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Temporal Constraint Propagation in Non-Preemptive Scheduling: Methods and Complexity Analysis

This overview delves into the mechanisms of temporal constraint propagation in non-preemptive scheduling scenarios. It covers essential variables such as start, end, and duration for activities, as well as various relations between these variables defined by temporal constraints. The propagation of time bounds is explored, in addition to methods for handling minimal and maximal distances between time points. Complexity analysis is provided, detailing computational efficiency for both consistent and inconsistent networks. This guide aims to equip readers with foundational knowledge for managing temporal constraints in scheduling.

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Temporal Constraint Propagation in Non-Preemptive Scheduling: Methods and Complexity Analysis

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  1. Temporal Constraint Propagation(Non-Preemptive Case)

  2. Outline • Variables • Relations between the variables • Temporal constraints • Time bounds • Minimal and maximal distances between time points

  3. Variables (definition) Three variables start(A) end(A) duration(A) for each activity A

  4. Variables (implementation) • Finite domain (bitvector) • The domain of each variable is a finite set • Interval domain (pair of numbers) • The domain of each variable is an interval startmin(A), startmax(A), endmin(A), endmax(A) durationmin(A), durationmax(A)

  5. Relation between the variables end(A) = start(A) + duration(A) endmin(A) = max(endmin(A), startmin(A) + durationmin(A)) endmax(A) = min(endmax(A), startmax(A) + durationmax(A)) startmin(A) = max(startmin(A), endmin(A) - durationmax(A)) startmax(A) = min(startmax(A), endmax(A) - durationmin(A)) durationmin(A) = max(durationmin(A), endmin(A) - startmax(A)) durationmax(A) = min(durationmax(A), endmax(A) - startmin(A))

  6. Temporal constraints • Simple precedences start(A) £ start(B) start(A) £ end(B) end(A) £ start(B) end(A) £ end(B)

  7. Temporal constraints • Precedences with minimal delays start(A) + delay £ start(B) start(A) + delay £ end(B) end(A) + delay £ start(B) end(A) + delay £ end(B)

  8. Temporal constraints • Precedences with fixed delays start(A) + delay = start(B) start(A) + delay = end(B) end(A) + delay = start(B) end(A) + delay = end(B)

  9. Temporal constraints • Maximal delays start(A) £ start(B) + delay start(A) £ end(B) + delay end(A) £ start(B) + delay end(A) £ end(B) + delay

  10. Propagation of time bounds var(A) + delay £ var(B) varmin(B) = max(varmin(B), varmin(A) + delay) varmax(A) = min(varmax(A), varmax(B) - delay) Complete propagation for bounded domains • Contradiction found when the constraints conflict • Best possible varmin(A) and varmax(A) found otherwise • Incremental variant of an operations research algorithm for project scheduling (PERT networks)

  11. Propagation of time bounds Complexity • For a consistent network: O(n*m) where n is the number of activities and m the number of constraints if constraints are propagated in the first-in first-out order • For an inconsistent network: O(h*n2) where h is the time horizon (can be reduced to O(n3) but not worth it in practice)

  12. Minimal and maximal distances • [x + dxy£ y] and [y + dyz£ z] implies [x + (dxy+ dyz) £ z] • Useful to solve disjunctions of temporal constraints [x - 5 £ y] [y + 2 £ z] [z + 4 £ x] OR [v + 3 £ w]

  13. Minimal and maximal distances • Matrix-based method Whenever dxy is modified, update dwz to max(dwz, dwx+ dxy+ dyz)

  14. Minimal and maximal distances Complexity • O(n2) after each modification of the constraint network • O(n3) to initialize the matrix

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