1. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University1 Flexible Methods for Multi-agent distributed resource Allocation by Exploiting Phase Transitions Weixiong Zhang (PI) Washington University St. Louis, MO 63130 zhang@cs.wustl.edu zhang@cs.wustl.edu - Modeling, Phase Transitions, Constraint and Complexity Analyses

2. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University2 Outline Project Objectives Constraint Analysis and Modeling –Constraint analysis –Hierarchical constraint models –Constraint minimization problem Approaches and Experimental Results –Complexity and phase transitions Satisfaction versus optimization –Methods for hierarchical constraint minimization Trading solution quality for less computation Hierarchical constraint models and lower-bound methods Summary and Discussions Future plans (research and integration plans)

3. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University3 Project Objectives Understanding resource allocation problems in ANTs applications. –Solution quality (e.g., lower bounds) –Complexity (e.g., polynomial vs. exponential) –Phase transitions Developing general and efficient algorithms for resource allocations –Phase-aware problem solver (e.g., move from exponential region to polynomial region) –Anytime methods for finding good-enough and soon enough solutions

4. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University4 Project Objectives (cont.) The research is to support CAMERA and its application –A difficult scheduling problem in military domain Use CAMERA application as a testbed and develop general approaches to complexity analysis and resource allocations

5. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University5 Outline Project Objectives Constraint Analysis and Modeling –Constraint analysis –Hierarchical constraint models –Constraint minimization problem Approaches and Experimental Results –Complexity and phase transitions Satisfaction versus optimization –Methods for hierarchical constraint minimization Trading solution quality for less computation Hierarchical constraint models and lower-bound methods Summary and Discussions Future plan (research and integration plans)

6. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University6 Flight Scheduling Constraints Schedule training missions for pilots under various constraints including –Mission codes Dependencies among missions codes –Resource restriction Ranges (space to fly) Equipment (e.g., airplanes and weapons) Supporting crew –Flying condition and duration Fly with coach Fly specific formation Day and/or night fly

7. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University7 Flight Scheduling Constraints (cont.) Range Plane & Support crew Pilot & coach Range Capabilities, etc. Facility Availability, etc Mission codes, Formation, Schedule types, etc. Complex constraints –Different types –Too many constraints

8. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University8 Flight Scheduling Objectives Maximizing pilot competence –Number of mission codes qualified by individual pilots –Average team combat readiness Minimizing training time (throughput of training site) –Multiple and individual schedules –Cope with dynamic environment (e.g., failed equipment) Minimizing resources required and used –Airplanes and weapons used –Supporting staff needed

9. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University9 Outline Project Objectives Constraint Analysis and Modeling –Constraint analysis –Hierarchical constraint models Approaches and Experimental Results –Constraint minimization problem –Complexity and phase transitions Satisfaction versus optimization –Methods for hierarchical constraint minimization Trading solution quality for less computation Hierarchical constraint models and lower-bound methods Summary and Discussions Future plans (research and integration plans)

10. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University10 Hierarchical Constraint Models - Motivation Ranges Planes & Support crew Pilots & coach Range Capabilities, etc. Facility Availability, etc Mission codes, Formation, Schedule types, etc. Too many constraints –Different types –Too many constraints Finding high quality solutions is not easy! Which constraints really matter?

11. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University11 Flight Scheduling Main Goal Training pilots to form squadrons (final exam) Squadron section 1 section 2section n leaderwingmanPilot m Code kCode 1Code 2 Organizational constraints Competence constraints Average pilot combat readiness % (CRP)

12. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University12 Code Dependencies - Critical Constraints Pilot 1 Competence Average pilot Combat Readiness Percentage (CRP) = + Pilot 2 Competence The most important factor is CRP: Pilot competence Code 1 Code 2Code n Prerequisites Competence depends on mission code qualified:

13. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University13 Code Dependence (Constraint) Graph (based on training manual) Nodes: mission codes; Links: dependencies among codes Dependence graph: A directed acyclic graph (DAG) defining a partial order among codes Pilot dependence graphs: subsets of code dependence graph Goal: schedule nodes in pilot graphs Nodes in code graph Nodes in pilot graph Similarly, range, facility and other constraint graphs can be constructed

14. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University14 Hierarchical Constraint Models Rationale: –Some (core) constraints (e.g., mission code dependencies) must be satisfied –Less important constraints (e.g., ranges and airplanes) may be satisfied later through negotiation –Unimportant constraints may be violated Constraint Solving Strategy: –Consider core constraints first –Progressively move up in the hierarchy mission code constraints range constraints facility constraint

15. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University15 Outline Project Objectives Constraint Analysis and Modeling –Constraint analysis –Hierarchical constraint models –Constraint minimization problem Approaches and Experimental Results –Complexity and phase transitions Satisfaction versus optimization –Methods for hierarchical constraint minimization Trading solution quality for less computation Hierarchical constraint models and lower-bound methods Summary and Discussions Future plans (research and integration plans)

16. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University16 Constraint Minimization Problem (CMP) Constraints have different importance –Constraints in different hierarchy play different roles –Constraints have different costs, importance, preferences, etc. Over-constrained –No solution satisfying all constraints –Finding a solution to minimize the overall weight of unsatisfied constraints It is an optimization problem –Finding best solutions within resource constraints Why CMP? –CAMERA is an optimization problem Various constraints play different roles and have different weights –Many real applications are optimization problems

17. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University17 Outline Project Objectives Constraint Analysis and Modeling –Constraint analysis –Hierarchical constraint models –Constraint minimization problem Approaches and Experimental Results –Complexity and phase transitions Satisfaction versus optimization –Methods for hierarchical constraint minimization Trading solution quality for less computation Hierarchical constraint models and lower-bound methods Summary and Discussions Future plans (research and integration plans)

18. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University18 Phase Transitions – Decision vs. Optimization (Previous Results) Decision problem –Finding an YES/NO answer –Easy-hard-easy phase transitions Optimization problem –Finding an optimal solution –Easy-hard phase transitions

19. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University19 Phase Transitions – Decision versus Optimization (A Closer Look) Different phase transition patterns –Decision problem has easy-hard-easy transition pattern –Optimization problem has easy-hard transition pattern Complexity discrepancy - Tighter constraints have different impact –They make a decision problem easier –They make an optimization problem more difficult Optimization is hard!

20. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University20 Outline Project Objectives Constraint Analysis and Modeling –Constraint analysis –Hierarchical constraint models –Constraint minimization problem Approaches and Experimental Results –Complexity and phase transitions Satisfaction versus optimization –Methods for hierarchical constraint minimization Trading solution quality for less computation Hierarchical constraint models and lower-bound methods Summary and Discussions Future plans (research and integration plans)

21. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University21 Methods for CMP – Quality/Complexity Tradeoff (The Ideas) Idea: Trade solution quality for reduced computation –Finding solutions with some constraints violated Method: Using decision problem for finding the best solution under limited computation –Decide if there is a solution with a fixed number of constraint violated. If yes, find it. –If more time allowed, revise the solution bound and repeat (progressive improvement)

22. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University22 Methods for CMP – Quality/Complexity Tradeoff (Experiments) New, shifted phase transitions in MAX-3SAT (25 variables)

23. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University23 Methods for CMP – Progressive approximation Anytime problem solving based on quality/complexity tradeoff –Start with a high solution cost bound –Iteratively reduce solution cost bound to find better approximation Solution cost Computation cost 2 4 6 8 Solution No solution

24. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University24 Methods for CMP – Hierarchical Constraint Minimization Hierarchical constraint models –Distinguish different constraints based on their importance - constraint hierarchy More important constraints are assigned larger weights –Group constraints based on their importance Problem solving using hierarchical models –Working way up from core constraints to the lest important constraints. Optimal solutions to a lower level hierarchy as approximation solutions mission code constraints range constraints facility constraint

25. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University25 Solving CMP – Experimental Results (Weight Distribution) Large number of constraints with small weights violated Constraints of small weights cause constraints of large weights to be violated in final solutions

26. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University26 Solving CMP – Experimental Results on Hierarchical CMP Problem solving in hierarchical constraint models –Finding the best solution to the core constraints –Reduced complexity and high quality solutions (e.g. using hierarchy threshold 5 below)

27. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University27 Methods for CMP – Hierarchical Models and Lower Bounds CAMERA constraints can be modeled hierarchically –Progressively consider less important constraints In the order of mission code constraints, range constraints, facility constraints, etc. –Mission code constraints have highest weights Use of hierarchical models – Lower bounds –When moving up to an upper hierarchy (where more constraints are considered) Solution quality increases Computational cost increases. –Solutions on a lower hierarchy are bounds on solutions on an upper hierarchy.

28. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University28 Outline Project Objectives Constraint Analysis and Modeling –Constraint analysis –Hierarchical constraint models –Constraint minimization problem Approaches and Experimental Results –Complexity and phase transitions Satisfaction versus optimization –Methods for hierarchical constraint minimization Trading solution quality for less computation Hierarchical constraint models and lower-bound methods Summary and Discussions Future plans (research and integration plans)

29. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University29 Summary of Results on CMP There are shifted phase transitions in CMPs with different solution cost bounds Phase transitions can be exploited to develop anytime algorithms –Using progressively tighter bounds on solutions –Progressively consider less important constraints Hierarchical constraint models can –Provide bounds on solutions and complexity –Reduce complexity

30. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University30 Features of Our Approaches – Discussion Avoiding difficult phases –Using hierarchical constraint models –Ignore less important constraints to shift a problem to an easy phase Good anytime performance –Progressively better solutions –Reduced computation Lower bounds –Help to decide how much to push to reduce complexity Generality –The approaches are general, and are especially suitable for scheduling problems

31. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University31 Future Plan (Research) Complete CAMERA scheduling modeling –Define constraint hierarchies based on real data Further constraint analysis –Analyze Solution quality and complexity at different level of constraint hierarchy of the model –Characterize solution and complexity lower bounds from different levels of constraint hierarchies More CMP algorithms –Exploit constraint structures –Combine phase transitions with constraint hierarchies More experimental results –Results of CAMERA models using real data

32. ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University32 Future Plan (Application & Integration) Applications in and close integration with CAMERA –Hierarchical models for Sortie (mission) generator To improve sortie quality and increase generator’s speed –Lower bounds for prediction Estimate the amount of time needed for a given solution quality Software development plan –Close integration with CAMERA source code Adding hierarchical constraint models Adding software for lower bounds and algorithms for CMP –Experiments on integrated system Phase transitions Real-time performance