objectives and approaches

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP 1

Flexible Methods for Multi-agent Distributed Resource Allocation by Exploiting Phase Transitions

Modeling and Analyzing Resource Allocation Problems Using Soft Constraint Satisfaction and Optimization

Weixiong Zhang (PI) Kenneth Swanson, Xiaotao Zhang,

Peng Wang, Michael P. Moran,Guandong Wang, Zhao Xing,

Zhongshen GuoComputational Intelligence Center and

Computer Science DepartmentWashington University in St. Louis


Objectives and Approaches

• Understanding and characterizing resource allocation problems in ANTs applications.– Modeling methods: soft constraint satisfaction/optimization– Phase transitions and backbones (sources of complexity)– Scalability (e.g. impact of problem structures)

• Developing general and efficient algorithms for resource allocations– Systematic search methods– Approximation methods– Distributed algorithms– Phase-aware problem solving for good enough/sooner

enough solutions


Work in this period

• EW challenge problem– Design and develop a moving target tracking

system in RadSim

– Preliminary working system

– Testbed for studying many technical difficulties

– (more to come at next PI meeting)

• Marbles pilot scheduling problems– (main focus of this presentation)


Current Work on EW Challenge Problem

• Technical issues under consideration– Scalability

• how problem structures and agent organization affect complexity

– Uncertainty in resource conflict resolution: uncertainty in measurement, communication error, etc.

– Scan scheduling for detecting new targets quickly with small amount of energy

– Irregular sensor layout• We have shown that triangle topology provides the best area

coverage• What if sensor layout is out of your control – how to quickly

form teams


Marbles Scheduling Problem

• Main focus of this period

• Some results


The Marbles Problem• Resource allocation in a task scheduling problem

– Schedule as many tasks as possible (to reduce the overall penalty of unscheduled tasks)

– Block resource requirement• Each task requires a set of resources• It cannot be schedule unless all resource requirements are fulfilled

– Exclusive resource contention• A shared resource may be applicable to multiple requirements• But it can be used to fulfill only one requirement

R1 R2 R3

T1

Q1,1 0 1 1

Q1,2 1 0 0

T2

Q2,1 1 1 0

Q2,2 1 1 0

Resource requirements

Resources

Tasks


The Problem is Difficult

• The problem is NP-hard– The decision version is NP-complete– Reduced from set packing (NP-complete)

• Set packing: – Given a collection S of finite sets of elements, a positive

integer K

– Decide: if S contains at least K mutually disjoint subsets

• Reduction:– Map an elements to a resource

– Map a subset to a task


Technical Content

• Hard and soft constraints• Modeling consideration and choices• Constraint models

– Models in optimization

– Models in satisfaction

• Experimental analysis (phase transitions)• Current and future work


Hard and Soft Constraints

• Task constraints (soft constraints) – Ctask: turning on tasks (typically, not all of them can be

satisfied at once)• Constraint (Ti = 1) to represent turning on task Ti

• Weight equal to 1 or its penalty

• Block resource requirements (hard constraints) – Creq: Fulfilling resource requirement of a task if it is on– Weight is more than the total weight of soft constraints

• Exclusive resource contention (hard constraints) – Cres: A resource can only be used by one requirement– Weight is more than the total weight of soft constraints


Main Modeling Considerations• Optimization vs. decision

– Optimization: try to turn on all tasks, and then find the maximal number of tasks that can be indeed turned on

– Decision: Guess the possible number of tasks that can be turned on, and then verify it. Do a binary search on the number of tasks. (caution: it may not work if tasks are weighted and it is to minimize the overall weight of unscheduled tasks.)

• General variables versus Boolean variables– CSP/COP (Constraint Optimization Problem) versus SAT/MAX-SAT– K-encoding issue

• Which choice to take and under what conditions?

Optimization Decision

General variables A COP model A set of CSP models

Boolean variables A MAX-SAT model A set of SAT models


Main Modeling Choices

• Variable versus values– Resources as variables and requirements as values– Or vice versa– Which one to use?

R1 R2 R3

T1

Q1,1 0 1 1

Q1,2 1 0 0

T2

Q2,1 1 1 0

Q2,2 1 1 0

Resource requirements

Resources

Tasks


Main Modeling Choices

• Expressiveness of a model• E.g. Two resources may be assigned to one

requirement (but one is used)• Should hidden constraints be made explicit?

