Valued Constraints

Islands of Tractability

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)

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

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.

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.

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).

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

Soft Constraint Languages

A voyage of Discovery

• In general the VCSP is NP-hard.

• It generalizes CSP.

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.

• …

Valued Constraint Languages

• For any domain D, and valuation structure a k-ary cost function is a mapping from Dk to .

• A valued constraint language (for D and ) is any set of cost functions.

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.

Example: MAX-CSP• The corresponding MAX-CSP instance for

a CSP instance can be obtained replacing each constraint <,with scope and relation , by the valued constraint <,> where:

• The VCSP problem for the language of 0/1 cost functions is just MAX-CSP.

Hard and Easy Languages

Boolean Not Equals

Two NP-hard Languages

Ternary Equality, and all Unary Cost functions

Variable:

Cost 1

Cost 0

Legend

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

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.

Submodular Cost Functions

• This cost function is submodular

• And this one is not.

Reasoning about Tractability

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]

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.

A Multimorphism 1: Technical

A Multimorphism 2: Definition

A Multimorphism 3: Example

A Multimorphism 4: Example

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).

Languages Characterised by Multimorphisms

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.

Boolean Not Equals

Two NP-hard Languages

Ternary Equality, and all Unary Cost functions

Variable:

Cost 1

Cost 0

Legend

These two languages have no

multimorphisms (to speak of)

Majority/Minority FunctionsCompletely characterised by a

multimorphism.

Max,Max FunctionsCompletely characterised by a

multimorphism.

ConstantCompletely characterised by a

multimorphism.

Min,Max FunctionsNearly characterised by a

multimorphism.

Open Questions

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.

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.