Hardness of Shops and Optimality of List Scheduling

1. Hardness of Shops and Optimality of List Scheduling Ola Svensson KTH Royal Institute of Technology Stockholm, Sweden

2. Gap Instance with r=3 and F=3

3. Jobs of different frequency “cannot” overlap Key-Lemma: Jobs of different frequencies overlap at most a fraction 1/r of their length.

4. 0 Reduction 1 2 • Jobs corresponding to an IS can be scheduled in parallel since no short-jobs interfer with the long-jobs Jobs can be scheduled in parallel iffthey correspond to an IS • Adjacent jobs cannot overlap by Key-Lemma

5. Completeness: If Graph is K-colorable then there is a schedule of makespan at most K*lb Soundness: If Graph has no IS of size n/Klog(K) then any schedule has makespan at least Klog(K) *lb Some point

6. Bounding #frequencies • Take “enough” and assign random frequencies to jobs • For a fixed K the graphs have bounded degree (Color in polytime using degree+1 colors, each color only needs one frequency)

7. Final Result SAT Graph Coloring/IS Job Shops • It is NP-hard to approximate job shops within any constant factor makespan K lb K colors no IS of size n/KlogK makespanklogK lb • Assuming NP-complete problems have no randomized quasi-polynomial then job shops have no better than log(lb) approximation.

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9. Final Comments • PTAS iff #machines and #operations per job bounded by a constant Jansen, Solis-Oba & Sviridenko’99 + Mastrolilli, S’08 • Preemptive case wide open • No non-constant gap instances • Best hardness 5/4 and no constant approx algos.

10. Expander Not Expander Not scheduled • 0.99n machines • 0.99n machines

11. Not Expander Expander d+1 2d • 0.99n machines • 0.99n machines

12. Building block (an instance of 1|prec|ΣwjCj) Instance of P|prec| Cmax processing time = 0 processing time = 1

13. Yes Case No Case d+1 2d • 0.99n machines • 0.99n machines

14. Do not have • Two extreme cases: • almost all predecessors have been completed • almost none of them have been completed

15. Final Result SAT Node Expansion P|prec|Cmax • Assuming 1|prec|ΣwjCj is hard to approximate within a factor less than 2 • Then P|prec|Cmax is hard to approximate within a factor less than 2 makespan d+1 makespan 2d

16. Is 1|prec|ΣwjCj difficult? (1/2) • Special Case of Vertex Cover • Removing Fixed Cost then equivalent to Vertex Cover • No PTAS unless NP-complete problems can be solved in randomized polynomial time Correa & Schulz’04 , Ambuhl & Mastrolilli’06

17. Is 1|prec|ΣwjCj difficult? (2/2) • Assuming a variant of the Unique Games Conjecture • It is NP-hard to approximate better than 2 Bansal & Khot’09 SAT Unique Games 1|prec|ΣwjCj P|prec|Cmax makespan d+1 makespan 2d Vertex Cover, Max Cut, Feedback arc set, rich family of CSPs…

18. Summary • Non-constant hardness for job shops that are essentially tight for acyclic job shops and general flow shops • Grahams list scheduling algorithm is optimal assuming a variant of the unique games conjecture • Master plan has been formalized for some classes of problems • Integrality gap of an SDP for CSPsimplies UG-hardness • Integrality gap of an LP for packing and covering problems implies UG-hardness Raghavendra’08 Kumar, Manokaran, Tulsiani, Vishnoi’11

19. Two open problems • Scheduling precedence-constrained jobs on related machines Q|prec|Cmax • log(m) approximation • No better hardness than for parallel machines • Scheduling on unrelated machines R||Cmax • 2 approximation • NP-hard to do better than 1.5 Chudak & Shmoys’97 Lenstra, Shmoys & Tardos’90