SweConsNet, May 24 2002

1. Activities of the Combinatorial Problem Solving GroupNicolas BeldiceanuSICSLägerhyddsvägen 1875237 Uppsalaemail: nicolas@sics.se SweConsNet, May 24 2002

2. Outline  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

3.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

4. Overview • Short Term Focus Explicit description of constraints Efficient and generic filtering algorithms • Long Term Perspective Synthetize filtering algorithms from the description of constraints • How to Proceed Interdisciplinary interactions (Applied Math, Algorithmic, Constraint) Implementation (make it available in SICStus Prolog) Feedback from applications (configuration, scheduling, biology, electronic market)

5. Overview • People N.Beldiceanu, M.Carlsson, P.Mildner, T.Szeredi, M.Ågren • Collaborations SICS Kista (M.Aronsson, E.Aurell, P.Kreuger) MPII Saarbrücken (S.Thiel) Uppsala University (Q.Guo)

6.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

7. Motivations for a Classification of Global Constraints • Find out the basic constituents of the global constraint, • Classify the properties of each basic constituent, • Understand how properties interact.

8. Main Idea of the Classification Global Constraints as: Graph Properties onStructuredNetwork ofElementary Constraints of theSame Type

9. nvalue(NVAL, VARIABLES ) : NVAL : dvar VARIABLES: collection(var-dvar) : NVAL 0 NVAL |VARIABLES| required(VARIABLES.var) : VARIABLES : IDENTITY : VARIABLES : CLIQUE : 2 : VARIABLES.var[1] =VARIABLES.var[2] : NSCC = NVAL • ARGUMENT • RESTRICTION(S) • VERTEXINPUT • VERTEXGENERATOR • EDGEINPUT • EDGEGENERATOR • EDGE ARITY • EDGE CONSTRAINT • GRAPH PROPERTY nvalue(4,{var-3,var-1,var-7,var-1,var-6})

10. V2 V1 V3 V5 V4 Collections of items: VARIABLES Vertices generator: IDENTITY V1 V2 V3 V4 V5 Edge generator: CLIQUE

11. Edge constraint: = = = = = = = = = = = = = = = = = 1 Graph property: V2 V2 = 3 7 = NSCC=NVAL V1 V1 V3 V3 = = = 6 1 V5 V5 V4 V4 nvalue(4,{var-3,var-1,var-7,var-1,var-6})

12. A Catalog of Global Constraints Alldifferent Alldifferent_except_0 Alldifferent_interval Alldifferent_modulo Alldifferent_partition Alldifferent_same_value Among Among_interval Among_modulo Among_seq Assign_and_count Assign_and_nvalue Balance Balance_modulo Balance_partition Bin_packing Binary_tree Cardinality_atleast Cardinality_atmost Change Change_continuity Change_pair Change_partition Circuit Circuit_cluster Circular_change Coloured_cumulative Coloured_cumulatives Common Common_interval Common_modulo Common_partition Connect_points Connected Count Crossing Cumulative Cumulative_2d Cumulative_product Cumulatives Cycle Cycle_card_on_path Cycle_cover Cycle_or_accessibility Cycle_resource Cyclic_change Cyclic_change_joker Cyclic_cumulative Derangement Diffn Diff_2 Diff_2_cyclic Diff_2_min_dist Disjoint Disjoint_tasks Distance_change Distance_less Distribute Domain_constraint Element Element_greatereq Element_lesseq Element_sparse Elements Elements_alldifferent Global_cardinality Golomb Graph_crossing Group Group_skip_isolated_item Inflexion Interval_and_count Interval_and_sum Inverse Longest_change Map Max_index Max_n Max_nvalue Maximum Maximum_modulo Maximum_pair Min_index Min_n Min_nvalue Minimum Minimum_except_0 Minimum_modulo Minimum_pair Nclass Nequivalence Ninterval Notallequal Npair Number_of_rest Nvalue Orchad Place_in_pyramid Polyomino Relaxed_sliding_sum Same Same_interval Same_modulo Same_partition Sliding_card_skip0 Sliding_sum Sliding_time_window Smooth Soft_alldifferent_ctr Soft_alldifferent_var

