foundations of constraint processing, fall 2005 september 12, 2005csp 101: continued1 foundations of...
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September 12, 2005 CSP 101: continued 1
Foundations of Constraint Processing, Fall 2005
Foundations of Constraint Processing
CSCE421/821, Fall 2005:
www.cse.unl.edu/~choueiry/F05-421-821/
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 123B
Tel: +1(402)472-5444
Constraint Satisfaction 101
September 12, 2005 CSP 101: continued 2
Foundations of Constraint Processing, Fall 2005
OutlineMotivating example, application areas
CSP: Definition, representation
Some simple modeling examples
More on definition and formal characterization
Basic solving techniques
Implementing backtrack search
Advanced solving techniques
Issues & research directions
CSP in a nutshell
Constraint Logic Programming (quickly)
September 12, 2005 CSP 101: continued 3
Foundations of Constraint Processing, Fall 2005
OutlineAdvanced solving techniques
Decomposition
Deep analysis (islands of tractability)
Distributed CSPs
Issues & research directions
CSP in a nutshell
Constraint Logic Programming (quickly)
September 12, 2005 CSP 101: continued 4
Foundations of Constraint Processing, Fall 2005
Decomposition• Decomposition
• Conjunctive
• Disjunctive
ConjunctiveDisjunctive
V1
V2
V3
V5
V6
V7
{ a, b, c, ...}
{ a, b, c, d}
V4
V3 V6
V7
V4
V1
V2V5
{ a, b, c, ...}
{ a, b, c, d}{ m, n, o, p}
{ .., .., .., ..}
{ .., .., .., ..}
{ .., .., .., ..}
{ .., .., .., ..}{ .., .., .., ..}
{ .., .., .., ..}
{ .., .., .., ..}{ .., .., .., ..}
{ m, n, o, p}
V4
{ .., .. }{ .., .. }
{b, c, ..}
{ .. }{ .., .. } V6
{ a, c, d}V5
{ n, o}V2
V3
V1
V1
V2
V3
V5
V6
V7
{ a, b, c, ...}
{ a, b, c, d}
V4{ m, n, o, p}
{ .., .., .., ..}{ .., .., .., ..}
{ .., .., .., ..} { .., .., .., ..}
{ .., .., .. } { .., .., .. }
V4{ a, c }
{ .., .. }
V7
V5
{ a, b }V1
V2
V3{ .., .. }
V6
{ m, p }
September 12, 2005 CSP 101: continued 5
Foundations of Constraint Processing, Fall 2005
Properties of decompositions
}P,,P,{P i21 P• Conjunctive or disjunctive?
• Consistent: No constraint is removed
• Simplifying: Size(Pi) < Size( P)
• Semi-complete: At least one solution is kept• Complete: No solution is lost
• Redundant: Solutions replicated in { Pi}
• Reducible: may be < Size( P)
September 12, 2005 CSP 101: continued 6
Foundations of Constraint Processing, Fall 2005
Deep analysis
Uncover particular properties, e.g.– bound the required level of consistency (islands of tractability)– predict ease/difficulty of solving a given instance
• Structure, topology of the constraint graph– tree, DAGs, chordal, bounded-width/induced width, k-trees, etc.
• Types, semantics of the constraints– linear inequalities, subsets of Allen's relations, functional,
monotonic, row-convex, all-diffs, etc.
• Order parameter (phase transition)
September 12, 2005 CSP 101: continued 7
Foundations of Constraint Processing, Fall 2005
Phase transition [Cheeseman et al. ‘91]
Cos
t of
sol
ving
Mostly solvable problems
Mostly un-solvable problems
Order parameterCritical value of order parameter
• Significant increase of cost around critical value• In CSPs, order parameter is constraint tightness & ratio
• Algorithms compared around phase transition
September 12, 2005 CSP 101: continued 8
Foundations of Constraint Processing, Fall 2005
Distributed CSPs• Mainly in search
– Asynchronous BT (e.g., work of Yokoo)– Fine grain local search (ERA, by Liu)– Privacy of constraints
• More purist multi-agent approaches– In scheduling & resource allocation– Based on decomposition of problem/solvers
• Let’s take a wider perspective than what is done today…
September 12, 2005 CSP 101: continued 9
Foundations of Constraint Processing, Fall 2005
Multi agent approach1. Computational tasks:
problem decomposed, replicated
2. Types of agent:broker, solver, problem, solver+problem
3. Architecture and authority:hierarchical, egalitarian, priority-based (e.g., vote)
4. Nature of communication:agent-to-agent, group, broadcast
5. Interaction strategies: cooperating vs. competing
transparent vs. secretivenegotiation + alliance/coalition formation
September 12, 2005 CSP 101: continued 10
Foundations of Constraint Processing, Fall 2005
Computational tasks• At one end of the spectrum, agents may be involved in
solving heterogeneous, distinct and completely independent problems and request other agents to supply specific functionalities for the completion of the individual tasks.
• At the other end of the spectrum, the same problem could be replicated and assigned to all agents, which can then share their individual results with other agents incrementally in order to speed up the execution of the global computational task.
