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Temporal Constraint Propagation (Preemptive Case)

This document outlines the framework for temporal constraint propagation when introducing new variables in a preemptive scheduling context. It defines the new variables such as set(A), WA(t), start(A), end(A), duration(A), and span(A), along with their implementation strategies. The relations between these variables are established, emphasizing how start and end times relate to the execution of tasks. Additionally, it covers different possible implementations, including explicit set variables and dynamic lists of intervals, to effectively manage temporal constraints within scheduling problems.

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Temporal Constraint Propagation (Preemptive Case)

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  1. Temporal Constraint Propagation(Preemptive Case)

  2. Outline • New variables • Definition • Implementation • Relations between the variables • Temporal constraints

  3. New variables (definition) • set(A) = {t such that A executes at time t} • WA(t) = 1 when tÎset(A), 0 otherwise • start(A) = mintÎset(A)(t) • end(A) = maxtÎset(A)(t + 1) • duration(A) = |set(A)| • span(A) = end(A) - start(A)

  4. New variables (implementation) • Three possible implementations for set(A) • Explicit set variable set(A) • Explicit Boolean variables WA(t) • Dynamic list of intervals Ii(A) = [si(A), ei(A)) with • W(Ii(A)) = 1 if "tÎ[si(A), ei(A)), tÎset(A) • W(Ii(A)) = 0 if "tÎ[si(A), ei(A)), tÏset(A) • W(Ii(A)) = unknown otherwise • Explicit or implicit integer variables for start(A), end(A), duration(A), and span(A)

  5. Relations between the variables • end(A) = start(A) + span(A) • duration(A) £ span(A) • duration(A) = |set(A)| cardinality constraint specific implementation for a list of intervals

  6. Relations between the variables • start(A) = mintÎset(A)(t) start(A)Îset(A) [t = startmin(A) = startmax(A)] implies [tÎset(A)] [t = startmin(A)Ïset(A)] implies [t < start(A)] [t = startmax(A)Ïset(A)] implies [start(A) < t] "tÎset(A), start(A) £ t [tÎset(A)] implies [start(A) £ t] [t < startmin(A)] implies [tÏset(A)]

  7. Relations between the variables • end(A) = maxtÎset(A)(t + 1) (end(A) - 1)Îset(A) [t = endmin(A) = endmax(A)] implies [(t - 1)Îset(A)] [t = (endmin(A) - 1)Ïset(A)] implies [(t + 1) < end(A)] [t = (endmax(A) - 1)Ïset(A)] implies [end(A) < (t + 1)] "tÎset(A), t < end(A) [tÎset(A)] implies [t < end(A)] [endmax(A) £ t] implies [tÏset(A)]

  8. Relation between the variables • pos(A) = {t such that WA(t) can be 1} • |{t'Îpos(A) such that t' < t}| < durationmin(A) implies [t < end(A)] • |{t'Îpos(A) such that t £ t'}| < durationmin(A) implies [start(A) < t]

  9. Temporal constraints • Constraints between start and end variables Similar to the non-preemptive case when start(A) and end(A) are explicit • Other constraints "tÎset(A), tÎset(B) (inclusion) "tÎset(A), tÏset(B) (exclusion) "tÏset(A), tÎset(B) (coverage)

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