Distributed Lagrangean Relaxation Protocol for the Generalized Mutual Assignment Problem

Katsutoshi Hirayama 　（平山 勝敏）

Faculty of Maritime Sciences 　（海事科学部）Kobe University 　（神戸大学）hirayama@maritime.kobe-u.ac.jp

Summary

This work is on the distributed combinatorial optimization rather than the distributed constraint satisfaction.

I present the Generalized Mutual Assignment Problem (a distributed fo

rmulation of the Generalized Assignment Problem) a distributed lagrangean relaxation protocol for the GMAP a “noise” strategy that makes the agents (in the protocol) qu

ickly agree on a feasible solution with reasonably good quality

Outline

Motivation distributed task assignment

Problem Generalized Assignment Problem Generalized Mutual Assignment Problem Lagrangean Relaxation Problem

Solution protocol Overview Primal/Dual Problem Convergence to Feasible Solution

Experiments Conclusion

Motivation: distributed task assignment

Example 1: transportation domain A set of companies, each having its own

transportation jobs. Each is deliberating whether to perform a job by

myself or outsource it to another company. Seek for an optimal assignment that satisfies their

individual resource constraints (#s of trucks).

Kobe

Kyoto

Tokyojob1

job2job3

Company1 has {job1} and 4 trucks

Company2 has {job2,job3} and 3 trucks

profit trucks

Co.1job1 5 2

job2 6 2

job3 5 1

Co.2job1 4 2

job2 2 2

job3 2 2

Motivation: distributed task assignment

Example 2: info gathering domain A set of research divisions, each having its own

interests in journal subscription. Each is deliberating whether to subscribe a journal by

myself or outsource it to another division. Seek for an optimal subscription that does not exceed

their individual budgets. Example 3: review assignment domain

A set of PCs, each having its own review assignment Each is deliberating whether to review a paper by

myself or outsource it to another PC/colleague. Seek for an optimal assignment that does not exceed

their individual maximally-acceptable numbers of papers.

Problem: generalized assignment problem (GAP)

These problems can be formulated as the GAP in a centralized context.

job1 job2 job3

Company1(agent1)

Company2(agent2)

(5,2)

(4,2)

(6,2)

(2,2)

(5,1)

(2,2)

43

(profit, resource requirement)

Assignment constraint:each job is assigned to exactlyone agent.

Knapsack constraint:the total resource requirementof each agent does not exceedits available resource capacity.

01 constraint:each job is assigned or notassign to an agent.

Problem: generalized assignment problem (GAP)

232221131211 224565 xxxxxx

}3,2,1{ },2,1{ },1,0{

3222

422

1

1

1

232221

131211

2313

2212

2111

jix

xxx

xxx

xx

xx

xx

ij

max.

s. t.

The GAP instance can be described as the integer program.

GAP: (as the integer program)

However, the problem must be dealt by the super-coordinator.

xij takes 1 if agent i is to perform job j; 0 otherwise.

assignmentconstraints

knapsackconstraints

Problem: generalized assignment problem (GAP)

Drawbacks of the centralized formulation Cause the security/privacy issue

Ex. the strategic information of a company would be revealed.

Need to maintain the super-coordinator (computational server)

Distributed formulation of the GAP: generalized mutual assignment problem (GMAP)

Problem: generalized mutual assignment problem (GMAP)

The agents (not the supper-coordinator) solve the problem while communicating with each other.

job1 job2 job3

Company1 (agent1) Company2 (agent2)

43

Problem: generalized mutual assignment problem (GMAP)

Assumption: The recipient agent has the right to decide whether it will undertake a job or not.

job1

43

job2 job3 job1 job2 job3

Sharing theassignmentconstraints

(profit, resource requirement)

Company1 (agent1) Company2 (agent2)

(5,2) (6,2) (5,1) (4,2) (2,2) (2,2)

Problem: generalized mutual assignment problem (GMAP)

The GMAP can also be described as a set of integer programs

232221 224 xxx

}3,2,1{ },1,0{

3222

1

1

1

2

232221

2313

2212

2111

jx

xxx

xx

xx

xx

j

max.

s. t.

131211 565 xxx

}3,2,1{ },1,0{

422

1

1

1

1

131211

2313

2212

2111

jx

xxx

xx

xx

xx

j

max.

s. t.

Agent1 decides x11, x12, x13 Agent2 decides x21, x22, x23

Sharing theassignmentconstraints

GMP1 GMP2

: variables of others

Problem: lagrangean relaxation problem

By dualizing the assignment constraints, the followings are obtained.

)1(2

)1(2

)1(2

224

23133

22122

21111

232221

xx

xx

xxxxx

}3,2,1{ },1,0{

3222

2

232221

jx

xxx

j

max.

s. t.

)1(2

)1(2

)1(2

565

23133

22122

21111

131211

xx

xx

xxxxx

}3,2,1{ },1,0{

422

1

131211

jx

xxx

j

max.

s. t.

Agent1 decides x11, x12, x13 Agent2 decide x21, x22, x23

LGMP1(μ) LGMP2(μ)

: variables of others),,( 321 : lagrangean multiplier vector

Problem: lagrangean relaxation problem

Two important features: The sum of the optimal values of {LGMPk(μ) | k in all of the a

gents} provides an upper bound for the optimal value of the GAP.

