Tractability by Approximating Constraint Languages

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

2. 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) Tractability by Approximating Constraint Languages

3. Question • What do the following problems have in common? • Stable Marriage Problem • Renamable HORN • Row Convexity Tractability by Approximating Constraint Languages

4. 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 Tractability by Approximating Constraint Languages

5. A Constraint Satisfaction Problem instance (CSP), P, is a triple hV,D,Ci 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 c2C 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 Constraint Satisfaction Problem Instances Tractability by Approximating Constraint Languages

6. 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 Tractability by Approximating Constraint Languages

7. Reasons for Tractability This is a maximal class of binary relations • 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 Tractability by Approximating Constraint Languages

8. 2 2 1 1 Definition of Max-Closure • We can display relations diagrammatically • Consider the ordered domain {1,2} where 1<2 • This relation is not max-closed • An n-ary relation, , over ordered domain {1, …, k} is said to be max-closed if whenever hd1,…,dni, he1,…,eni are in  then so is their pointwise maximum • hmax(d1,e1),…,max(dn,en)i • The set of all max-closed relations forms a tractable constraint language • However, this relation is max-closed Tractability by Approximating Constraint Languages

9. Approximating Tractable Languages

10. 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 Tractability by Approximating Constraint Languages

11. 2 2 1 1 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: {hkeep,swapi,hswap,keepi,hswap,swapi} Tractability by Approximating Constraint Languages

12. Approximating Tractable Languages • Let  be a constraint language, P = hV,D,Ci 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 Tractability by Approximating Constraint Languages

13. 2 2 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 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} If I swapone side I must also swap the other {hkeep,keepi, hswap,swapi} hkeep,keepi {keep,swap} keep 2 keep 2 V1 V2 The Lifted CSP Permuted CSP A CSP {hkeep,swapi, hswap,keepi, hswap,swapi} I must swap one side only I must swap at least one side {hkeep,swapi, hswap,keepi} hkeep,swapi hswap,keepi swap 2 1 V3 Tractability by Approximating Constraint Languages

14. Applications … of the new theory

15. 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 Tractability by Approximating Constraint Languages

16. 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 Tractability by Approximating Constraint Languages

17. 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 Tractability by Approximating Constraint Languages

18. 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 Tractability by Approximating Constraint Languages

19. Domain Values: We deduce Binary constraints Variables: Tractability by Approximating Constraint Languages

20. 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 Tractability by Approximating Constraint Languages

21. 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) Tractability by Approximating Constraint Languages

22. 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) Tractability by Approximating Constraint Languages

23. Closing Remarks

24. 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 Tractability by Approximating Constraint Languages

25. 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? Tractability by Approximating Constraint Languages