LECTURE 6 CONSTRAINT PROGRAMMING & GLOBAL CONSTRAINTS

Transcription

1 LECURE 6 CONSRAIN PROGRAMMING & GLOBAL CONSRAINS

2 AI (C.S.P.) AI languages (Alice, Constraints) Logic Programming Historical Account of Constraint Programming Constraint (Logic) Programming AI (CSP) Operations Research Efficient Constraint Solvers Operation research heorem Proving Etended Unification (, AC, ) Concurrent Logic Languages (Parlog, CP, GHC/KL, ) Industrial applications Concurrent Constraint Languages

3 Constraint Programming In the 990 s, Constraint Solving echniques has been integrated in several programming paradigms after LP : OO (C++, Java), functional (Lisp,ML) Also in the 000 s several specific modeling languages appeared: OPL, Comet, MiniZinc,... Need for data + programming abstractions herefore CP is the combination of: Host language (to state the constraints) Constraint Vocabulary (arithmetic, symbolic, global, ) Constraint Solver(s)

4 Modeling / Programming / Solving Modeling corresponds to state the problem with a certain number of constraints Programming consists in generating these constraints in a efficient and/or elegant way Solving amounts to find & generate a solution to the original problem Of course modeling, programming and solving are related A Constraint Programming system feature all

5 Modeling Modeling problems with constraints depends on the constraint vocabulary available e.g. only arithmetic constraints? or also symbolic constraints such as all_different, etc? Depends on the vocabulary, depends on the solver Also : Different views of the problem lead to different models Different model can use different set of variables It is important to compare different models And sometime to combine different models

6 Eample: the element constraint Synta: element(i,l,x) meaning that element I of list L to be equal to X Similar to arrays : L[i] = X Note that the constraint is a multi-directional relation hus the values of I and/or X can be constrained

7 An eample Simple assignment problem: four workers w,w,w,w and four products p,p,p,p. Assign workers to products to make profit >= 9 Profit matri: p p p p w 7 w 8 5 w 7 w 6

8 Basic Operations Research Model (linear equations) 6 Boolean variables Bij meaning worker i is assigned product j i [,]: i [,]: 7B B 8B P B P 9 j j B B B B B B ij ij B 5B 7B 6B B B B B p p p p w 7 w 8 5 w 7 w 6 prim. constraints B 8 choices to find all four solutions

9 Better Model (symbolic constraints) Make use of disequalities and symbolic constraints Four variables,,, corresponding to workers alldifferent([,,, ]) element (,[7,,,], P) element (,[8,,5,], P) element (,[,,7,], P) element (,[,,6,], P) P P P P P P 9 p p p p w 7 w 8 5 w 7 w 6 7 prim. constraints choices to find all four solutions

10 Different Model (also with symbolic constraints) Four variables,,, corresponding to products alldifferent([,,, ]) element(,[7,8,,], P) element(,[,,,], P) element(,[,5,7,6], P) element(,[,,,], P) P P P PP P 9 p p p p w 7 w 8 5 w 7 w 6 7 prim. constraints 7 choices to find all four solutions

11 Comparing Models B B B B Relative efficiency comes from more direct mapping to primary constraints fewer variables usually requires empirical evaluation Other criteria fleibility (new constraints) e.g. worker works on product > worker 0, B 0 B B B B B B?????

12 Combining Models Combine models by relating the variables and there values in each model e.g.. B = means = means = Combined models can gain more information through propagation Can use reified constraints: (V=D) (V=D)

13 Combined Model ),[,,,], ( ),[,5,7,6], ( ),[,,,], ( ),[7,8,,], ( ]),,, ([ P P P P P P element P element P element P element alldifferent 9 ),[,,6,], ( ),[,,7,], ( ),[8,,5,], ( ),[7,,,], ( ]),,, ([ P P P P P P P element P element P element P element alldifferent ) ), ( ), ( ), ( ( ), ), ( ), ( ), ( ( ), ), ( ), ( ), ( ( ), ), ( ), ( ), ( ( 9 prim. constraints. 5 choices to find all solutions

14 Programming CSP Models can be huge and hard to write E.g. huge data from which to generate the constraints Need for: loops, recursion, etc data abstraction I/O (e.g. data from file) Sub-problems / sub-programs hus including CSP facilities into a programming language

15 Constraint Programming Systems GNU Prolog: CLP language (see also Sictus Prolog, B-Prolog, ) Comet (Dynadec): CP modeling language + solver, also includes Local Search Gecode : CP library for C++ IBM ILOG CP CPLEX Optimizer: both OPL modeling language and also a CP library for C++, also include MIP (CPLEX) & some Local Search library

16 God save the Queens Place 8 queens on a chessboard s.t. no two queens attack each other Modeling with constraints : 8 variables {Q,...,Q8} value of Qi = column of queen on line i Domains : {,...,8} constraints : i [,8] j [,8] s.t. j > i Qi Qj Qi Qj + i -j Qi Qj - i + j alternatively : all_different({q,...,q 8 }), all_different({q +,...,Q 8 +8}), all_different({q -,...,Q -8}),

17 GNU Prolog queens(n,l):- fd_set_vector_ma(n), length(l,n), fd_domain(l,,n), safe(l), fd_labelingff(l). safe([]). safe([x L]):- noattack(l,x,), safe(l). noattack([],_,_). noattack([y L],X,I):- diff(x,y,i), I #= I+, noattack(l,x,i). diff(x,y,i):- X#\=Y, X#\=Y+I, X+I#\=Y.