If all of the optimal solutions to {LGMPk(μ) | k in all of the agents} satisfy the assignment constraints for some values of μ, then these optimal solutions constitute an optimal solution to the GAP.

LGMP1(μ) LGMP2(μ)

Opt.Value1 Opt.Value2

Opt.Sol1 Opt.Sol2

solve solve

GAP

+ Opt.Value

Opt.Sol (if Opt.Sol1 and Opt.Sol2 satisfy the assignment constraints)

=

Solution protocol: overview

The agents alternate the following in parallel while performing P2P communication until all of the assignment constraints are satisfied. Each agent k solves LGMPk(μ), the primal problem, using a k

napsack solution algorithm. The agents exchange solutions with each other. Each agent k finds appropriate values for μ (solves the (lagr

angean) dual problem) using the subgradient optimization method.

Agent1 Agent2 Agent3sharing sharing

Solve dual & primal prlms Solve dual & primal prlms

Solve dual & primal prlms Solve dual & primal prlms Solve dual & primal prlms

Solve dual & primal prlms

exchange

time

Solution protocol: primal problem

Primal problem: LGMPk(μ) Knapsack problem Solved by an exact method (i.e., an optimal solution is nee

ded)

)2,2

5( 1

)1(2

)1(2

)1(2

565

23133

22122

21111

131211

xx

xx

xxxxx

}3,2,1{ },1,0{

422

1

131211

jx

xxx

j

max.

s. t.

LGMP1(μ)

job1

agent1

4

job2 job3

)2,2

6( 2 )1,

25( 3

(profit, resource requirement)

Solution protocol: dual problem

Dual problem The problem of finding appropriate values for μ Solved by the subgradient optimization method

Subgradient Gj for the assignment constraint on job j

Updating rule for μj

agents

1i

ijj xG

jtjj Gl

tl : step length at time t

Solution protocol: example

When )0,0,0(),,( 321

)2,2

5( 1

job1

agent1

4

job2 job3

)2,2

6( 2 )1,

25( 3

3

job1 job2 job3

agent2

)2,2

4( 1 )2,

22( 2 )1,

22( 3

Select {job1,job2} Select {job1}

)1,0,1(),,( 321 GGG

1tland

Therefore, in the next,

)1,0,1(),,( 321

Note: the agents involved in job j must assign μj to a common value.

Solution protocol: convergence to feasible solution

A common value to μj ensures the optimality when the protocol stops. However, there is no guarantee that the protocol will eventually stop.

You could force the protocol to terminate at some point to get a satisfactory solution, but no feasible solution had been found. In a centralized case, lagrangean heuristics are usually devis

ed to transform the “best” infeasible solution into a feasible solution.

In a distributed case, such the “best” infeasible solution is inaccessible, since it belongs to global information.

I introduce a simple strategy to make the agents quickly agree on a feasible solution with reasonably good quality.

Noise strategy: let agents assign slightly different values to μj

Solution protocol: convergence to feasible solution

Noise strategy The updating rule for μj is replaced by

jtjj GlN )1(

N : random variable whose value is uniformly distributed over ],[

This rule diversifies agents’ views on the value of μj, and being able to break an oscillation in which agents repeat “clustering and dispersing” around some job.

For δ≠0, the optimality when the protocol stops does not hold.

For δ=0, the optimality when the protocol stops does hold.

Solution protocol: rough image

optimal

feasible region

value of theobject functionof the GAP

• Controlled by multiple agents• No window, no altimeter, but a touchdown can be detected.

Experiments

Objective Clarify the effect of the noise strategy

Settings Problem instances (20 in total)

feasible instances #agents ∈ {3,5,7}; #jobs = 5×#agents profit and resource requirement of each job: an integer randomly

selected from [1,10] capacity of each agent = 20 Assignment topology: chain/ring/complete/random

Protocol Implemented in Java using TCP/IP socket comm. step length lt=1.0 δ∈{0.0, 0.3, 0.5, 1.0} 20 runs of the protocol with each value of δ for each instance; c

utoff a run at (100×#jobs) rounds

Experiments

Measure the followings for each instance Opt.Ratio: the ratio of the runs where optimal solutions were f

ound Fes.Ratio: the ratio of the runs where feasible solutions were f

ound Avg/Bst.Quality: the average/best value of the solution qualiti

es Avg.Cost: the average value of the numbers of rounds at which

feasible solutions were found

optimal

feasible

value ofobject function

o

f o

falitySolutionQu

Experiments

Observations The protocol with δ= 0.0 failed to find an optimal solution fo

r almost all of the instances. In the protocol with δ ≠ 0.0, Opt.Ratio, Fes.Ratio, and Avg.C

ost were obviously improved while Avg/Bst.Quality was kept at a “reasonable” level (average > 86%, best > 92%).

In 3 out of 6 complete-topology instances, an optimal solution was never found at any value of δ.

For many instances, increasing the value of δ may generally have an effect to rush the agents into reaching a compromise.

Conclusion

I have presented Generalized mutual assignment problem Distributed lagrangean relaxation protocol Noise strategy that makes the agents quickly agree on a fea

sible solution with reasonably good quality Future work

More sophisticated techniques to update μ The method that would realize distributed calculation of an u

pper bound of the optimal value.