1. Non-clairvoyant Precedence Constrained Scheduling
July 9th, 2019
Sahil singla
Princeton University and Institute for Advanced Study
Joint work with naveen Garg, Anupam gupta, And Amit Kumar

2. What is Scheduling?
𝐦 identical machines: Each processes 1 unit per time
𝐧 jobs: (Release time r j , Processing time p j , Weight w j )
Goal: Execute jobs to minimize:
Total completion time: j C j
Total time in system (flow time): j ( C j − r j )
Time of last job (makespan): ma x j C j
𝒓 𝟏
time
𝒓 𝟐
Weighted versions:
j w j C j & j w j ( C j − r j )
At any moment execute
at most 1 job per machine
at most 1 machine per job
m machines
n jobs
Preemption & Migration allowed

3. Non-Clairvoyant Scheduling
What if (r j , p j , w j ) not known? (non-clairvoyant)
Don’t know existence of job before r j
Don’t know size p j until executed p j
Compare to offline optimum solution
Single machine do shortest job first, but in general NP hard
Cannot achieve offline optimum
E.g, two jobs on a single machine
(Weighted) Round-robin gives O(1) approx
p 1 =1
p 2 =2
Preemption & Migration allowed

4. Precedence Constraintsj
Some jobs can only start after finishing another: i≺j
Directed acyclic dependency-graph (DAG)
Wtd. Completion time captures Makespan
1 job with weight >0 depending on all others
Applications to parallel/distributed systems
Multi-processing modeled as DAGs
Non-homogenous jobs
Big data systems
Graham’66
Grandl et al.’16
Apache Tez and Apache Spark

5. can we do online with precedence?
What is Known with Precedence?
Offline:
Wtd Completion: Constant factor algorithms Hall-Schulz-Shmoys-Wein’97
Wtd Flow: Speedup necessary & constant factor with constant speedup Kulkarni-Li’18
Online/Non-clairvoyant:
Not much is known
Some works assume each DAG is a “hyper-job”: Greedy works
Same release date
Only one value for the entire “hyper-job”
Agrawal-Li-Lu-Moseley’16
Robert-Schabanel’08
can we do online with precedence?

6. Main results
Thm [GGKS'19]: For Wtd Completion Time and DAG Preced: 𝟏𝟎 approximation
All jobs in a DAG released together
Thm [GGKS'19]: For Wtd Flow Time and DAG with No-Surprise Preced: 𝐎(𝟏) approximation with a small “speedup”
Remark: The assumptions are necessary and we give a scalable algorithm, i.e. O(1/ ϵ 2 ) approx even with (1+ϵ)-speedup.

7. OUTLINE Introduction: Precedence Constrained Scheduling
Our Algorithm: Independence Graph & Nash Social Welfare
Primal-Dual Analysis
Other Results and Open Problems

8. Weighted Round-Robin Algorithm
What if no-precedence?
Run proportional to weights
All job rates ≤1 and total rate ≤𝑚
What if only chains or trees?
Cannot be uniform over all chains
Weight according to subtree weight
For inverted stars, who should get j’s weight?
𝑤 𝑗 =∞
Easy
Chains
Medium
Trees j
Who should j blame?

9. How to Distribute Speeds?
Use a dependency Graph i
How to Distribute Speeds?
Cannot arbitrarily distribute:
Want some kind of “fair” algorithm
For every t imagine a Dependency Graph G t
Indep jobs I t and All jobs J t
Edge (i,j) if i≺j or i=j
Inverted star j
Indep
jobs 𝐈 𝐭
All jobs 𝐉 𝐭 i j
Machines

10. Maximize Nash Social Welfare (NSW)
Want a “Fair” Solution
NSW used previously by Munagala et al. (without precedence)
Convex Program
max j∈ J t w j ⋅log R j t
L i t := j z ij t ∀i∈ I t
R j t := i z ij t ∀j∈ J t
L i t ≤ ∀i∈ I t
i L i t ≤m
0≤z ij t ≤1
Machines i
Real Rate i
Virtual Rate j
𝐈 𝐭
𝐉 𝐭
Generalizes round-robin
Similar to Eisenberg-Gale for Fisher markets
Combinatorial Water-Filling algo

11. OUTLINE Introduction: Precedence Constrained Scheduling
Our Algorithm: Independence Graph & Nash Social Welfare
Primal-Dual Analysis
Other Results and Open Problems

12. Proof approach Assume our machines at twice speed
Thm: Weighted Compl time with DAG Preced has 𝟏𝟎 approx
Assume our machines at twice speed
Two lower bounds on OPT :
Show ALG is comparable to I + II
ALG ≤ j w j ⋅ chai n j +2⋅ r j +2⋅LP
Chain Lower Bound
C j ≥chai n j the length of max chain to j
C j ≥ r j the release date
LP Lower Bound
Write an LP lower bounding OPT
Any feasible solution to the dual j

13. Primal-Dual LP Thm: Weighted Compl time with DAG Preced has 𝟏𝟎 approx
x jt denotes processing j gets at t
min j,t w j ⋅t⋅ x j,t p j t≥ r j x j,t p j ≥1 ∀j j x j,t ≤m ∀t s≤t x i,s p i ≥ s≤t x j,s p j ∀t, i≺j
max j α j −m⋅ t β t
α j , β t ≥ ∀j,t
α j − w j ⋅t+ s≥t γ s,j out − γ s,j in ≤ p j ⋅ β t ∀j,t≥ r j
Jobs finished
Bounded Speed
Precedence

