130 September 2003 Tractability by Approximating Constraint Languages Martin Green and David Cohen CP 2003 Ninth International Conference on Principles and Practice of Constraint Programming

230 September 2003Tractability by Approximating Constraint Languages Outline of the Talk  Background information (4 minutes)  Approximating tractable languages (8 minutes)  Applications of this new theory (6 minutes)  Remarks and directions for future research (2 minutes)

330 September 2003Tractability by Approximating Constraint Languages Question  What do the following problems have in common?  Stable Marriage Problem  Renamable HORN  Row Convexity

430 September 2003Tractability by Approximating Constraint Languages Answer  We search for and apply domain permutations (approximations) to each problem variable  We can use this technique to approximate:  SMP instance ) Max-Closed  Renamable HORN ) HORN-SAT  Permutably Row Convex ) Row Convex  … and the approximations are tractable

530 September 2003Tractability by Approximating Constraint Languages Constraint Satisfaction Problem Instances  A Constraint Satisfaction Problem instance (CSP), P, is a triple h V,D,C i where:  V is a set of variables  D is any set, called the domain of the instance  C is a set of constraints  Each constraint c 2 C is a pair h ,  i where  is a list of distinct variables of V and  is a |  |-ary relation over D  A solution to P is a mapping  such that   Informally:  We describe V as a set of questions that need to be answered  D is the set of all possible answers that can be given to these questions  A constraint in C is a rationality condition that limits the answers that may be simultaneously assigned to some groups of questions  A solution is then a satisfactory set of answers to all of the questions

630 September 2003Tractability by Approximating Constraint Languages Complexity of Constraint Satisfaction  The decision problem for the general constraint satisfaction problem is:  Given a CSP, P, does P have a solution?  For general CSPs this is NP-complete  However, there are restrictions to the set of allowed instances that make the constraint satisfaction problem tractable  It turns out that there are many tractable subproblems of the general constraint satisfaction problem

730 September 2003Tractability by Approximating Constraint Languages Reasons for Tractability  Structural  Actually there is just the acyclic structure  … and approximations  Relational or Language-based  There are many known examples of language-based tractable subproblems (for example, max-closed)  Here we give an approximation technique which, for instance, allows us to extend the maximal tractable binary max-closed language binary This is a maximal class of binary relations

830 September 2003Tractability by Approximating Constraint Languages Definition of Max-Closure  An n -ary relation, , over ordered domain  1, …, k  is said to be max-closed if whenever h d 1, …,d n i, h e 1, …,e n i are in  then so is their pointwise maximum  h max (d 1,e 1 ), …, max (d n,e n ) i  The set of all max-closed relations forms a tractable constraint language  We can display relations diagrammatically  Consider the ordered domain  1,2  where 1<2  This relation is not max-closed  However, this relation is max-closed 22 11

930 September 2003 Approximating Tractable Languages

1030 September 2003Tractability by Approximating Constraint Languages Permuting the Domain  Suppose we have a tractable language  and that P is a CSP not in CSP (  ), that is, some relation is not in   If we can find permutations of the domain (independently) for each variable, that make P into an instance of CSP (  ), then we can solve the instance P using the algorithm for   We first permute the domains  Then we apply the algorithm for  on the permuted instance  Finally we permute the domains back again for any discovered solution  It is this approximation technique for (tractable) constraint languages that we will discuss in the remainder of this talk

1130 September 2003Tractability by Approximating Constraint Languages Permuting the Domain (2)  We can see whether a relation can be permuted into a relation in  by testing combinations of permutations  This gives rise to a lifted relation  Example:  There are only two permutations over  1,2    1 ! 1,2 ! 2  ( keep ) and  1 ! 2,2 ! 1  ( swap )  We can independently apply these permutations to both sides of the relation  We might obtain a max-closed relation by applying one of the permutations to each domain, e.g., swap to both sides  The lifted relation is:  h keep,swap i  h swap,keep i, h swap,swap i 