18 GeCode (basic version) class Queens : public Script { public: IntVarArray q; Queens(bool share, Queens& s) : Script(share,s) { } q.update(*this, share, s.q); Queens(const SizeOptions& opt) : q(*this,opt.size(),0,opt.size()-) { const int n = q.size(); for (int i = 0; i<n; i++) for (int j = i+; j<n; j++) { } rel(*this, q[i]!= q[j]); rel(*this, q[i]+i!= q[j]+j); rel(*this, q[i]-i!= q[j]-j); virtual Space* copy(bool share) { return new Queens(share,*this); } branch(*this, q, IN_VAR_SIZE_MIN, IN_VAL_MIN); }

19 Comet (basic version) import cotfd; Solver<CP> cp(); int n = 5; range S =..n; var<cp>{int} Q[i in S,j in S](cp,0..); solve<cp>{ forall(i in S){ } cp.post(sum(j in S) Q[i,j] == ); cp.post(sum(j in S) Q[j,i] == ); forall(i in S){ cp.post(sum(j in..i) Q[j,j+(n-i)] <= ); cp.post(sum(j in..i) Q[j+(n-i),j] <= ); cp.post(sum(j in..i) Q[-j+i+,j] <= ); cp.post(sum(j in..i) Q[n-j+,n-i+j] <= ); } } using{ } forall(i in S,j in S){ try<cp> cp.post(q[i,j]==0); cp.post(q[i,j]==); } forall(i in S) cout << all(j in S) Q[n-i+,j] << endl; cout << cp.getnfail() << endl;

20 IBM ILOG Solver (basic version)

21 OPL Studio IBM Ilog Solver (using alldifferent constraints) int n << "number of queens:"; range Domain..n; var Domain queens[domain]; solve { alldifferent(queens); forall(ordered i,j in Domain) { abs(queens[i]-queens[j])<> abs(i-j) ; }; };

22 Mini Zinc Modeling Language developed by NICA (Australia) now standard in Constraint Programming community Intermediate language between Zinc and FlatZinc array [..n] of var..n: q; predicate noattack(int: i, int: j, var int: qi, var int: qj) = qi!= qj /\ qi + i!= qj + j /\ qi - i!= qj - j; constraint forall (i in..n, j in i+..n) ( noattack(i, j, q[i], q[j]) ); solve satisfy; output [ "queens:", show(q), "\n"];

23 Global Constraints hat does it mean to be global? No real definition of a global constraint but: Involves several variables (more global view of subproblem) Has some specific propagation/filtering mechanism (more efficient than basic AC or BC) Somehow a global contraint regroups several basic constraints and treat them together Basic eample: all_different(x, Xn) (to be detailed later)

24 Global Constraints () [from Van Hentenryck 009]

25 Global Constraints () [from Van Hentenryck 009]

26 Global Constraint Catalogue A unified catalogue regrouping many GC, their definition, semantics and filtering algorithms hy those constraints? Found to be useful for modeling applications some level of abstraction more than 5 different Global Constraints!

27

28 Global Constraints & dedicated filtering algorithms (from [van Hoeve 00])

29 An Eample of Specialized Filtering : all_different constraint. all_different(x, Xn) replaced by n(n+)/ disequations not much filtering, e.g. X Y, Y Z, Z X with domains {0,}. Reasoning with number of values in the union of domains of variables ok for previous eample, but... all_different(x,y,z,) with D =D y =D z ={,} and D ={,,,5}...??. Same, but consider all subsets of variables. Graph algorithm [Regin 99]: Maimal bipartite matching (here: size ) X Y Z 5

30 Filtering for all_different [from Van Hoeve 00]

31 Filtering for all_different () [from Van Hoeve 00]

32 Filtering for all_different () [from Van Hoeve 00]

33 Another version of all_different n j n,, ),, ent( all_differ Conve hull relaation [Hooker, illiams & Yan 00] (strongest possible linear relaation) J j j n j j n J n J J J n n with,, all ), ( ) ( For n = :,,,,,,,, 6 6, 6, 6, 0

34 he sequence constraint Eample: Nurse Roistering in hospital [from Van Hoeve 00]

35 he sequence constraint () [from Van Hoeve 00]

36 he sequence constraint () Best algorithm for this type of filtering is O(n³) [from Van Hoeve 00]

37 Cumulative Scheduling From disjunctive to cumulative scheduling Capacity duration = energy eample [,] : or

38 Eample cumulative resource problem eample: Scheduling problem with precedence Maimal capacity of 5 (shared resource) he 6 tasks have to fit in a bo of maimal height = 5 Here, makespan =

39 he cumulative constraint Cumulative([S,..,Sn], ([D,..,Dn], ([R,..,Rn], C) Si = start time of task i Di = duration of task i Ri = resource usage of task i (per time unit) C = resource capacity (limit) Eample: cumulative([a,b,c,d,e,f], [,6,,,5,6], [,,,,,],5)

40 he cumulative constraint () Specialized filtering can be done by considering compulsory parts Basic idea for one task: hen combine for all tasks

41 he cumulative constraint () More efficient (but more comple) filtering schemes than checking compulsory part only are possible Edge-Finding: Identify pairs (S,i) such that task i cannot precede (resp. follow) any task in subset S in any feasible schedule Update the earliest starting date (resp. latest finishing date) accordingly Problem: perform edge-finding efficiently Best algorithm is in polynomial time: O(n²)

42 Eperiment by yourself! GNU Prolog Comet Gecode IBM ILOG CP CPLEX Optimizer