14. Setting Duals and LP Feasibility
Use the CP solution to set LP duals
Only active jobs count at t
Feasibility:
Waiting jobs: getting money from I t
Running jobs: rate proportional to total weight
Dual objective comparable: ALG ≤2⋅LP+ j w j ⋅(chai n j + 2⋅r j )
max j α j −m⋅ t β t
α j , β t ≥ ∀j,t
α j − w j ⋅t+ s≥t γ s,j out − γ s,j in ≤ p j ⋅ β t ∀j,t≥ r j
Jobs that help
Thm: Weighted Compl time with DAG Preced has 𝟏𝟎 approx

15. OUTLINE Introduction: Precedence Constrained Scheduling
Our Algorithm: Independence Graph & Nash Social Welfare
Primal-Dual Analysis
Other Results and Open Problems

16. Weighted Flow Time
All jobs in a DAG released together
Thm [GGKS'19]: For Wtd Flow Time and DAG with No-Surprise Preced: 𝐎(𝟏) approximation with a small “speedup”
Remark: We give a scalable algorithm, i.e., 𝐎(𝟏) approx even with (1+𝜖)-speedup
No surprise assump necessary (even with speedup & single machine)
At t=0, release n indep jobs of size 1.
At t=1, release n 3 new jobs of size 0 depending on a random j.
Assumption allows to define a Total Order:
Different DAGs w.r.t. r j and within DAG w.r.t. C j
Slightly change the LP and CP objectives
Set duals using CP solution

17. Conclusion Questions? Scheduling with Precedence
No “hyper-job” assump: 10 approx for Wtd Compl. Nash-Social Welfare
Primal-Dual analysis: Set LP duals using the CP solution
O 1/ ϵ 2 approx for Wtd Flow Time with (1+ϵ) speedup No-Surprise Assump
Open Problems
Improve approximation factors
For related machines can we get O(k⋅ log n )?
Questions?

18. Further SLIDES

19. Relationship to Fisher Markets
max j∈ J t w j ⋅log R j t
L j t := j ′ z j j ′ t ∀j∈ I t
R j t := j ′ z j j ′ t ∀j∈ J t
L j t ≤ ∀j∈ I t
j L j t ≤m
z j j ′ t ≥0
Fisher Markets
m divisible goods and n buyers
Buyer j has budget B j and utility u ij for item i
Find equilibrium item prices p i s.t. market clears
Comparison to Eisenberg-Gale Convex Program
Difficult: Additional constraint overall speed ≤m
Simpler: Unit-weights, i.e, u ij =1
Combinatorial algorithms
Complicated for Fisher-Markets
Simple Water-Filling for our convex program
max j∈Buyers B j ⋅log U j
L i := j z ij ∀i∈Items
U j := i u ij ⋅ z ij ∀j∈Buyers
L i ≤ ∀i∈Items
z ij ≥0

20. Primal-Dual LP Thm: Wtd Compl time with DAG Preced has 𝟏𝟎 approx.
min j,t w j ⋅t⋅ x j,t p j t≥ r j x j,t p j ≥1 ∀j j x j,t ≤m ∀t s≤t x j,s p j ≥ s≤t x j ′ ,s p j ′ ∀t, j≺j′
max j α j −m⋅ t β t
α j , β t ≥ ∀j,t
α j − w j ⋅t+ s≥t j≺j′ γ s,j→ j ′ − j′≺j γ s, j ′ →j ≤ p j ⋅ β t ∀j,t≥ r j
Jobs finished
Bounded Speed
Precedence
Alternately,
max j α j −m⋅ t β t
α j , β t ≥ ∀j,t
α j − w j ⋅t+ s≥t γ s,j out − γ s,j in ≤ p j ⋅ β t ∀j,t≥ r j
x jt is processing j gets at t

21. Dual Feasibility (Idea)
max j α j −m⋅ t β t
α j , β t ≥ ∀j,t
α j − w j ⋅t+ s≥t γ s,j out − γ s,j in ≤ p j ⋅ β t ∀j,t≥ r j
Waiting Jobs J t ∖ I t
Getting money from 𝐼 𝑡
Running Jobs I t
Running rate proportional to how many jobs depend on it
Careful but simple case analysis
Thm [GGKS'19]: For Wtd Completion Time and DAG Preced: 𝟏𝟎 approximation

22. Related Machines Machines of k speed classes:
n i machines of speed s i
Cannot do better than factor k:
Minimize makespan: Suppose s 1 ≫ s 2 ≫…≫ s k
Let n 1 s 1 ≪ n 2 s 2 ≪…≪ n k s k with n i jobs of size s i
Cannot do better than factor log n loglog n =: 1 ϵ
Objective: Makespan for 1/ϵ-ary tree of depth 1/ϵ
Machines: One fast & many slow: s 1 =1= n 1 & s 2 =ϵ, n 2 =∞
Sizes: P is random root-leaf path. Jobs incident to P size 1, otherwise 0
OPT is 1 ϵ but any ALG is 1 ϵ 2
Can we get O(1) approx? 1