1230 September 2003Tractability by Approximating Constraint Languages Approximating Tractable Languages  Let  be a constraint language, P = h V,D,C i a CSP and G a set of permutations of D  If there exists a permutation of the domain, from G, for each variable of P, such that the permuted CSP has constraint relations all in  then we say that P is G -approximately over   For a given  and G the problem of determining whether an instance is G -approximately over  is called the approximation problem for  and G  For any set of CSP instances over D, we may ask whether the approximation problem (for  and G ) is tractable

1330 September 2003Tractability by Approximating Constraint Languages Approximating Tractable Languages (2)  We can determine whether an approximation exists for a given instance by considering a lifted CSP with the same structure but whose domains are permutations  Domain: 1,21,2 V1V1 V2V2 V3V If I swap one side I must also swap the other I must swap one side only I must swap at least one side  h keep,keep i, h swap,swap i   h keep,swap i, h swap,keep i   h keep,swap i, h swap,keep i, h swap,swap i  h keep,keep i h keep,swap ih swap,keep i keep swap  keep,swap  A CSPThe Lifted CSP Permuted CSP

1430 September 2003 Applications … of the new theory

1530 September 2003Tractability by Approximating Constraint Languages Is Approximating Tractable?  It may be hoped that tractable languages have tractable approximations  Clearly approximating const-0 is not tractable  Is approximating the binary max-closed language tractable?  Consider all binary relations over a domain of size three (there are 512)  We wish to lift them into the binary max-closed language  The lifted language has 458 distinct relations  We can use Polyanna to determine tractability  They are intractable! It is rarely tractable to approximate even tractable languages Luckily there are useful tractable approximations

1630 September 2003Tractability by Approximating Constraint Languages Novel Classes of Tractable CSPs  Theorem  Let  be any constraint language over D, G be a set of two permutations of D, and R be the set of all binary relations  Then the G -approximation problem for  is tractable  Proof  Any lifted relation is binary two valued  The approximation problem for R is 2-SAT

1730 September 2003Tractability by Approximating Constraint Languages Novel Classes of Tractable CSPs (Example)  Let  D be the ordered domain  1, …,k ,   be the set of binary max-closed relations and  G  keep,swap   This approximation problem is tractable  This tractable class includes  all binary max-closed CSPs  all binary min-closed CSPs  … and some others  This class is clearly hybrid

1830 September 2003Tractability by Approximating Constraint Languages Stable Marriage Problem  We have a set W of n women, and a set M of n men  Each woman w has a preference order for all the men given by  w  Similarly, each man m has a preference ordering,  m, that ranks the women  We are to form n marriages such that every pair of marriages is stable

1930 September 2003Tractability by Approximating Constraint Languages Variables: Domain Values: We deduce Binary constraints

2030 September 2003Tractability by Approximating Constraint Languages Stable Marriage Problem (2)  It turns out that every SMP instance is approximately max-closed  We order the men (domain) according to the preference list for each woman  This completely explains the known solution algorithm

2130 September 2003Tractability by Approximating Constraint Languages Renamable HORN  A set of clauses is Renamable HORN if there is a replacement of some literals, uniformly in all clauses, with their negated versions, which makes all clauses into HORN-clauses  This approximation problem is tractable (because the lifted language is majority closed)

2230 September 2003Tractability by Approximating Constraint Languages Row Convexity  A CSP instance is said to be Row Convex if, after some permutation of each domain, each relation is Row Convex  This approximation problem is tractable (because the lifted language has only unary relations)

2330 September 2003 Closing Remarks

2430 September 2003Tractability by Approximating Constraint Languages Conclusions  We have identified a novel, hybrid, class of tractable subproblems of the general constraint satisfaction problem  The theory also gives a unifying explanation for the tractability of:  the constraint approach to the Stable Marriage Problem;  recognising instances of Renamable HORN;  finding domain permutations for Row Convex CSP instances

2530 September 2003Tractability by Approximating Constraint Languages Future Research  We want to determine whether we can tractably find the domain permutations for instances of the Stable Marriage Problem for which we do not know the preference orderings  We wish to discover if it is tractable to identify approximately Connected Row Convex instances  We hope to discover or explain other tractable classes for which the approximation problem is tractable Any Questions?