13. A Catalog of Global Constraints Stretch Stretch_circuit Stretch_path Symmetric_alldiff Temporal_path Tree Tree_resource Used_by Used_by_interval Used_by_modulo Used_by_partition • Old constraints which were not inside: element, ... • Invented constraints: disjoint, ... • Catalog updated as new constraints are presented • domain_constraint [REFALO, CP2000] • stretch [PESANT, CP2001] • soft alldifferent [PETIT, RÉGIN, CP2001] • Modifications in the way of describing constraints • possibility to have constraint (like global cardinality, stretch) where • different limits are associated to different values • avoid properties which can’t be evaluated in polynomial time when all • parameters of the constraint are fixed (e.g maximum clique)

14.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

15. Filtering Algorithms for Families of Constraints Alldifferent Alldifferent_except_0 Alldifferent_interval Alldifferent_modulo Alldifferent_partition Alldifferent_same_value Among Among_interval Among_modulo Among_seq Assign_and_count Assign_and_nvalue Balance Balance_modulo Balance_partition Bin_packing Binary_tree Cardinality_atleast Cardinality_atmost Change Change_continuity Change_pair Change_partition Circuit Circuit_cluster Circular_change Coloured_cumulative Coloured_cumulatives Common Common_interval Common_modulo Common_partition Connect_points Connected Count Crossing Cumulative Cumulative_2d Cumulative_product Cumulatives Cycle Cycle_card_on_path Cycle_cover Cycle_or_accessibility Cycle_resource Cyclic_change Cyclic_change_joker Cyclic_cumulative Derangement Diffn Diff_2 Diff_2_cyclic Diff_2_min_dist Disjoint Disjoint_tasks Distance_change Distance_less Distribute Domain_constraint Element Element_greatereq Element_lesseq Element_sparse Elements Elements_alldifferent Global_cardinality Golomb Graph_crossing Group Group_skip_isolated_item Inflexion Interval_and_count Interval_and_sum Inverse Longest_change Map Max_index Max_n Max_nvalue Maximum Maximum_modulo Maximum_pair Min_index Min_n Min_nvalue Minimum Minimum_except_0 Minimum_modulo Minimum_pair Nclass Nequivalence Ninterval Notallequal Npair Number_of_rest Nvalue Orchad Place_in_pyramid Polyomino Relaxed_sliding_sum Same Same_interval Same_modulo Same_partition Sliding_card_skip0 Sliding_sum Sliding_time_window Smooth Soft_alldifferent_ctr Soft_alldifferent_var

16.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

17. Applications of Sweep Algorithms Within theGeometry Literature Database, more than 100 references: • Voronoi diagram • Map overlay • Nearest objects • Triangulations • Hidden surface removals • Rectangles intersection • Shortest path sweep line event point y (1) sweep line status (2) (1)Move to next eventpoint (2)Updatesweep line status x But not yet used within constraint programming !

18. Applications of Sweep Algorithms within Constraint Programming Pruning for the following constraint patterns: • A conjunction of constraints with two sharedvariables • Several conjunction of constraints with one sharedvariable • The cardinality operator with two sharedvariables • The non-overlapping constraint between polygons • A multi-resource cumulatives constraint

19. Y 4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 Examples ofForbiddenandSafeRegions Y Y Y Y X X X X X 0X4 0Y4 0R9 0X4 0Y4 2Z3 0X4 0Y4 1S6 0X4 0Y4 0T0 1U2 0X4 0Y4 X+1T T+1X Y+1U U+4Y alldifferent( {X,Y,4-Y,R}) |X-Y|>Z X+2YS X+Y0 (mod 2)

20. 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 alldifferent({X,Y,4-Y,R}) An Example |X-Y|>Z PROBLEM: Adjust minimum of X according to Y and to the fact that 4 or 5 constraints should hold: X+2YS 0X4 0Y4 2Z3 1S6 0T0 1U2 alldifferent({X,Y,4-Y, R}) |X-Y|>Z X+2YS X+1T  T+1XY+1U  U+4Y X+Y0 (mod 2) X+1T  T+1X Y+1U  U+4Y X+Y0 (mod 2) Deduction: X>1 Y Y Y X=0 X=1 X=2

21.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

22. resource consumption 4  4 3 4 3 2 1 1 2 0 time resource consumption 7  0 6 Cumulated profiles 4 5 2  0 1 1 2 2 3 3 4 4 5 5 6 6 3 1 time 1 0 -1 2 1 0 -1 -2 The cumulative Constraint The originalcumulative constraint [Aggoun & Beldiceanu 92]: Restrict the resource consumption at each point in time. The generalizedcumulatives constraint [Beldiceanu & Carlsson 01]: A pool of cumulative resources , Height of a task can be negative, Maximum or minimum resource consumption , Holds for time-points crossed by at least one task.

