Asynchronous Backtracking for Asymmetric Constraints

1
Asynchronous Backtracking for Asymmetric
Constraints

• Roie Zivan and Amnon Meisels
• Dept. of Computer Science
• Ben-Gurion University

2
ABT Strong Assumption

• All constraints are held by the lower priority
  agents involved in them
• Formally
• For every binary constraint Rij between agents Ai
  and Aj, jgti. Aj holds Rij

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Different Agents, Different Constraints
First the mission, then the ladies
Mister Anderson?
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Different Agents, Different Constraints
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Asymmetric CSPs (Partially Known Constraints
model (Brito and Meseguer03)

• Every Binary constraint Cij is split into two
  parts
• Ci(j) held by agent Ai
• C(i)j held by agent Aj
• Ci(j) U C(i)j Cij
• Finding a consistent assignment for an
  asymmetric constraint requires that both sides
  approve with the other sides assignment

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Asymmetric DisCSP example
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Two Phase ABT (DFC_PKCP Brito and Meseguer03)

• Phase 1 Run ABT on one direction checking only
  the constraints Ci(j) , iltj.
• Phase 2 When a solution is found check it in
  the opposite direction (constraints C(i)i, iltj).
  
• If the solution found is consistent
• Report
• else - record the relevant Nogoods and return
  to phase 1.

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ABT_ASC

• A single phase algorithm for Asymmetric DisCSPs
• Every agent sends ok messages to all its
  neighbors
• Agents replace their assignments only if they
  conflict with higher priority assignments
• For conflicts with lower priority agents, the
  assignment is kept and a notifying Nogood is sent

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ABT_ASC
A3,4
A3
A2
A1,2,A3,1
A1
A3,4
A4
A1,2,A3,1
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Backtrack in ABT
AgentView of Agent A2 A12, A31
Nogood
A1
A12
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Backtrack in ABT
AgentView of Agent A2 A31
Nogood
A21,A31
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Correctness proof (outline)

• Assumption Standard ABT is correct
• Constraints generate Explicit Nogoods
• All Nogoods are created by unions of Explicit
  Nogoods
• ABT_ASC cannot create Explicit Nogoods that are
  not created by standard ABT

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ABT_ASC
A2
A3
A21,A31
A21,A31
A1,2,A3,1
A2,1,A4,1
A2 ? A3
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ABT_ASC
A2
A3
A21,A31
A1,2
A2,1
A2 ? A3
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ABT_ASC vs 2_phase ABT (p1 0.4)
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ABT_ASC vs 2_phase ABT (p1 0.4)
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ABT_ASC vs 2_phase ABT (p1 0.7)
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ABT_ASC vs 2_phase ABT (p1 0.7)
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Number of changes of phase
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Privacy discussion

• Explicit Nogoods are sent to lower priority
  agents, possible breach of privacy of
  constraints
• Possible solutions
• Agents can send forbidden values - DFC method of
  Brito and Meseguer
• Use an anonymous message delivery system (for
  Nogoods) conceal the identity of the Nogoods
  sender (may require additional messages)
• Single direction solutions are not exposed

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Conclusions

• Standard ABT assumes the constraints graph is a
  DAG
• This assumption is not compatible with many real
  world problems
• Previous solutions to Asymmetric DisCSPs included
  running two phases of a single direct algorithm
• ABT_ASC is a one phase algorithm which
  outperforms the two phase algorithm by a large
  factor
• This improvement does not generate additional
  privacy loss

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A Distributed Constraint Network
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Asynchronous Backtracking

• Motivation
• Concurrent computatio
• HighLights
• Agents always hold an assignment which is
  consistent with their view of the system.
• On backtrack operations, agents change their
  views by eliminating inconsistent values.

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Asynchronous BackTracking - ABT

• Agents are totally ordered
• Each agent assigns its variable and sends it in
  ok? messages to agents ordered after it
• All received messages are stored in the AgentView
• Agents check their assignments against their
  AgentView
• Nogood messages received eliminate agents
  assignment as long as they are consistent with
  the current AgentView.
• Yokoo 1998, 2000

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Asynchronous Backtrack (ABT)
A3
A2
A1
A31
A4
A12
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Backtrack in ABT
AgentView of Agent A4 A12, A31
Nogood
Current Assignment
A3
A12, A31
A41