Distributed Scheduling

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Distributed Scheduling
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What is Distributed Scheduling?

• Scheduling
• A resource allocation problem
• Often very complex set of constraints
• Tied directly to planning the binding between
  the plan and the resources
• Distributed
• Each node has only partial information
  (non-global)
• No centralized algorithm processing is
  distributed to each node

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Dynamic vs. Static

• Many agent systems include elements of
  uncertainty
• lack of progress on tasks (also early completion)
• resource failure
• newly available resources or plans
• incomplete information
• quality of solution
• Proactive robust schedules provide alternatives
  and redundancy for dynamism
• Reactive schedules are repaired in response to
  changes

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Approaches

• Guaranteed optimality
• search
• constraint propagation
• dynamic programming
• Best effort
• heuristic based search (often domain specific)
• iterative improvement (local search/genetic
  algorithms)

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Job Shop Scheduling

• Allocation of m tasks to n resources
• Each task has a release time, duration, resource
  requirements, and ordering constraints
• Each resource has a capacity constraint
• A solution to the problem is a feasible schedule
  that includes start times and resource
  assignments for each task

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Job Shop Scheduling (cont)

• An objective function can be used to evaluate the
  goodness of a schedule
• May include weights for the completion of each
  task by a certain deadline and penalties for
  lateness. May also include rewards for early
  completion.
• Considered one of the hardest CSPs

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Constraint Satisfaction Problems (CSP)

• Definition
• V A set of variables
• Di A domain of values for Vi
• C A set of boolean constraints on V
• Order of Constraints
• Unary V1 ! red
• Binary V1!V2
• N-ary max(V1, V2, V3) gt 5
• NP-Complete!
• Can represent 3-SAT as a CSP

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Techniques for Solving CSPs

• Depth-first search
• Backtracking
• Checks for constraint violations before
  generating successors for DFS
• Arc-consistency
• When an assignment is made, all inconsistent
  values are removed from the possible assignments
  for other variables (often used in preprocessing)
• Heuristics
• Value ordering
• Most Constrained Variable
• Least Constraining Value Assignment
• Iterative improvement
• Hill climbing/annealing
• Heuristic repair min-conflicts

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Map Coloring as CSP

• Constraint Graph
• DWAred, green, blue

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Map Coloring Example
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Distributed CSP

• Variables and constraints distributed among
  agents
• No agent controls all variables or knows all
  constraints
• Naturally captures many problems in multiagent
  systems
• Coordination tasks, conflicts between actions,
  etc
• DisCSP first formalized by Yokoo et al 1998.
• Why is distribution needed?
• Privacy
• No central control
• Communication costs
• Autonomy
• Robustness

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Meeting Scheduling Problem

• A set of agents want to schedule some meetings
• M1 A1, A3
• M2 A1, A2
• M3 A2, A4, A5
• Timeslots 8-9am, 9-10am, 10-11am, ...
• Each agent can attend one meeting at a time

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MSP (cont)
Variables represent what time slot an agent
attends a meeting Equality constraints for
variables of the same meeting Inequality
constraints for variables of the same agent
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Distributed CSP Algorithms (cont)

• Asynchronous Weak Commitment
• dynamic prioritization of variables
• uses the min-conflict heuristic
• Distributed Breakout
• mutual exclusion among neighbors

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Distributed CSP Algorithms

• SynchronousBacktracking
• essentially the original backtracking algorithm
• Asynchronous Backtracking
• total order of nodes

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Problems with CSP

• What if no solution exists?
• What if multiple solutions exist?
• In both cases we want to know which assignment is
  the best
• Generalizes into the Constraint Optimization
  Problem
• Constraints are no longer boolean
• The goal is now to obtain the highest global
  utility

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Constraint Optimization Problem (COP)

• Definition
• V A set of variables
• Di A domain of values for Vi
• U A set of utility functions on V
• Goal is to optimize global utility
• can also model minimal cost problems by using
  negative utilities

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COP (cont)