Algorithmic and Domain Centralization in Distributed Constraint Optimization Problems

1
Algorithmic and Domain Centralization in
Distributed Constraint Optimization Problems

• John P. Davin
• Carnegie Mellon University
• June 27, 2005
• Committee
• Manuela Veloso, Co-Chair
• Pragnesh Jay Modi, Co-Chair
• Scott Fahlman
• Stephen F. Smith
• Carnegie Mellon University
• School of Computer Science
• In partial fulfillment of the 5th year master's
  degree.
• Full paper http//www.cs.cmu.edu/jdavin/thesis/

2
DCOP

• DCOP - Distributed Constraint Optimization
  Problem
• Provides a model for many multi-agent
  optimization problems (scheduling, sensor nets,
  military planning).
• More expressive than Distributed Constraint
  Satisfaction.
• Computationally challenging (NP Complete).

3
Centralization in DCOPs

• Centralization aggregating information about
  the problem in a single agent.
• ? resulting in a larger local search space.
• We define two types of centralization
• Algorithmic a DCOP algorithm actively
  centralizes parts of the problem through
  communication.
• ? allows a centralized search procedure.
• Domain the DCOP definition inherently has parts
  of the problem already centralized.
• ? eg., multiple variables per agent (ex.
  scheduling)

4
Motivation

• Current state of DCOP research
• Two existing DCOP algorithms Adopt OptAPO
  exhibit differing levels of centralization.
• DCOP has been applied to several domains (eg,
  meeting scheduling).
• Only 1 variable per agent problems used in
  existing work.
• Still Needed
• It is unclear exactly how Adopt OptAPO are
  affected by centralization.
• Domains with multiple variables per agent have
  not been explored

5
Thesis Statement

• Questions
• How does algorithmic centralization affect
  performance?
• How can we take advantage of domain
  centralization?

6
Spectrum of centralization
7
Outline

• Introduction
• Part 1 Algorithmic centralization in DCOPs
• Part 2 Domain centralization in DCOPs
• Conclusions

8
Part 1 Algorithmic Centralization in DCOP
Algorithms
9
Evaluation Metrics

• How does algorithmic centralization affect
  performance?
• How do we measure performance?

10
Evaluation Metrics Cycles

• Cycle one unit of algorithm progress in which
  all agents process incoming messages, perform
  computation, and send outgoing messages.
• Independent of machine speed, network conditions,
  etc.
• Used in prior work Yokoo et al., Mailler et al.

11
Evaluation Metrics Constraint Checks

• Constraint check the act of evaluating a
  constraint between N variables.
• Provides a measure of computation.
• Concurrent constraint checks (CCC) maximum
  constraint checks from the agents during a cycle.

12
Problems with previous measures

• Cycles do not measure computational time (they
  dont reflect the length of a cycle).
• Constraint checks alone do not measure
  communication overhead.

• ? We need a metric that combines both of the
  previously used metrics.

13
CBR Cycle-Based Runtime

• The length of a cycle is determined by
  communication and computation

• (t time for one constraint check.)

Define ccc(m) as the total constraint checks
14
CBR parameters

• L communication latency (time to communicate in
  each cycle).
• ? can be parameterized based on the system
  environment. Eg., L1, 10, 100, 1000.
• We assume t1, since constraint checks are
  usually faster than communication time.
• ? L is defined in terms of t (L10 indicates
  communication is 10 times slower than a
  constraint check).

• L

15
Comparing Adopt and OptAPO

• Algorithmic centralization
• Adopt is non-centralized.
• OptAPO is partially centralized.
• Prior work Mailler Lesser has shown that
  OptAPO completes in fewer cycles than Adopt for
  graph coloring problems at density 2n and 3n.
• But how do they compare when we use CBR to
  measure both communication time and computation?

16
DCOP Algorithms Adopt

• ADOPT, Jay Modi, et al.
• Variables are ordered in a priority tree (or
  chain).
• Agents pass their current value down the tree to
  neighboring agents using VALUE messages.
• Agents send COST messages up to their parents.
  Cost messages inform the parent of the lower and
  upper bounds at the subtree.
• These costs are dependent on the values of the
  agents ancestor variables.

17
DCOP Algorithms OptAPO
OptAPO Optimal Asynchronous Partial Overlay.
Mailler and Lesser

• cooperative mediation an agent is dynamically
  appointed as mediator, and collects constraints
  for a subset of the problem.
• OptAPO agents attempt to minimize the number of
  constraints that are centralized.
• The mediator solves its subproblem using Branch
  Bound centralized search Freuder and Wallace.

x1
x2
x5
mediator
x3
Values constraints
18
Results DavinModi, 05

• Tested on graph coloring problems, D3
  (3-coloring).
• Variables 8, 12, 16, 20, with link density
  2n or 3n.
• 50 randomly generated problems for each size.

CCC
Cycles
? OptAPO takes fewer cycles, but more constraint
checks.
19
How do Adopt and OptAPO compare using CBR?
Density 2
20
How do Adopt and OptAPO compare using CBR?
Density 3
? For L values lt 1000, Adopt has a lower CBR than
OptAPO. ? OptAPOs high number of constraint
checks outweigh its lower number of cycles.
21
How much centralization occurs in OptAPO?

• OptAPO sometimes centralizes all of the problem
  structure.

22
How does the distribution of computation differ?

• We measure the distribution of computation during
  a cycle as
• This is the ratio of the maximum computing agent
  to the total computation during a cycle.
• A value of 1.0 indicates one agent did all the
  computation.
• Lower values indicate more evenly distributed
  load.

23
How does the distribution of computation differ?

• Load was measured during the execution of one
  representative graph coloring problem with 8
  variables, density 2

• OptAPO has varying load, because one agent (the
  mediator) does all of the search within each
  cycle.
• Adopts load is evenly balanced.

