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. 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 iff they 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
makespan K lb
K colors
no IS of size
n/KlogK
makespan klogK lb
It is NP-hard to approximate job shops within any constant factor
Assuming NP-complete problems have no randomized quasi-polynomial then job shops have no better than log(lb) approximation.

8. Performance Guarantee
Gap Instance
Hardness Result
Job Shops
Õ(log2(lb))
Shmoys, Stein, Wein’94
Goldberg,Paterson,Srinivisan, Sweedyk’97
Ω(log(lb))
Feige & Scheideler’98
Mastrolilli,S’08
Acyclic Job Shops
Õ(log(lb))
Generalized Flow Shops
Mastrolilli,S’09
Flow Shops
Mastrolilli, S’09
5/4
Williamson, Hall, Hoogeven, Lenstra, Sevastianov, Shmoys’97 ~ ~ ~ ~ ~ ~ ~

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 2d
d+1
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 2d
d+1
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
makespan d+1
makespan 2d
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

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
P|prec|Cmax
makespan d+1
makespan 2d
1|prec|ΣwjCj
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 CSPs implies 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