1. Valued Constraints Islands of Tractability

2. Agenda The soft constraint formalism (5 minutes) Valued Constraint Languages (5 minutes) Hard and Easy Languages (10 minutes) Reasoning about Tractability (10 min) Languages and multimorphisms (15 min) Open Questions (5 minutes)

3. Soft Constraints Classical constraint satisfaction problems answer questions about feasibility. We can give costs to tuples in constraint relations – crisp case just 0 and 1. This allows us to compare complete assignments by aggregating costs for individual constraints …and so to answer optimization questions

4. Soft Constraint Problem Instance A set of problem variables; A domain of values; A set of constraints; A set of costs (valuation structure) Each constraint has a: –Scope: list of concerned variables; –Cost Function: cost of each assignment.

5. Assignment Costs: Axioms ? is the best value. > is the worst value. ­ models projection and is a commutative, associative and idempotent. © models aggregation and is commutative and associative; 8 a : (a ­ > = a) Æ (a © ? ) = a; 8 a : (a ­ ? = ? ) Æ (a © > ) = > ; © distributes over ­ : 8 a,b,c : (a © (b ­ c) = (a © b) ­ (a © c)). We then define: (a · b), (a ­ b = b). With respect to · we can show ­ and © are monotonic.

6. VCSP framework Here we insist · is totally ordered. Then the costs are a valuation structure. We write: –0 to mean ? (the best value); – 1 to mean > (the worst value); –Projection ( ­ ) becomes minimum; If © is strictly monotonic then we can also subtract costs (we get ª ).

7. Valued CSP Instance A set of problem variables; A domain of values; A set of constraints; A set of costs (valuation structure) Each constraint has a: –Scope: list of concerned variables; –Cost Function: cost of each assignment. A totally ordered set with a strictly monotonic aggregation operator

8. Soft Constraint Languages

9. A voyage of Discovery In general the VCSP is NP-hard. It generalizes CSP.

10. Possible Islands The constraint scopes form a hypergraph. The cost functions are a set of functions from the domain to a valuation structure. We could restrict the hypergraph structure or the types of cost functions of a set of instances to find an island of tractability. …

11. Valued Constraint Languages For any domain D, and valuation structure  a k-ary cost function is a mapping  from D k to . A valued constraint language (for D and  ) is any set  of cost functions.

12. Example: Relations A relation can be seen as a cost function that only takes the values 0 and 1. …So the VCSP obtained by restricting to functions with values 0 and 1 is the classical CSP. This gives the first few islands of tractability.

13. Example: MAX-CSP The corresponding MAX-CSP instance for a CSP instance can be obtained replacing each constraint where: The VCSP problem for the language of 0/1 cost functions is just MAX-CSP.

14. Hard and Easy Languages

15. Boolean Not Equals Two NP-hard Languages Ternary Equality, and all Unary Cost functions Variable: Cost 1Cost 0 Legend

16. Submodular Set Functions Let S be any set and  a real valued function. We say that  is submodular if –  (X) +  (Y) ¸  (X [ Y) +  (X Å Y) We can use these functions to express optimization problems. We know that this optimization (minimization) problem is tractable (seventh power of problem size). For example:  (X) = |X| For example:  (X) = 5

17. Submodular Cost Functions We can represent a submodular function on a set as a cost function on a list of Boolean (0/1) variables (valued constraint): –Union becomes MAX; –Intersection becomes MIN. We can extend the definition to non- Boolean ordered domains. This (finite cost) language is still tractable.

18. Submodular Cost Functions This cost function is submodular And this one is not.

19. Binary Submodular If we restrict our attention to submodular functions that are binary then we obtain: This result is obtained by decomposing binary submodular functions into so called generalized interval functions. The complexity of solving an optimisation problem whose constraints are binary submodular is cubic.

20. Generalized Interval Functions We solve a VCSP whose cost functions are GI functions by finding the minimal cut in a particular weighted digraph. The weighted graph has O(nd) vertexes and hence this takes O(n 3 d 3 ) steps.

21. Reasoning about Tractability

22. Tractability? We have a complete characterization of tractable Boolean MAX-SAT languages. –There are just three maximal tractable languages: 0-valid, 1-valid or 2-monotone [Creignou 1995] We have a characterization of the tractability of crisp constraint languages. –They have a non-trivial polymorphism [Jeavons, Cohen, Gyssens 1996]

23. Tractability? We generalise the notion of a polymorphism to a multimorphism. The maximal tractable MAX-SAT languages are characterised by single multimorphisms. So this is a good place to search for islands of tractability.

24. A Multimorphism 1: Technical

25. A Multimorphism 2: Definition

26. A Multimorphism 3: Example

27. A Multimorphism 4: Example

28. Expressibility If multimorphisms are to be able to capture complexity then it has to be the case that those cost functions expressed by  have the multimorphisms of . Since valued languages extend crisp languages it had better be the case that polymorphisms lead to analogous multimorphisms (and vice-versa).

29. Languages Characterised by Multimorphisms

30. Characterisation It means much to say that every known example of a tractable language indeed has a multimorphism. It means more still to observe that they are all characterised by single multimorphisms. It means even more to observe that the intractable languages have no multimorphisms.

31. Boolean Not Equals Two NP-hard Languages Ternary Equality, and all Unary Cost functions Variable: Cost 1Cost 0 Legend These two languages have no multimorphisms (to speak of)

32. Majority/Minority Functions Completely characterised by a multimorphism.

33. Max,Max Functions Completely characterised by a multimorphism.

34. Constant Completely characterised by a multimorphism.

35. Min,Max Functions Nearly characterised by a multimorphism.

36. Open Questions

37. Expressibility and Multimorphisms Do multimorphisms capture expressibility? –We have done some work on this and cannot show that it is not true! Do multimorphisms capture complexity? (or are we just lucky?) –In the submodular case we have no proof for non-binary that allowing infinite costs is tractable.

38. Algebra of Multimorphisms If multimorphisms are the right thing to study then have they been studied before? We achieved a great deal by discovering the (known) work on clones and polymorphisms.