Integer vs. constraint programming. IP vs. CP: Language

Transcription

1 Discrete Math for Bioinformatics WS 0/, by A. Bockmayr/K. Reinert,. Januar 0, 0:6 00 Integer vs. constraint programming Practical Problem Solving Model building: Language Model solving: Algorithms IP vs. CP: Language IP Variables (mostly) 0- Finite domain Constraints Linear equations Arithmetic constraints and inequalities Symbolic/global constraints CP Example Variables: x,...,x n {0,...,m } Constraint: Pairwise different values Example () Integer programming: Only linear equations and inequalities x i x j x i < x j x i > x j x i x j x i x j + Eliminating disjunction x i x j + my, x j x i + my, y + y =, y,y {0,}, 0 x i,x j m, New variables: z ik = iff x i = k, i =,...,n, k = 0,...,m z i0 + + z im =, z k + + z nk, Constraint programming symbolic constraint alldifferent(x,...,x n ) Symbolic/global constraints alldifferent([x,...,x n ]) cumulative([s,...,s n ],[d,...,d n ],[r,...,r n ],c,e). n tasks: starting time s i, duration d i, resource demand r i resource capacity c, completion time e

2 00 Discrete Math for Bioinformatics WS 0/, by A. Bockmayr/K. Reinert,. Januar 0, 0:6 B C C A A B A B C cumulative([,,], [,,], [,,], ) cumulative([,,], [,,], [,,], ) cumulative([,,], [,,], [,,], ) Beldiceanu/Contejean 9 Diffn Constraint Nonoverlapping of n-dimensional rectangles [O,...,O n,l,...,l n ], where O i (resp. L i ) denotes the origin (resp. length) in dimension i diffn([[o,...,o n,l,...,l n ],...,[O m,...,o mn,l m,...,l mn ]]) diffn([[,,,],[,,,],[,,,]]) General form: diffn(rectangles,min_vol,max_vol,end,distances,regions) IP vs. CP: Algorithms IP CP Inference Linear programming Domain filtering Cutting planes Constraint propagation Search Branch-and-relax Branch-and-bound Branch-and-cut Bounds on Two-sided One-sided the objective function Local vs. global reasoning Linear arithmetic constraints x + y, y + x, x + y = z, x,y {0,...,}

3 Discrete Math for Bioinformatics WS 0/, by A. Bockmayr/K. Reinert,. Januar 0, 0:6 00 y CP x,y, z LP x,y, z. IP x,y, z 0 0 x Global reasoning in CP? global constraints! Global reasoning in CP Example x,x,x {0,} pairwise different values Local consistency. disequalities: x x,x x,x x x,x,x {0,}, i.e., no domain reduction is possible Global constraint: alldifferent(x,x,x ) detects infeasibility (uses bipartite matching) Global reasoning in CP: inside global constraints Summary ILP CP(FD) Language Linear arithmetic Arithmetic constraints Symbolic constraints Global consistency (LP) Local consistency Algorithms Cutting planes Domain reduction Branch-and-bound User-defined enumeration Branch-and-cut Symbolic constraints more expressivity + more efficiency Unifying framework for CP and IP: Branch-and-infer (Bockmayr/Kasper 98),..., SCIP Discrete Tomography Binary matrix with m rows and n columns Horizontal projection numbers (h,...,h m ) Vertical projection numbers (v,...,v n )

4 00 Discrete Math for Bioinformatics WS 0/, by A. Bockmayr/K. Reinert,. Januar 0, 0:6 Properties Horizontal convexity (h) Vertical convexity (v) Connectivity (polyomino) (p) H V Complexity (Woeginger 0) polynomial: ( ), (p,v,h) NP-complete: (p,v), (p,h), (v,h), (v), (h), (p) IP Model Variables x ij = { 0 cell(i, j) is labeled white cell(i, j) is labeled black Constraints I: Projections n j= x ij = h i, m i= x ij = v j Constraints II: Convexity hi= x ik x i(k+) n m h i x ik + x il h i, v j x kj + x lj v j, l=k+h i l=k+v j IP Model (contd) Constraints III: Connectivity j+h i j+h i x ik x i+,k h i k=j k=j hi= h i+ = x i Various linear arithmetic models possible, e.g. convexity Enormous differences in size and running time, e.g. day vs. < sec Large number of constraints ( mn in the above model) Finite Domain Model Variables x i start of horizontal convex block in row i, for i m y j start of vertical convex block in column j, for j n

5 Discrete Math for Bioinformatics WS 0/, by A. Bockmayr/K. Reinert,. Januar 0, 0:6 00 X Y H V Domain x i [,...,n h i + ], for i m y j [,...,m ] v j +, for j n Conditional Propagation Projection/Convexity modelled by FD variables Compatibility of x i and y j x i j < x i + h i y j i < y j + v j for i m and j n row i Conditional propagation if x i j then (if j < x i + h i then (y j i, i < y j + v j )) Finite Domain Model (contd) Connectivity row i row i+ x i x i + h i - n n x i+ + h i+ - x i+ Block i must start before the end of block i + x i x i+ + h i+, for i m Block i + must start before the end of block i x i+ x i + h i, for i m Cumulative

6 006 Discrete Math for Bioinformatics WS 0/, by A. Bockmayr/K. Reinert,. Januar 0, 0:6 6 6 d and d Diffn Model 6 6 Propositional satisfiability x,...,x n {0,} 0- variables (or atomic formulae in propositional logic) A literal L is a 0- variable x or its negation x = x. A clause is set of literals C = {L,...,L k } corresponding to the logical disjunction L L k or the clausal inequality L + + L k. A clause set is a set of clauses S = {C,...,C m } corresponding to the logical conjunction C C m. A clause set S is satisfiable if there exists an assignment I : {x,...,x n } {0,} making the logical formula true (equivalently, if the system of clausal inequalities has a 0- solution). Examples. Clause set S = { {x,x }, {x,x } } Corresponding logical formula: (x x ) (x x ) Corresponding system of clausal inequalities: x + x, x x Satisfying assignments: I(x ) =,I(x ) = 0 and I (x ) = 0,I (x ) =.. The clause set S = { {x,x }, {x,x } {x,x }, {x,x } } is unsatisfiable.

7 Discrete Math for Bioinformatics WS 0/, by A. Bockmayr/K. Reinert,. Januar 0, 0:6 00 SAT problem SAT problem: Given a set of clauses S, is S satisfiable? Theorem (Cook ): SAT is NP-complete. There exist highly efficient SAT solvers. Enormous progress has been made during the last 0- years, see e.g. SAT competitions SAT is a third general approach for solving constraint satisfaction/optimization problems (in addition to IP and CP) Davis-Putnam procedure function Satisfiable(S) return boolean /* unit resolution */ repeat for each clause {L} in S of length do delete from S every clause containing L delete L in every clause of S containing L od if S = /0 then true else if S contains the empty clause C = /0 then false fi until no further changes /* branching */ choose a literal L occuring in S if Satisfiable(S {L}) then true else if Satisfiable(S {L}) then true else false fi end function Example Let S be the clause set { x, x, x, x, x }, { x, x, x, x, }, { x, x, x, x }, { x, x, }, { x, x, }, { x, x, x }, { x }. Satisfying assignment: I(x ) = 0,I(x ) = I(x ) =,I(x ) = 0.