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Communication Tradeoffs of Centralization

• How does centralization affect performance under
  a range of communication latencies?

• ?Adopt has the lowest CBR at L1,10,100, and
  crosses over between L100 and 1000.
• OptAPO outperforms Branch Bound at density 2
  but not at density 3.

25
Part 2 Domain Centralization in DCOPs
26
How can we take advantage of domain
centralization?

• Adopt treats variables within an agent as
  independent pseudo-agents.
• Does not take advantage of partially centralized
  domains.

x1
agent2
x2
agent1
x3
27
How can we take advantage of domain
centralization?

• Instead, we could use centralized search to
  optimize the agents local variables. We call
  this AdoptMVA, for Multiple Variables per Agent.
• Potentially more efficient
• Adopts variable ordering heuristics do not apply
  to agent orderings
• Need agent ordering heuristics for AdoptMVA.
• Also need intra-agent heuristics for the local
  search.

agent2
x1
agent1
x2
x3
28
Extending Adopt to AdoptMVA

• In Adopt, a context is a partial solution of the
  form (xj,dj),(xk,dk).
• We define a context where S is the set
  of all possible assignments to an agents local
  variables.
• We must modify Adopts cost function to include
  the cost of constraints between variables in s

Intra-agent cost
Inter-agent cost

• Use Branch Bound search to find the
  that minimizes the local cost.

29
Results

• Setup Randomly generated meeting scheduling
  problems
• Based on an 8-hour day (D 8).
• Number of attendees (A) per meeting is randomly
  chosen from a geometric progression.
• All meeting scheduling problems generated were
  fully schedulable.
• We compared using a lexicographic agent ordering
  for both algorithms.

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How does AdoptMVA perform vs Adopt?
High density meeting scheduling (4 meetings per
agent)
CBR
Cycles
20 problems per datapoint.
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How does AdoptMVA perform vs Adopt?
Graph Coloring with 4 variables per agent, link
density 2
CBR
Cycles
? AdoptMVA uses fewer cycles than Adopt, and has
a lower CBR at L1000.
32
Adopt and AdoptMVA Variable ordering
Original Problem
Adopt hierarchy
AdoptMVA hierarchy

• AdoptMVAs order has a reduced granularity from
  Adopts order. Adopt can interleave the variables
  of an agent, while AdoptMVA can only order
  agents.

33
Inter-agent ordering heuristics

• We tested several heuristics for ordering the
  agents
• Lexicographic
• Inter-agents links order by of links to other
  agents.
• AdoptToMVA-Max compute the Brelaz ordering over
  variables, and convert it to an agent ordering
  using the maximum priority variable within each
  agent.
• AdoptToMVA-Min AdoptToMVA-Max but using the
  minimum priority variables.

34
Comparison of agent orderings

• Intra-agent ordering MVA-HigherVars

Low Density meeting scheduling
High Density meeting scheduling
Graph Coloring with 4 variables per agent
35
Comparison of agent orderings

• Intra-agent ordering MVA-HigherVars

Low Density meeting scheduling
High Density meeting scheduling

• Ordering makes an order of magnitude difference
  in some cases.
• AdoptToMVA-Min was the best on 8 out of 11 cases,
  but with high variance.

36
Intra-agent Branch Bound Heuristics

• Best-first Value ordering heuristic we put the
  best domain value first in the value ordering.
• Variable ordering heuristics
• Lexicographic
• Random
• Brelaz Graph Coloring only order by number of
  links to other variables within the agent.
• MVA-AllVars order by of links to external
  variables.
• MVA-LowerVars MVA-AllVars but only consider
  lower priority variables.
• MVA-HigherVars MVA-AllVars but only consider
  higher priority variables.

37
Comparison of intra-agent search heuristics

• Goal Reduce constraint checks used by Branch
  Bound.
• Metric Average CCC per Cycle (TotalCCC / Total
  Cycles).

High density Meeting Scheduling
Cycles
Avg CCC

• Nearly all differences are statistically
  significant. Excepting MVA-AllVars vs
  MVA-HigherVars
• MVA-HigherVars is the most computationally
  efficient heuristic.
• ?Low density Meeting Scheduling produced similar
  results.

38
Comparison of intra-agent search heuristics
Graph Coloring
Avg CCC
Cycles
? Confirms Brelaz is the most efficient heuristic
for graph coloring.
39
How does Meeting Scheduling scale as agents and
meeting size are increased?
Original Data
Outliers removed
(A avg of attendees (meeting size))

• Original data had several outliers (two standard
  deviations away from the mean) so they were
  removed for easier interpretation.
• ? Meeting size has a large effect on solution
  difficulty.

40
Thesis contributions

• Formalization of algorithmic vs. domain
  centralization.
• Empirical comparison of Adopt OptAPO showing
  new results.
• CBR - a performance metric which accounts for
  both communication computation in DCOP
  algorithms.
• AdoptMVA - a DCOP algorithm which takes advantage
  of domain centralization. We also contribute an
  efficient intra-agent search heuristic for
  meeting scheduling.
• Empirical analysis of the meeting scheduling
  problem.
• Impact of Problem Centralization in DCO
  Algorithms, AAMAS 05, Davin Modi.

41
Future Work

• Improve AdoptMVA
• Agent ordering heuristics is there a heuristic
  which will work better than the ones tested?
• Intra-agent search heuristics develop a
  heuristic that is both informed and randomly
  varied.
• Test DCOP algorithms in a fully distributed
  testbed.

42
The End

• Questions?

My future plans AAMAS in July and MSN Search
(Microsoft) in the Fall. Contact jdavin_at_cmu.edu