable precedences»,...

45 23 / 28 Global constraint disjunctive : propagation II «Edge-Finding» detects if job k should be executed before (after) all jobs in set Ω. We can verify all possible pairs (Ω, k) in time O(n log n). (Carlier & Pinson, 94). «Not-First/Not-Last» detects if job k should be execute before (after) at least one job in set Ω. We can verify all possible pairs (Ω, k) in time O(n log n). (Vilím, 04). «Detectable precedences»,...

46 23 / 28 Global constraint disjunctive : propagation II «Edge-Finding» detects if job k should be executed before (after) all jobs in set Ω. We can verify all possible pairs (Ω, k) in time O(n log n). (Carlier & Pinson, 94). «Not-First/Not-Last» detects if job k should be execute before (after) at least one job in set Ω. We can verify all possible pairs (Ω, k) in time O(n log n). (Vilím, 04). «Detectable precedences»,...

47 24 / 28 Global constraint Cumulative r rd p 1 cumulative(x 1,..., X n, p 1,..., p n, rd 1,..., rd n, r) Jobs should not overlap ; + each job i consumes rd i units of the resource ; + at each time moment we cannot use more than r units of the resource. It is a generalisation of disjunctive, for which rd i = 1, i, et r = 1.

48 24 / 28 Global constraint Cumulative r rd p 1 cumulative(x 1,..., X n, p 1,..., p n, rd 1,..., rd n, r) Jobs should not overlap ; + each job i consumes rd i units of the resource ; + at each time moment we cannot use more than r units of the resource. It is a generalisation of disjunctive, for which rd i = 1, i, et r = 1.

49 25 / 28 Global constraint cumulative : example f (6) source d (2) e (5) puits a (2) b (6) c (2) If X c 9, X e 4, X f 14, then, after propagation of precedence constraints D(X a ) = [0, 1], D(X b ) = [2, 3], D(X c ) = [8, 9], D(X d ) = [0, 2], D(X e ) = [2, 4], D(X f ) = [0, 14].

50 25 / 28 Global constraint cumulative : example f (6) source d (2) e (5) puits a (2) b (6) c (2) If X c 9, X e 4, X f 14, then, after propagation of precedence constraints D(X a ) = [0, 1], D(X b ) = [2, 3], D(X c ) = [8, 9], D(X d ) = [0, 2], D(X e ) = [2, 4], D(X f ) = [0, 14].

51 26 / 28 Cumulative : Time-table propagation (1) e f a d b c Obligatory parts date de début au plus tôt partie obligatoire p i date de fin au plus tard

52 26 / 28 Cumulative : Time-table propagation (1) e f a d b c Obligatory parts date de début au plus tôt partie obligatoire p i date de fin au plus tard

53 26 / 28 Cumulative : Time-table propagation (1) e f d a b c Obligatory parts date de début au plus tôt partie obligatoire p i date de fin au plus tard

54 27 / 28 Cumulative : Time-table propagation (2) f a e b c Complexity Time-table propagation can be done in timeo(n log n) (Lahrichi, 82)

55 27 / 28 Cumulative : Time-table propagation (2) f a e b c Complexity Time-table propagation can be done in timeo(n log n) (Lahrichi, 82)

56 28 / 28 Constraint Cumulative : other «propagations» «Edge-Finding» detects if job k should start before (finish after) all the jobs in set Ω. We can verify all possible pairs (Ω, k) in time O(n 2 ) (Kameugne et al, 11) or in time O(rn log n) (Vilim, 09) «Not-First/Not-Last» detects if job k should be execute after (before) at least one job in set Ω. We can verify all possible pairs (Ω, k) in time O(n 3 ). (Baptiste et al., 01).

57 28 / 28 Constraint Cumulative : other «propagations» «Edge-Finding» detects if job k should start before (finish after) all the jobs in set Ω. We can verify all possible pairs (Ω, k) in time O(n 2 ) (Kameugne et al, 11) or in time O(rn log n) (Vilim, 09) «Not-First/Not-Last» detects if job k should be execute after (before) at least one job in set Ω. We can verify all possible pairs (Ω, k) in time O(n 3 ). (Baptiste et al., 01).