T–79.4201 Search Problems and Algorithms T–79.4201 Search Problems and Algorithms Constraint satisfaction: algorithms ◮ For some classes of constraints there are efficient special purpose algorithms (domain specific methods/constraint solvers). Lecture 6: Constraint satisfaction: algorithms ◮ But now we consider general methods consisting of ◮ constraint propagation techniques and ◮ Basic algorithmic framework for solving CSPs ◮ search methods. ◮ DPLL algorithm for Boolean constraints in CNF ◮ We discuss the basic structure of such a general method (procedure Solve below), the components used, and their interaction. ◮ The DPLL algorithm for solving Boolean constraint satisfaction problems given in CNF is presented as an example of such a general method. I.N. & P .O. Autumn 2007 1 I.N. & P .O. Autumn 2007 2 T–79.4201 Search Problems and Algorithms T–79.4201 Search Problems and Algorithms Constraint Programming: Basic Framework Solve procedure Solve: ◮ The procedure Solve takes as input a constraint satisfaction var continue: Boolean ; problem (CSP) and transforms it until it is solved. continue:= TRUE; ◮ It employs a number of subprocedures (Happy, Preprocess, while continue and not Happy do Constraint Propagation, Atomic, Split, Proceed by Cases). Preprocess; ◮ The subprocedures Happy and Atomic test the given CSP to Constraint Propagation; check the termination condition for Solve. if not Happy then ◮ The subprocedures Preprocess and Constraint Propagation if Atomic then transforms the given CSP to another one that is equivalent to it. continue:= FALSE else ◮ Split divides the given CSP into two or more CSPs whose union Split; is equivalent to the CSP . Proceed by Cases ◮ Proceed by Cases specifies what search techniques are used to end process the CSPs generated by Split. end ◮ The subprocedures will be explained in more detail below. end I.N. & P .O. Autumn 2007 3 I.N. & P .O. Autumn 2007 4
T–79.4201 Search Problems and Algorithms T–79.4201 Search Problems and Algorithms Equivalence of CSPs Example. ◮ To understand Solve we need the notion of equivalence. ◮ Consider the following CSPs: P 1 = �{ x 1 < x 2 } ; x 1 ∈ { 1 , 3 } , x 2 ∈ { 1 , 3 }� ◮ Basically, CSPs P 1 and P 2 are equivalent if they have the same P 2 = �{ x 1 < x 3 , x 3 ≤ x 2 } ; x 1 ∈ { 1 , 3 } , x 2 ∈ { 1 , 3 } , x 3 ∈ { 1 , 2 , 3 }� set of solutions. ◮ However, transformations can add new variables to a CSP and ◮ P 1 and P 2 are not equivalent (because they have different sets for then equivalence is understood w.r.t. the original variables. variables). ◮ We say that two value assignments T and T ′ agree on a set of ◮ However, they are equivalent on variables X = { x 1 , x 2 } as variables X if T ( x ) = T ′ ( x ) for all x ∈ X . ◮ for the unique solution T 1 = { x 1 �→ 1 , x 2 �→ 3 } of P 1 there is a ◮ CSPs P 1 and P 2 are equivalent w.r.t. a set of variables X iff corresponding solution T 21 = { x 1 �→ 1 , x 2 �→ 3 , x 3 �→ 3 } of P 2 ◮ for every solution T 1 of P 1 there is a solution T 2 of P 2 such that T 1 such that T 1 and T 21 agree on variables X and and T 2 agree on variables X and ◮ for the solutions T 21 and T 22 = { x 1 �→ 1 , x 2 �→ 3 , x 3 �→ 2 } of P 2 T 1 ◮ for every solution T 2 of P 2 there is a solution T 1 of P 1 such that T 2 is a corresponding solution of P 1 agree on X . and T 1 agree on variables X . ◮ For instance, CSP ◮ Union of CSPs P 1 ,..., P m is equivalent w.r.t. X to a CSP P 0 iff P 0 = �{ x 1 < x 2 ; x 1 ∈ { 1 ,..., 10 } , x 2 ∈ { 1 ,..., 10 }� ◮ for every solution T 0 of P 0 there is a solution T i of P i for some is equivalent w.r.t. { x 1 , x 2 } to the union of 1 ≤ i ≤ m such that T 0 and T i agree on variables X and P 01 = �{ x 1 < x 2 } ; x 1 ∈ { 1 ,..., 5 } , x 2 ∈ { 1 ,..., 10 }� ◮ for each 1 ≤ i ≤ m for every solution T i of P i there is a solution T 0 P 02 = �{ x 1 < x 2 } ; x 1 ∈ { 6 ,..., 10 } , x 2 ∈ { 1 ,..., 10 }� of P 0 such that T i and T 0 agree on variables X . I.N. & P .O. Autumn 2007 5 I.N. & P .O. Autumn 2007 6 T–79.4201 Search Problems and Algorithms T–79.4201 Search Problems and Algorithms Solved and Failed CSPs Transformations ◮ For termination we need to defined when a CSP has been solved ◮ In the following we represent transformations of CSPs by means and when it is failed, i.e., has no solution. of proof rules. ◮ Let C be a constraint on variables y 1 ,..., y k with domains ◮ A rule D 1 ,..., D k ( C ⊆ D 1 ×···× D k ). P 0 ◮ C is solved if C = D 1 ×···× D k and C � = / 0 . P 1 ◮ A CSP is solved if transforms the CSP P 0 to the CSP P 1 . — all its constraints are solved ◮ A rule — no domain of it is empty. P 0 ◮ A CSP is failed if P 1 | ··· | P 1 — it contains the false constraint ⊥ , or transforms the CSP P 0 to the set of CSPs P 1 ,..., P n . — one of its domains or constraints is empty. I.N. & P .O. Autumn 2007 7 I.N. & P .O. Autumn 2007 8
T–79.4201 Search Problems and Algorithms T–79.4201 Search Problems and Algorithms Happy Preprocess This is a test applied to the current CSP to see whether the goal ◮ Bring constraints to desired syntactic form. conditions set for the original CSP have been achieved. Typical ◮ Example: Constraints on reals. conditions include: Desired syntactic form: no repeated occurrences of a variable. ax 7 + bx 5 y + cy 10 = 0 ◮ a solution has been found, ax 7 + z + cy 10 = 0 , bx 5 y = z ◮ all solutions have been found, ◮ a solved form has been reached from which one can generate all (Notice a new variable is introduced.) solutions, Atomic ◮ it is determined that no solution exists (the CSP is failed), ◮ This is a test applied to the current CSP to see whether the CSP ◮ an optimal solution w.r.t. some objective function has been found, is amenable for splitting. ◮ Typically a CSP is considered atomic if the domains of the ◮ all optimal solutions have been found. variables are either singletons or empty. Example. For a CSP ◮ But a CSP can be viewed atomic also if it is clear that search � x 1 + x 2 = x 3 , x 1 − x 2 = 0 ; x i ∈ D i � ‘under’ this CSP is not needed. the solved form could be, for example, For example, this could be the case when the CSP is “solved” or � x 1 = x 2 , x 3 = 2 x 2 ; x i ∈ D i � an optimal solution can be computed directly from the CSP . I.N. & P .O. Autumn 2007 9 I.N. & P .O. Autumn 2007 10 T–79.4201 Search Problems and Algorithms T–79.4201 Search Problems and Algorithms Split — a domain Split x ∈ D ◮ D finite (Enumeration) : x ∈ { a } | x ∈ D −{ a } ◮ After Constraint Propagation, Split is called when the test Happy x ∈ { a 1 ,..., a k } ◮ D finite (Labeling) : fails but the CSP is not yet Atomic. x ∈ { a 1 } | ... | x ∈ { a k } ◮ A call to Split replaces the current CSP P 0 by CSPs P 1 ,..., P n x ∈ [ a .. b ] ◮ D interval of reals (Bisection) : such that the union of P 1 ,..., P n is equivalent to P 0 , i.e., the rule x ∈ [ a .. a + b 2 ] | x ∈ [ a + b 2 .. b ] P 0 Split — a constraint P 1 | ··· | P n ◮ Disjunctive constraints like is applied Start[task1] + Duration[task1] ≤ Start[task2] ∨ Start[task2] + Duration[task2] ≤ Start[task1] ◮ A split can be implemented by splitting domains or constraints. can be split using the rule: C 1 ∨ C 2 ◮ For efficiency an important issue is the splitting heuristics, i.e. C 1 | C 2 which split to apply and in which order to consider the resulting ◮ Constraints in "compound" form: CSPs. | p (¯ x ) | = a p (¯ x ) = a | p (¯ x ) = − a I.N. & P .O. Autumn 2007 11 I.N. & P .O. Autumn 2007 12
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