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Constraint Programming Michael Trick (actually 75% Pascal Van Hentenryck, 20% Irv Lustig, 5% Trick) Carnegie Mellon

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Outline Motivation An Overview of Constraint Programming Constraint Programming at Work Getting Started Sports Scheduling Manufacturing Perspectives

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Combinatorial Optimization Many, many practical applications Resource allocation, scheduling, routing Properties Computationally difficult Technical and modeling expertise needed Experimental in nature Important ($$$) in practice Many solution techniques Integer programming Specialized methods Local search/metaheuristics Constraint programming

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Began in 1980s from AI world Prolog III (Marseilles, France) CLP(R) CHIP (ECRC, Germany) Application areas Scheduling, sequencing, resource and personnel allocation, etc. etc. Active research area Specialized conferences (CP, CP/AI-OR, …) Journal (Constraints) Companies

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Constraint Programming Two main contributions A new approach to combinatorial optimization Orthogonal and complementary to standard OR methods Combinatorial versus numerical A new language for combinatorial optimization Rich language for constraints Language for search procedures Vertical extensions

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The Tutorial Goal: to provide an introduction What is constraint programming? What is it good for? How does it compare to integer programming? How easy is it to use? What is the underlying technology?

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Constraint Programming Constraint programming by example Illustrate rich language Contrast with integer programming Illustrate some underlying technologies Disclaimers Can’t cover all of CP I want to make you curious Language/system used Could use many; choose OPL

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Modeling in Constraint Programming A rich constraint language Arithmetic, higher-order, logical constraints Global constraints for natural substructures Specification of a search procedure Definition of search tree to explore Specification of search strategy

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Comparison of CP/IP Branch and Prune Prune: eliminate infeasible configurations Branch: decompose into subproblems Prune Carefully examine constraints to reduce possible variable values Branch Use heuristics based on feasibility info Main focus:constraints and feasibility Branch and Bound Bound: eliminate suboptimal solutions Branch: decompose into subproblems Bound Use (linear) relaxation of problem (+ cuts) Branch Use information from relaxation Main focus: objective function and optimality

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Illustrative artificial example Color a map of (part of) Europe: Belgium, Denmark, France, Germany, Netherlands, Luxembourg No two adjacent countries same color Is four colors enough?

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OPL example enum Country {Belgium,Denmark,France,Germany,Netherlands,Luxem bourg}; enum Colors {blue,red,yellow,gray}; var Colors color[Country]; solve { color[France] <> color[Belgium]; color[France] <> color[Luxembourg]; color[France] <> color[Germany]; color[Luxembourg] <> color[Germany]; color[Luxembourg] <> color[Belgium]; color[Belgium] <> color[Netherlands]; color[Belgium] <> color[Germany]; color[Germany] <> color[Netherlands]; color[Germany] <> color[Denmark]; }; Variables non- numeric Constraints are non-linear Looks nothing like IP! Perfectly legal CP

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Constraint Programming Domain store For each variable: what is the set of possible values? If empty for any variable, then infeasible If singleton for any variable, then solution Constraints Capture interesting and well studied substructures Need to Determine if constraint is feasible WRT the domain store Prune “impossible” values from the domains

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Constraints Can have differing techniques to “handle” a constraint type: 3x+10y+2z + 4w = 4 x in {0,1}, y in {0,1,2}, z in {0,1,2}, w in {0,1} Simple bound on sizes gives y in {0} More complicated handling gives x in {0}, y in {0}, z in {0,2}, w in {0,1}

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Constraint Solving General algorithm is Repeat select a constraint c if c is infeasible wrt domain store return infeasible else apply pruning algorithm of c Until no value can be removed

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Branching Once the constraint solving is done, if the problem is not infeasible nor are the domains singletons, then apply the search method Choose a variable x with non-singleton domain (d1, d2, … di) Foreach d in (d1, d2, … di) add constraint x=di to problem and solve

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Show OPL solving coloring problem

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Strength of CP Since there is no need for a linear relaxation, the language can represent much more directly (no need for big-M IP formulations.

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Examples of formulation abilities Facility location: want a constraint that customer j can be assigned to warehouse i only if warehouse open. (y[i]=1 if warehouse i open) IP: x[i,j] is 1 if cust j assigned to i x[i,j] <= y[i] CP:x[j] is the warehouse cust j assigned to (not a 0,1 variable) y[x[j]] = 1;

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Similar example Routing type constraints. Let x[i] be the ith customer visited and d[i,j] be distance from i to j sum (i in 1..n) d[x[i],x[i+1]] gives total distance traveled

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Formulation strengths Logical requirements: if A=1 and B =3 or D=1. Really painful in IP. Straightforward in CP: ((A=1) &(B ((C>=3)\/(D=1))

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Global Constraints Recognize that some types of constraints come up often Create specialized routines to handle Strong pruning Efficient handling Extend system to include these

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Global constraint: alldifferent Most well known and studied constraint. alldifferent(x,y,z) states that x, y, and z take on different values. So x=2, y=1, z=3 would be ok, but not x=1, y=3, z=1. Clear uses in routing (x[i] is i th customer visited, alldifferent[x] says each customer visited at most once), very useful in many other situations.