• Interaction between modeling considerations and choices and search algorithms


COP/CSP Models


COP1(requirements as variables)

CSP1(requirements as variables)

COP2(resources as variables)

CSP2(resources as variables)

COP3(resources as variables,

more explicit than COP2)

CSP3(resources as variables,

more explicit than CSP2)MAX-SAT4 SAT4

MAX-SAT5(more explicit than MAX SAT4)

SAT5(more explicit than SAT4)




Original ISI Marbles model


COP2 Model: Resources as Variables

t: # of tasksqi: # of resource requirements of task ir: # of resourcesTi: Boolean variable for task iRk = { Qij | task i, requirement j }

Ctask = ^k=1..t (Ti = 1)

Cblock = ^k=1..t Cblock(Ti)

Cblock(Ti)= ^j=1..qi ((Ti=0) Vk=1..r(Rk=Qij))

R1 R2 R3

T1

Q1,1 0 1 1

Q1,2 1 0 0

T2

Q2,1 1 1 0

Q2,2 1 1 0

Ctask = (T1 = 1) ^ (T2 = 1) (T1=0) V(R2=Q11) V(R3=Q11)Cblock = (T1=0) V(R1=Q12) (T2=0) V(R1=Q21) V(R2=Q11) (T2=0) V(R1=Q22) V(R2=Q22)


COP3 Model: More Explicit than COP2t: # of tasksqi: # of resource requirements of task ir: # of resourcesTi: Boolean variable for task iRk = { Qij | task i, requirement j}

Ctask = ^k=1..t (Ti = 1)

Cblock = ^k=1..t Cblock(Ti) ^k=1..t C'block(Ti)

Cblock(Ti)= ^j=1..qi ((Ti=0) Vk=1..r(Rk=Qij))

C'block(Ti) = ^((Ru ≠ Qij) V (Rv ≠ Qij))

R1 R2 R3

T1

Q1,1 0 1 1

Q1,2 1 0 0

T2

Q2,1 1 1 0

Q2,2 1 1 0

Ctask = (T1 = 1) ^ (T2 = 1) (T1=0) V(R2=Q11) V(R3=Q11) Cblock = (T1=0) V(R1=Q12) (T2=0) V(R1=Q21) V(R2=Q11) (T2=0) V(R1=Q22) V(R2=Q22) (R2 ≠ Q11) V(R3 ≠ Q11)C'block = (R1 ≠ Q21) V(R2 ≠ Q21) (R1 ≠ Q22) V(R3 ≠ Q22)


Phase Transitions

• Marbles problems (8 tasks with 2 requirements each)

02

46

86

7

8

9

100

0.2

0.4

0.6

0.8

1

# tasks to be turned on

8 tasks, 2 resource requirement/task, density 0.3

# resources

poss

ibili

ty t

hat

task

s ca

n be

tur

ned

on


Phase Transitions (2)

• Marbles problems (8 tasks with 2 requirements each)

0.10.15

0.20.25

0.30.35

6

7

8

9

100

0.2

0.4

0.6

0.8

1

density

8 tasks, 2 resource requirement/task

# resources

poss

ibili

ty 4

tas

ks t

urne

d on


Experimental Results:Systematic search and local search









more explicit than CSP2)MAX-SAT4 SAT4







Experimental Results: Complete Algorithm

• MAX SAT Models


Experimental Results: Complete Algorithm

• SAT Models


Experimental Analysis: Complete Algorithm

• Summary


Model 4 3

MAX SAT4

2

SAT4

Model 5 4

MAX SAT5

1

SAT5


Experimental Analysis: Local Search

• Experiment setup– A WalkSAT-like algorithm for all models– Given all models the same total amount of CPU time

(adjusted by the numbers of moves and restarts– Measure the final solution quality– CPU time the best solution found the first time

• Results– Optimization models are better than decision models– COP1 is the best– COP2 is better than COP3









more explicit than CSP2)

MAX-SAT4 SAT4




Summary

• Marbles problems are indeed difficult

• There are phase transition phenomena in Marbles problems

• Modeling and search algorithms affect each other

• Good modeling methods can greatly reduce problem-solving time


Next Steps

• Marbles scheduling problem– More accurate results on phase transitions– More efficient search algorithms– Large problems– Timed Marbles problems

• for a long period, e.g., days,weeks and months

• EW challenge problem– Scalability

– Uncertainty

– Scan scheduling (larger coverage, less energy)

– Irregular layout

objectives and approaches

Documents

resource conflict resolution

difficultthe problem

scan scheduling

phase transitions modeling

modeling methods

task tiweight equal

xiaotao zhang

agent organization