September 12, 2005 CSP 101: continued 11
Foundations of Constraint Processing, Fall 2005
Types of agentsAn agent in the system can be any of the following:
• an agent that collects data from the environment and formulates the CSP
• a reasoning module with specific computational characteristics (e.g., various search algorithms)
• brokers that facilitate matching between a service seeker and a number of service providers (e.g., CORBA brokers).
September 12, 2005 CSP 101: continued 12
Foundations of Constraint Processing, Fall 2005
Agent architecture..
.. and how authority is granted• Agents could be organized in a strict hierarchy in which
a given agent has full control over the activities of the agents that lie underneath it in the hierarchy. It decides how the lower level agents may cooperate while ensuring coordination with the higher-level agent.
• Agents could be in a strictly flat structure competing for services and rewards, either chaotically or according to some strict priority policy, for example, based on voting or time-responsiveness.
September 12, 2005 CSP 101: continued 13
Foundations of Constraint Processing, Fall 2005
Communication environment
Communications among agents may be conducted according to:
• a one-to-one schema
• multi-cast (i.e., one-to-group),
• or broadcast, where all agents in the environment have access to the content of the communicated information.
September 12, 2005 CSP 101: continued 14
Foundations of Constraint Processing, Fall 2005
Type of supported interactions
• Agents may be cooperative, pooling their resources and capabilities to achieve a common, global objective, or they could be competitive trying to win rewards and optimize their individual gain.
• They could also adopt a midway strategy, dynamically forming coalitions and gathering support to acquire more resources and realize greater gains.
• Also, agents may be transparent about their intentions, resources, needs, and constraints or may be secretive, hiding one or the other of their strengths or weaknesses.
September 12, 2005 CSP 101: continued 15
Foundations of Constraint Processing, Fall 2005
Outline
Advanced solving techniques
Issues & research directions
CSP in a nutshell
Constraint Logic Programming (quickly)
September 12, 2005 CSP 101: continued 16
Foundations of Constraint Processing, Fall 2005
Research directions• Preceding (i.e., search, backtrack, iterative repair, V/V/ordering,
consistency checking, decomposition, symmetries & interchangeability, deep analysis)
• Evaluation of algorithms– worst-case analysis vs. empirical studies– random problems vs. real-world problems
• Cross-fertilization: – DB, SAT & theoretical computer science (TCS), mathematical
programming, interval mathematics, logical inference, applications, etc.
• Modeling & Reformulation• Extensions
– Non-binary, conditional, soft constraints & preferences, etc
• Multi agents– Distribution and negotiation– decomposition & alliance formation
September 12, 2005 CSP 101: continued 17
Foundations of Constraint Processing, Fall 2005
Outline
Advanced solving techniques
Issues & research directions
CSP in a nutshell
Constraint Logic Programming (quickly)
September 12, 2005 CSP 101: continued 18
Foundations of Constraint Processing, Fall 2005
CSP in a nutshell (I)Definition: P = (V, D, C )
• Constraint graph, constraint network
• Finite domains
• Binary constraints, universal constraints
Examples: map coloring, puzzles, resource allocation, temporal reasoning, product configuration, databases, spreadsheets, graphical layouts, graphical user-interfaces, bioinformatics, etc.
{ 1, 2, 3, 4 }
{ 3, 5, 7 }{ 3, 4, 9 }
{ 3, 6, 7 }
v2 > v4
V4
V2
v1+v3 < 9
V3
V1
v2 < v3
v1 < v2
nVVV ,,, 21 V nVVV DDD ,,, 21 D
lCCC ,,,21 C
September 12, 2005 CSP 101: continued 19
Foundations of Constraint Processing, Fall 2005
CSP in a nutshell (II)
Solution technique: Search
Enhancing search:
constructive
iterative repair
Intelligent backtrack Variable/value ordering Consistency checkingHybrid search Symmetries Decomposition
September 12, 2005 CSP 101: continued 20
Foundations of Constraint Processing, Fall 2005
CSP in a nutshell (III)Deep analysis:
exploit problem structure
Research
Graph topology Constraint semanticsPhase transition
- k-ary constraints, soft constraints- continuous vs. finite domains - evaluation of algorithms (empirical) - cross-fertilization (mathematical program.) - preferences and soft constraints reformulation and approximation architectures (multi-agent, negotiation)
September 12, 2005 CSP 101: continued 21
Foundations of Constraint Processing, Fall 2005
Outline
Advanced solving techniques
Issues & research directions
CSP in a nutshell
Constraint Logic Programming (quickly)
September 12, 2005 CSP 101: continued 22
Foundations of Constraint Processing, Fall 2005
Constraint Logic Programming (CLP)
A merger of
Constraint solving
Logic Programming, mostly Horn clauses e.g., Prolog)
Building blocks• Constraint: primitives but also user-defined
– cumulative/capacity (linear ineq), MUTEX, cycle, etc.– domain: Booleans, natural/rational/real numbers, finite
• Rules (declarative): a statement is a conjunction of constraints and is tested for satisfiability before execution proceeds further
• Mechanisms: satisfiability, entailment, delaying constraints