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Alldifferent feasibility and pruning Feasibility? Given domains, create domain/variable bipartite graph x1 x2 x3 x4 x5 1 2 3 4 5

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Alldifferent feasibility and pruning Pruning? Which edges are in no matching? x1 x2 x3 x4 x5 1 2 3 4 5 Domain is sharply reduced

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Global constraints Many different types of constraints have specialized routines distribute(card,value,base): the number of times value[i] appears in base is card[i] circuit(succ) : the values in succ form a hamiltonian circuit (so if you follow the sequence 1, succ[1], succ[succ[1]] etc, you will get a loop through 1..n.

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Global constraints Many others, and new ones being created all the time Strengthen and expand the language Make modeling easier and more natural System is faster at finding solutions Details hidden to user

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Vertical language extensions Can add constraints and definitions to make modeling even more natural Ideas remain the same: there are domains and constraints; constraints check for feasibility and prune domains; a search strategy guides the system in finding solutions

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Scheduling Want concepts of jobs, machines, “before”, “after”, jobs requiring machines, and so on. Easy to extend

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Example of scheduling forall(j in Jobs) forall(t in 1..nbTasks-1) task[j,t] precedes task[j,t+1]; forall(j in Jobs) forall(t in Tasks) task[j,t] requires tool[resource[j,t]];

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Search Strategy Combined with model, search strategies are integral to constraint systems. Allow choice of branching variables or more powerful search strategies Can be key in solving problems Two steps Specify tree to search Specify how to explore the tree

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Example of Search Strategies forall(s in Stores ordered by increasing regretdmax(cost[s])) tryall(w in Warehouses ordered by increasing supplyCost[s,w]) supplier[s] = w; }; implements a maximum regret ordering (find a store with maximum regret then order the warehouses by increasing cost)

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Example Problem Painting cars (from Magnanti and Sokel). Sequence cars to minimize paint changeover Cars cannot be sequenced too far out of order

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Small example Small Example: 10 cars in sequence. The order for assembly is 1, 2,..., 10. A car must be painted within 3 positions of its assembly order. For instance, car 5 can be painted in positions 2 through 8 inclusive. Cars 1, 5, and 9 are red; 2, 6, and 10 are blue; 3 and 7 green; and 4 and 8 are yellow. Initial sequence 1, 2,... 10 corresponds to color pattern RBGYRBGYRB and has 9 purgings. The sequence 2,1,5,3,7,4,8,6,10,9 corresponds to color pattern BRRGGYYBBR and has 5 purgings.

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Constraint Program int n=…; int rnge=…; int ncolor=…; range Slots 1..n; var Slots slot[1..n]; var Slots revslot[1..n]; int color[1..n]= …; minimize sum (j in 1..n-1) (color[revslot[j]] <> color[revslot[j+1]]) subject to { forall (i in Slots) i-rnge<=slot[i] <= i+rnge; /*Must be in range */ alldifferent(slot); /*must choose different slots */ forall (i in Slots) revslot[slot[i]] = i; };

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Personal use Tremendous help in my work on sports scheduling: much easier to formulate idiosyncratic constraints Very fast to create prototypes Competitive (at least!) to IP approaches

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Result Formulation is much easier than IP formulation Gets good solutions much faster than IP Is competitive in proving optimality

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Finding optimal solutions Constraint programs can find optimal solutions. Typically works by finding a feasible solution and adding a constraint that future solutions must be better than it. Repeat until infeasible: the last solution found is optimal

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Perspectives Many solution techniques Integer programming Constraint programming Local search Combinations Which to use?

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Comparing IP and CP Complementary technologies Integer programming Objective function: relaxations Constraint programming Feasibility: domain reductions Might need to experiment with both CP particularly useful when IP formulation is hard or relaxation does not give much information

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Combining Methods Local and Global Search Use CP/IP for very large neighborhood search (take a solution, remove large subset, find optimal completion) Combining CP and IP Use LP as constraint handler Use CP as subproblem solver in branch and price ……

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Conclusions Constraint programming should become a part of every OR person’s toolkit Combinations of CP and IP represent a “big thing” in future techniques Blurring of lines between optimization and heuristics This talk at http://mat.gsia.cmu.edu/INFORMS/cp.ppt

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