Constraint Satisfaction 101 Foundations of Constraint Processing CSCE

Constraint Satisfaction 101 Foundations of Constraint Processing CSCE 421/821, Fall 2012 www. cse. unl. edu/~choueiry/S 09 -421 -821/ All questions Piazza Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 360 Tel: +1(402)472 -5444 Foundations of Constraint Processing CSP 101: continued 1

Outline Motivating example, application areas CSP: Definition, representation Some simple modeling examples More on definition and formal characterization Basic solving techniques Implementing backtrack search ü Advanced solving techniques ü Issues & research directions ü CSP in a nutshell ü Constraint Logic Programming (quickly) Foundations of Constraint Processing CSP 101: continued 2

Outline ü Advanced solving techniques üDecomposition üDeep analysis (islands of tractability) üDistributed CSPs ü Issues & research directions ü CSP in a nutshell ü Constraint Logic Programming (quickly) Foundations of Constraint Processing CSP 101: continued 3

Decomposition • Decomposition [Freuder & Hubbe, CP 93] Conjunctive Disjunctive • Conjunctive • Disjunctive Foundations of Constraint Processing CSP 101: continued 4

Properties of decompositions • Conjunctive or disjunctive? • • • Consistent: Simplifying: Semi-complete: Complete: Redundant: Reducible: No constraint is removed Size(Pi) < Size( P) At least one solution is kept No solution is lost Solutions replicated in { Pi} may be < Size( P) Foundations of Constraint Processing CSP 101: continued 5

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) Foundations of Constraint Processing CSP 101: continued 6

Cost of solving Phase transition Mostly solvable problems [Cheeseman et al. ‘ 91] Mostly un-solvable problems Critical value of order parameter Order parameter • Significant increase of cost around critical value • In CSPs, order parameter is constraint tightness & ratio • Algorithms compared around phase transition Foundations of Constraint Processing CSP 101: continued 7

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 – – Applications: scheduling & resource allocation Based on decomposition of problem/solvers Emerging area: Social Choice (Voting, auctions) Let’s take a wider perspective than what is done today… Foundations of Constraint Processing CSP 101: continued 8

Multi-agent approach 1. 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. secretive negotiation + alliance/coalition formation Foundations of Constraint Processing CSP 101: continued 9

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. Foundations of Constraint Processing CSP 101: continued 10

Types of agents An 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). Foundations of Constraint Processing CSP 101: continued 11

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. Foundations of Constraint Processing CSP 101: continued 12

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. Foundations of Constraint Processing CSP 101: continued 13

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. Foundations of Constraint Processing CSP 101: continued 14

Outline ü Advanced solving techniques ü Issues & research directions ü CSP in a nutshell ü Constraint Logic Programming (quickly) Foundations of Constraint Processing CSP 101: continued 15

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 Foundations of Constraint Processing CSP 101: continued 16

Outline ü Advanced solving techniques ü Issues & research directions ü CSP in a nutshell ü Constraint Logic Programming (quickly) Foundations of Constraint Processing CSP 101: continued 17

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. Foundations of Constraint Processing CSP 101: continued 18

CSP in a nutshell (II) Solution technique: Search Enhancing search: constructive iterative repair Intelligent backtrack Variable/value ordering Constraint propagation Hybrid search Symmetries Decomposition Foundations of Constraint Processing CSP 101: continued 19

CSP in a nutshell (III) Deep analysis: exploit problem structure Tractability studies Graph topology Constraint semantics Phase transition Other directions -k-ary constraints (representation, efficient propagators) - continuous vs. finite domains - evaluation of algorithms (theoretical vs empirical) - cross-fertilization (mathematical program. ) - preferences and soft constraints reformulation and approximation architectures (multi-agent, negotiation) Foundations of Constraint Processing CSP 101: continued 20

Outline ü Advanced solving techniques ü Issues & research directions ü CSP in a nutshell ü Constraint Logic Programming (quickly) Foundations of Constraint Processing CSP 101: continued 21

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 Foundations of Constraint Processing CSP 101: continued 22