Distributed Scheduling

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

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

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

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

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

Constraint Satisfaction Problems (CSP) Definition: –V = A set of variables –D i = A domain of values for V i –C = A set of boolean constraints on V Order of Constraints: –Unary: V 1 != red –Binary: V 1 !=V 2 –N-ary: max(V 1, V 2, V 3 ) > 5 NP-Complete! –Can represent 3-SAT as a CSP

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

Map Coloring as CSP Constraint Graph D WA ={red, green, blue}

Map Coloring Example

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 Why is distribution needed? –Privacy –No central control –Communication costs –Autonomy –Robustness

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

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

Distributed CSP Algorithms (cont’) Asynchronous Weak Commitment –dynamic prioritization of variables –uses the min-conflict heuristic Distributed Breakout –mutual exclusion among neighbors

Distributed CSP Algorithms Synchronous Backtracking –essentially the original backtracking algorithm Asynchronous Backtracking –total order of nodes

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

Constraint Optimization Problem (COP) Definition: –V = A set of variables –D i = A domain of values for V i –U = A set of utility functions on V Goal is to optimize global utility –can also model minimal cost problems by using negative utilities

COP (cont’)