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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.