D. Gillies, "Algorithms to schedule tasks with AND/OR precedence constraints," Ph.D. thesis, University of Illinois, Urbana, IL, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to be consistent with the notion of edges in a directed blocking graph. Often the meaning of the directed edge is reversed in scheduling literature. and C blocks A in a third. For this reason, we have made no a priori assumptions about the structure of the precedence relations. The work of Refs. [23,24] studies a larger variety of settings, including multiple processors, deadlines, and individual processing times. They prove the NP hardness of finding feasible schedules in many settings that are polynomially solvable with more traditional and precedence constraints, however they do not consider ....

D. Gillies, "Algorithms to schedule tasks with AND/OR precedence constraints," Ph.D. thesis, University of Illinois, Urbana, IL, 1993.

....for all tasks. It is worth noting that with classical and precedence constraints, this problem of minimizing the number of scheduled tasks can be solved exactly, in polynomially time by computing a depth order. A model for scheduling with and or precedence constraints has been studied earlier [15, 16], but with one key difference. The precedence constraints for an instance can be represented as a directed graph, with each node additionally tagged as either an and node or an or node; in this previous work, they assume that there is no cycle in this precedence graph, as that would make the ....

....the existence of a valid solution, and in fact cycles will often exist as it may be the case that part A blocks B in one direction, B blocks C in another, and C blocks A in a third. For this reason, we make no a priori assumptions about the structure of the precedence relations. Gillies and Lin [15, 16] prove the NP hardness of many variants of the problem, however they do not consider the approximability of the hard problems. Notation and Definitions. The input contains a set of tasks, T . Each task, t i 2 T , is labeled as either an and task or an or task, and has an associated set of tasks, P ....

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D. Gillies. Algorithms to schedule tasks with AND/OR precedence constraints. Ph.D. thesis, University of Illinois, Urbana, IL, 1991.

....for all tasks. It is worth noting that with classical and precedence constraints, this problem of minimizing the number of scheduled tasks can be solved exactly, in polynomially time by computing a depth order. A model for scheduling with and or precedence constraints has been studied earlier [19, 20], but with one key difference. The precedence constraints for an instance can be represented as a directed graph 6 with each node additionally tagged as either an and node or an or node; in this previous work, they assume that there is no cycle in this precedence graph, as that would make the ....

....for the existence of a valid solution, and in fact cycles will often exist as it may be the case that part A blocks B in one direction, B blocks C in another, and C blocks A in a third. For this reason, we make no apriori assumptions about the structure of the precedence relations. Gillies and Lin [19, 20] prove the NPhardness of many variants of the problem, however they do not consider the approximability of the hard problems. It is important to note that in context of assembly sequencing, the precedence constraints for this scheduling problem could be more naturally modeled as DNF scheduling, ....

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D. Gillies. Algorithms to schedule tasks with AND/OR precedence constraints. Ph.D. thesis, University of Illinois, Urbana, IL, 1991.

....notion of edges in a directed blocking graph. Often the meaning of the directed edge is reversed in scheduling literature. CHAPTER 5. AND OR SCHEDULING 39 precedence constraints. However, a series of papers by Gillies et al., study several variants of scheduling with and or precedence constraints [24, 25]. Our model for this problem was chosen to be similar to [24, 25] however with one key difference. As mentioned in Section 5.2, the precedence constraints for an instance can be represented as a directed graph. In this previous work, only the case of partial order precedence constraints is ....

....the directed edge is reversed in scheduling literature. CHAPTER 5. AND OR SCHEDULING 39 precedence constraints. However, a series of papers by Gillies et al., study several variants of scheduling with and or precedence constraints [24, 25] Our model for this problem was chosen to be similar to [24, 25], however with one key difference. As mentioned in Section 5.2, the precedence constraints for an instance can be represented as a directed graph. In this previous work, only the case of partial order precedence constraints is considered. Notice that in traditional scheduling, with and precedence ....

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D. Gillies. Algorithms to Schedule Tasks with AND/OR Precedence Constraints. Ph.D. thesis, University of Illinois, Urbana, IL, 1993.

....for this exact approximation problem. Minimizing the number of ones in satisfying a 3CNF formula is known to be n 0.5 # hard to approximate [29] and related minimization problems are studied in [34] Previous Work. A model for scheduling with and or precedence constraints was introduced in [14,15], however with one key di#erence. In this previous work, only the case of partial order precedence constraints is considered. Notice that in traditional scheduling, with and precedence constraints, the existence of a cycle in the precedence constraints makes the scheduling problem infeasible, and ....

....drawn from assembly sequencing, as it may be the case that part A blocks part B in one direction, part B blocks part C in another, and part C blocks part A in a third direction. For this reason, we make no apriori assumptions about the structure of the precedence constraints. The work of [14,15] studies a larger variety of settings, including multiple processors, deadlines, and individual processing times. They prove the NP hardness of finding feasible schedules in many setting which were polynomially solvable with more traditional and precedence constraints, however they do not consider ....

D. Gillies. Algorithms to Schedule Tasks with AND/OR Precedence Constraints. Ph.D. thesis, University of Illinois, Urbana, IL, 1993.

....this exact approximation problem. Minimizing the number of ones in satisfying a 3CNF formula is known to be n 0:5 Gammaffl hard to approximate [29] and related minimization problems are studied in [34] Previous Work. A model for scheduling with and or precedence constraints was introduced in [14,15], however with one key difference. In this previous work, only the case of partial order precedence constraints is considered. Notice that in traditional scheduling, with and precedence constraints, the existence of a cycle in the precedence constraints makes the scheduling problem infeasible, and ....

....drawn from assembly sequencing, as it may be the case that part A blocks part B in one direction, part B blocks part C in another, and part C blocks part A in a third direction. For this reason, we make no apriori assumptions about the structure of the precedence constraints. The work of [14,15] studies a larger variety of settings, including multiple processors, deadlines, and individual processing times. They prove the NP hardness of finding feasible schedules in many setting which were polynomially solvable with more traditional and precedence constraints, however they do not consider ....

D. Gillies. Algorithms to schedule tasks with AND/OR precedence constraints. Ph.D. thesis, University of Illinois, Urbana, IL, 1993.

....least slack first, and some generalizations of the algorithms in [4] neglect to compare the deadlines among different in trees. In the worst case these algorithms may meet only p n deadlines when it is possible to meet n out of n 1 deadlines. For more information the reader is referred to [11] [12]. 3.2. Scheduling to Minimize Completion Time. We now consider the problem of scheduling AND OR unskipped tasks with arbitrary processing times on m processors to meet a common deadline. This problem is equivalent to that of scheduling to minimize the overall completion time. Ullman has shown ....

....can simulate this other model. There are also situations where both OR skipped and OR unskipped tasks are present in a single in tree. With slight modifications our AND OR skipped heuristic can be used to handle such mixed task systems. Details of these transformations and algorithms appear in [12]. During this investigation we reached several conclusions about the complexity of AND OR scheduling. Contrary to our intuition, the skipped problems we considered were generally of higher complexity than the corresponding unskipped problems. This can be seen by comparing Table 1 and Table 2, and ....

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D. W Gillies, Algorithms to schedule tasks with AND/OR precedence constraints, Ph.D. Thesis, Department of Computer Science, Univ. of Illinois, Urbana, 1993.

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