23. For all tasks : For all time-points crossed by task : Let be the machine where task is assigned , For all tasks which both cross point , and are assigned to : The sum of the height of tasks is not greater (less) than the capacity of machine . 1 2 3 4 5 6 1 0 -1 2 1 0 -1 -2 t i t D E F I N I T I O N m t s i m s m resource consumption 7  0 6 Machine 2 E X A M P L E 4 5 2  0 3 1 Machine 1 time

24.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

25. Include Objective Within Constraint

26. FILTERING ALGORITHMS Algorithms sum_of...(Variables, Values, Cost) find BOUNDS for Cost PROPAGATE from bounds of Costto Variables Lower bound LB: domination Remove val from var iff: LB+lower_regret(var,val)>max(Cost) Upper bound UB: matching Remove val from var iff: UB-upper_regret(var,val)<min(Cost)

27. Algorithms (main results) Lower bound LB: domination O(n log n + m) for a tight bound (when intervals) n:number of variables m: number of values O(n log n + m) for computing all the exact lower regret of all values (when intervals) n:number of variables m: number of values Upper bound UB: matching O(m log m + c e) for a tight bound m: number of values c:cardinality of max.matching e:nb.of edges in bipartite graph O(e) for computing all the exact upper regret e:number of edges in bipartite graph

28.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

29. Engineering Integrate standard constraints from other systems Global cardinality, sort, ... Reuse algorithms from literature Alldifferent [Mehlhorn, Thiel], Knapsack (dynamic programming [Trick]) Incremental algorithms Special cases when go down, optimizations valid for several global constraints (target, source) Reingenering of the code Revisiting filtering algorithms for inserting explanations Making the system available on several platforms

30.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

31. Different Aspects of Trace Control of Execution Posting constraints, demons, waking constraints, entailment, failure, choice points Locating Information Situate a piece of information according to the context it originally came Domain Modification Intendedpruning, domain modification, domain Declarative Aspect Explanation(s) for pruning, failing, adding a constraint, waking a demon Procedural Aspect Method used or not, nested block structure, method dependant explanations

32.  OVERVIEW  CLASSIFICATION OF GLOBAL CONSTRAINTS  GENERIC FILTERING ALGORITHMS  SWEEP BASED FILTERING ALGORITHMS  RESOURCES CONSTRAINT  COST FILTERING ALGORITHMS  ENGINEERING OF FILTERING ALGORITHMS  CONSTRAINT DEBUGGING  IMPLIED CONSTRAINTS  SUMMARY AND CONCLUSION

33. Implied Constraints (as relations between graph properties) QUESTION For a directed graph, relation between:  NV: number of vertices,  NA: number of arcs,  NCC: number of connected components. ANSWER INTUITION The maximum number of arcs is achieved by having a complete clique in each connected components and the minimum number of connected components. PROPAGATION RULES

34. Implied Constraints (as relations between graph properties) QUESTION For a directed graph, relation between:  NV: number of vertices,  NA: number of arcs,  NCC : number of connected components,  NSCC: number of strongly connected components. GUESS

35. Summary and Conclusion • Provide a Constraint Classification • Generic Propagation Algorithms for several Families of Constraints • Use Algorithms from Graph and Geometry • Several Constaints Implemented within SICStus and Used within Demonstrators Collaboration required with: applied mathematics, design of algorithms

36. Further Sources of Information Papers (some at http://www.sics.se/libindex.html) • Sweep as a Generic Pruning Technique Applied to the Non-Overlapping Rectangles Constraint, [Beldiceanu,Carlsson 2001] . • Sweep as a Generic Pruning Technique Applied to Constraint Relaxation, [Beldiceanu,Carlsson 2001] . • Non-overlapping Constraints between Convex Polytopes, [Beldiceanu,Guo,Thiel 2001] . • A New Multi-Resource cumulatives Constraint with Negative Heights, [Beldiceanu,Carlsson 2001] . • Sweep Synchronization as a Global Propagation Mechanism[Beldiceanu,Carlsson, Thiel 2002] . (submitted) • Cost-Filtering Algorithms for the two Sides of the Sum of Weights of Distinct Values Constraint, [Beldiceanu,Thiel 2002] . (in preparation) Implementation (http://www.sics.se/sicstus/) • Alldifferent, Assignment, Case, Circuit, Count, Cumulative, Cumulatives, Disjoint1, Disjoint2, Element, Global_cardinality, Knapsack, Scalar_product, Serialized, Sort, Sum, Sum_of_weights_of_distinct_values.