ants pi meeting, nov. 29, 2000w. zhang, washington university1 flexible methods for multi-agent...

ANTs PI Meeting, Nov. 29, 2000W. Zhang, Washington University 1

Flexible Methods for Multi-agent distributed resource Allocation by Exploiting Phase Transitions

Weixiong Zhang (PI)Washington UniversitySt. Louis, MO [email protected]

- Modeling, Phase Transitions, Constraint and Complexity Analyses

mailto:[email protected]


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)


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


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


Outline






• Summary and Discussions• Future plan (research and integration plans)


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


Flight Scheduling Constraints (cont.)

Range Plane &Support

crew

Pilot &coach

Range Capabilities,

etc.

FacilityAvailability,

etc

Mission codes,Formation,

Schedule types,etc.

• Complex constraints– Different types– Too many constraints


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


Outline


– Constraint analysis – Hierarchical constraint models

• Approaches and Experimental Results– Constraint minimization problem– Complexity and phase transitions





Hierarchical Constraint Models - Motivation

Ranges Planes &Support

crew

Pilots &coach

Range Capabilities,

etc.

FacilityAvailability,

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?


Flight Scheduling Main GoalTraining pilots to form squadrons (final exam)

Squadron

section 1 section 2 section n

leader wingman Pilot m

Code kCode 1 Code 2

Organizationalconstraints

Competenceconstraints

Average pilot combat readiness

% (CRP)


Code Dependencies - Critical Constraints

Pilot 1Competence

Average pilot Combat Readiness Percentage (CRP)

= +Pilot 2

Competence

• The most important factor is CRP:

Pilot competence

Code 1 Code 2 Code n

Prerequisites

• Competence depends on mission code qualified:


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


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 codeconstraints

rangeconstraints

facility constraint


Outline








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


Outline








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


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!


Outline








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)


Methods for CMP – Quality/Complexity Tradeoff (Experiments)

• New, shifted phase transitions in MAX-3SAT (25 variables)


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

So l

uti o

n co

st

Computation cost

2

4

6

8 Solution

No solution


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 codeconstraints

rangeconstraints

facility constraint


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


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)


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.


Outline








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


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


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


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

ants pi meeting, nov. 29, 2000w. zhang, washington university1 flexible methods for multi-agent...

Documents

complexity analysis

phase transitionssatisfaction

experimental resultscomplexity

resource allocationsw

lowerbound methodssummary

etcmission codes

lower boundscomplexity

integration plansw