Preemptive Scheduling of Intrees on Two Processors

1. Preemptive Scheduling of Intrees on Two Processors Coffman, E. G., Jr., Columbia University Matsypura, D., Oron, D., Timkovsky, V. G., University of Sydney Marseille, CIRM, May 12-16, 2008

2. My Hidden Co-Authors Wife: Natasha Daughter: Linda Anastasia

3. The Problem of Interest Target: Integer release times

4. FRAGMENTS 1 3 1 2 1 2 3/2 schedule Previous Results O(n2), Baptiste-Timkovsky, 2001 O(n2), Brucker-Hurink-Knust, 2002; O(n log n), Huo-Leung, 2005 O(n2), Lushchakova, 2006 O(n2), Coffman-Seturaman-Timkovsky, 2003 O(n2), Muntz-Coffman, 1969 Half-Integrality Proof, Sauer-Stone, 1987 • Any optimal schedule is a concatenation • of these fragments • Each job is preempted at most once • in the middle

5. Schedule Fractionality • The fractionality of a preemptive schedule is the greatest reciprocal 1/k such that the interval between every two event times (i.e., start times, completion times, or preemption times) in the schedule is a multiple of 1/k . • Preemptive schedules of fractionality 1/2 are simply half-integer schedules. • We say that a preemptive scheduling problem has a bounded fractionality if there exist constant k and optimal preemptive schedules of fractionality 1/k for all its instances, or an unbounded fractionality otherwise.

6. Fractionality Conjecture Weak Conjecture [mid 80s]: This problem has bounded fractionality 1/m Strong Conjecture [mid 80s]: This problem has bounded fractionality 1/p(m), where p is a polynom BOTH ARE NOT TRUE

7. Three-Macine Example of Unbounded Fractionality The Sauer-Stone Theorem [1987]: For any fixed m>2 there exists an instance of this problem, i.e., precedence constraints, whose all optimal schedules are of fractionality 1/mn. Sauer, N. W., Stone, M. G., Rational preemptive scheduling, Order 4 (1987) 195-206

8. These three jobs in the trunk are needed to prevent a delay of all preceding jobs release times M1 M2 0 Two-Machine Example of Unbounded Fractionality 1 2 3 4 5 6 7 8 3/2 schedule

9. The Example Description • 3n+3 jobs in triplets: Aj, Bj, Cj, j = 1,2,…,n,n+1 • Release times: 0, 0, 0 for j = 1, • Release times: 2j-3, 2j-2, 2j-2 for 1< j<n+1 • Relaese times: 2j-3, 2j-2, 2j-1 for j = n+1 • Precedence constraints: Aj Aj+1, Bj Aj+1, Cj Aj+1, j<n+1 An+1 Bn+1 Cn+1

10. preemptive 0 1 2 3 4 5 6 7 8 half-integer 0 1 2 3 4 5 6 7 8 nonpreemptive 0 1 2 3 4 5 6 7 8 9 Schedules Comparison

11. Minimum Maximum and Total Completion Times for the Example Minimum Maximum and Total Completion Time: Unbounded Fractionality Minimum Maximum and Total Completion Time: Half-Integer Minimum Maximum and Total Completion Time: Nonpreemptive

12. NP Preemption Hypothesis • Recognition versions of preemptive problems in the classification belong to NP. • In other words, there exist solutions to these problems that can be checked in polynomial (in problem size) time. • IS THIS TRUE? • The problems are strong candidates in finding counterexamples to the hypothesis.

13. Half-Integer Solution This problem reduces to serching a minimum path in a directed graph and has an O(n15) algorithm. What about

14. Thank you • v.timkovsky@econ.usyd.edu.au