Distributed Lagrangean Relaxation Protocol for the Generalized Mutual Assignment Problem

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Distributed Lagrangean Relaxation Protocol for
the Generalized Mutual Assignment Problem

• Katsutoshi Hirayama (?? ??)

Faculty of Maritime Sciences (?????) Kobe
University (????) hirayama_at_maritime.kobe-u.ac.jp
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Summary

• This work is on the distributed combinatorial
  optimization rather than the distributed
  constraint satisfaction.
• I present
• the Generalized Mutual Assignment Problem (a
  distributed formulation of the Generalized
  Assignment Problem)
• a distributed lagrangean relaxation protocol for
  the GMAP
• a noise strategy that makes the agents (in the
  protocol) quickly agree on a feasible solution
  with reasonably good quality

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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

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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).

Kyoto
job2
job3
Kobe
Tokyo
job1
Company1 has job1 and 4 trucks
Company2 has job2,job3 and 3 trucks
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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.

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Problem generalized assignment problem (GAP)

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

job1
job2
job3
Assignment constraint each job is assigned to
exactly one agent. Knapsack constraint the
total resource requirement of each agent does not
exceed its available resource capacity. 01
constraint each job is assigned or not assign to
an agent.
(5,1)
(6,2)
(5,2)
(2,2)
(2,2)
(4,2)
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3
Company2 (agent2)
Company1 (agent1)
(profit, resource requirement)
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Problem generalized assignment problem (GAP)

• The GAP instance can be described as the integer
  program.

GAP (as the integer program)
max.
s. t.
assignment constraints
knapsack constraints
xij takes 1 if agent i is to perform job j 0
otherwise.
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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)
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Problem generalized mutual assignment problem
(GMAP)

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

Company1 (agent1)
Company2 (agent2)
job1
job2
job3
4
3
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Problem generalized mutual assignment problem
(GMAP)

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

Company1 (agent1)
Company2 (agent2)
job1
job2
job3
job1
job2
job3
(5,2)
(6,2)
(5,1)
(4,2)
(2,2)
(2,2)
Sharing the assignment constraints
4
3
(profit, resource requirement)
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Problem generalized mutual assignment problem
(GMAP)

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

Agent1 decides x11, x12, x13
Agent2 decides x21, x22, x23
GMP1
GMP2
max.
max.
s. t.
s. t.
Sharing the assignment constraints
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Problem lagrangean relaxation problem

• By dualizing the assignment constraints, the
  followings are obtained.

Agent1 decides x11, x12, x13
Agent2 decide x21, x22, x23
LGMP1(µ)
LGMP2(µ)
max.
max.
s. t.
s. t.
lagrangean multiplier vector
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Problem lagrangean relaxation problem

• Two important features
• The sum of the optimal values of LGMPk(µ) k in
  all of the agents 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)
solve
solve
Opt.Value1
Opt.Value2
Opt.Sol1
Opt.Sol2
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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 knapsack solution algorithm.
• The agents exchange solutions with each other.
• Each agent k finds appropriate values for µ
  (solves the (lagrangean) dual problem) using the
  subgradient optimization method.

Agent1
Agent2
Agent3
time
sharing
sharing
Solve dual primal prlms
Solve dual primal prlms
Solve dual primal prlms
exchange
Solve dual primal prlms
Solve dual primal prlms
Solve dual primal prlms
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Solution protocol primal problem

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

LGMP1(µ)
job1
job2
job3
max.
4
s. t.
agent1
(profit, resource requirement)
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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

step length at time t
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Solution protocol example
When
and
job1
job2
job3
job1
job2
job3
Therefore, in the next,
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3
agent1
agent2
Select job1
Select job1,job2
Note the agents involved in job j must
assign µj to a common value.
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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 devised 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
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Solution protocol convergence to feasible
solution

• Noise strategy
• The updating rule for µj is replaced by

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 d?0, the optimality when the protocol stops
  does not hold.
• For d0, the optimality when the protocol stops
  does hold.

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Solution protocol rough image
value of the object function of the GAP

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

optimal
feasible region
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Experiments

• Objective
• Clarify the effect of the noise strategy
• Settings
• Problem instances (20 in total)
• feasible instances
• agents ? 3,5,7 jobs 5agents
• 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 lt1.0
• d?0.0, 0.3, 0.5, 1.0
• 20 runs of the protocol with each value of d for
  each instance cutoff a run at (100jobs) rounds

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Experiments

• Measure the followings for each instance
• Opt.Ratio the ratio of the runs where optimal
  solutions were found
• Fes.Ratio the ratio of the runs where feasible
  solutions were found
• Avg/Bst.Quality the average/best value of the
  solution qualities
• Avg.Cost the average value of the numbers of
  rounds at which feasible solutions were found

value of object function
o
optimal
feasible
f
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Experiments

• Observations
• The protocol with d 0.0 failed to find an
  optimal solution for almost all of the instances.
• In the protocol with d ? 0.0, Opt.Ratio,
  Fes.Ratio, and Avg.Cost were obviously improved
  while Avg/Bst.Quality was kept at a reasonable
  level (average gt 86, best gt 92).
• In 3 out of 6 complete-topology instances, an
  optimal solution was never found at any value of
  d.
• For many instances, increasing the value of d may
  generally have an effect to rush the agents into
  reaching a compromise.

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Conclusion

• I have presented
• Generalized mutual assignment problem
• Distributed lagrangean relaxation protocol
• Noise strategy that makes the agents quickly
  agree on a feasible solution with reasonably good
  quality
• Future work
• More sophisticated techniques to update µ
• The method that would realize distributed
  calculation of an upper bound of the optimal
  value.