Stochastic Load Balancing on Unrelated Machines Viswanath Nagarajan - - PowerPoint PPT Presentation

▶
stochastic load balancing on unrelated machines
SMART_READER_LITE
LIVE PREVIEW

Stochastic Load Balancing on Unrelated Machines Viswanath Nagarajan - - PowerPoint PPT Presentation

Dec 24, 2023 153 likes •793 views

Stochastic Load Balancing on Unrelated Machines Viswanath Nagarajan Industrial & Operations Engineering Department University of Michigan Joint work with Anupam Gupta (CMU), Amit Kumar (IIT Delhi), Xiangkun Shen (UM) V. Nagarajan (UM)

slide-1
SLIDE 1

Stochastic Load Balancing on Unrelated Machines

Viswanath Nagarajan

Industrial & Operations Engineering Department University of Michigan

Joint work with Anupam Gupta (CMU), Amit Kumar (IIT Delhi), Xiangkun Shen (UM)

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 1 / 25

slide-2
SLIDE 2

Outline

1

Introduction Motivation Related Work Results

2

Techniques Effective Size Algorithm

3

Conclusion

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 2 / 25

slide-3
SLIDE 3

Load Balancing Problem

Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66].

A B C D AB ACD D BC ABCD D AC

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 3 / 25

slide-4
SLIDE 4

Load Balancing Problem

Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66].

A B C D AB ACD D BC ABCD D AC

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 3 / 25

slide-5
SLIDE 5

Load Balancing Problem

Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66].

A B C D AB ACD D BC ABCD D AC

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 3 / 25

slide-6
SLIDE 6

Load Balancing Problem

Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66].

A B C D AB ACD D BC ABCD D AC

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 3 / 25

slide-7
SLIDE 7

Load Balancing Problem

Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66].

A B C D makespan AB ACD D BC ABCD D AC

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 3 / 25

slide-8
SLIDE 8

Load Balancing (Formally)

n jobs and m machines. pij = size of job j on machine i. Identical machines: pij = pj. Related machines: (pij) matrix rank 1. Unrelated machines: general case. Find an assignment to minimize makespan: min

J1,···Jm m

max

i=1

  • j∈Ji

pij. Ji = set of jobs assigned to machine i.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 4 / 25

slide-9
SLIDE 9

Optimization under Uncertainty

In many situations precise input data unknown. Various models to deal with uncertainty. Stochastic: input drawn from some distribution. Robust: input drawn from some uncertainty set. Online: no prior information about input.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 5 / 25

slide-10
SLIDE 10

Optimization under Uncertainty

In many situations precise input data unknown. Various models to deal with uncertainty. Stochastic: input drawn from some distribution. Robust: input drawn from some uncertainty set. Online: no prior information about input. Here: stochastic setting.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 5 / 25

slide-11
SLIDE 11

Stochastic Load Balancing

n jobs and m machines. Random variable Xij is size of job j on machine i. Arbitrary distributions. Independent and known upfront. Find an assignment {Ji}m

i=1 to minimize expected makespan:

E   m max

i=1

  • j∈Ji

Xij   . ? ? ? ? ?

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 6 / 25

slide-12
SLIDE 12

Stochastic Load Balancing

n jobs and m machines. Random variable Xij is size of job j on machine i. Arbitrary distributions. Independent and known upfront. Find an assignment {Ji}m

i=1 to minimize expected makespan:

E   m max

i=1

  • j∈Ji

Xij   . ? ? ? ? ?

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 6 / 25

slide-13
SLIDE 13

Stochastic Load Balancing

n jobs and m machines. Random variable Xij is size of job j on machine i. Arbitrary distributions. Independent and known upfront. Find an assignment {Ji}m

i=1 to minimize expected makespan:

E   m max

i=1

  • j∈Ji

Xij   . ? ? ? ? ? ? ? ? ? ?

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 6 / 25

slide-14
SLIDE 14

Stochastic Load Balancing

n jobs and m machines. Random variable Xij is size of job j on machine i. Arbitrary distributions. Independent and known upfront. Find an assignment {Ji}m

i=1 to minimize expected makespan:

E   m max

i=1

  • j∈Ji

Xij   . ? ? ? ? ?

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 6 / 25

slide-15
SLIDE 15

Stochastic Load Balancing

n jobs and m machines. Random variable Xij is size of job j on machine i. Arbitrary distributions. Independent and known upfront. Find an assignment {Ji}m

i=1 to minimize expected makespan:

E   m max

i=1

  • j∈Ji

Xij   .

minimize expected makespan

? ? ? ? ?

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 6 / 25

slide-16
SLIDE 16

Natural Approach for Stochastic Optimization

1 Replace each random variable X in the stochastic problem by

deterministic surrogate d(X).

2 Solve the resulting deterministic problem. 3 Return the same solution for the stochastic problem.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 7 / 25

slide-17
SLIDE 17

Deterministic Surrogate for Load Balancing?

Expected size?

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 8 / 25

slide-18
SLIDE 18

Deterministic Surrogate for Load Balancing?

Expected size? Type 1: size 1 (deterministic). Type 2: size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

m − √m jobs: expectation 1 m jobs: expectation

1 √m m − √m machines m − √m machines √m machines √m √m

E[mkspan] =

log m log log m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 8 / 25

slide-19
SLIDE 19

Deterministic Surrogate for Load Balancing?

Expected size? Type 1: size 1 (deterministic). Type 2: size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

m − √m jobs: expectation 1 m jobs: expectation

1 √m m − √m machines m − √m machines √m machines √m √m

E[mkspan] =

log m log log m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 8 / 25

slide-20
SLIDE 20

Deterministic Surrogate for Load Balancing?

Expected size? Type 1: size 1 (deterministic). Type 2: size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

m jobs: expectation

1 √m m − √m machines m − √m machines √m machines

E[mkspan] =

log m log log m

m − √m machines m − √m machines √m machines

E[mkspan] ≤ 2

√m √m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 8 / 25

slide-21
SLIDE 21

Deterministic Surrogate for Load Balancing?

Expected size? Type 1: size 1 (deterministic). Type 2: size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

m jobs: expectation

1 √m m − √m machines m − √m machines √m machines

E[mkspan] =

log m log log m

m − √m machines m − √m machines √m machines

E[mkspan] ≤ 2

√m √m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 8 / 25

slide-22
SLIDE 22

Deterministic Surrogate for Load Balancing?

Expected size? Type 1: size 1 (deterministic). Type 2: size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

m jobs: expectation

1 √m m − √m machines m − √m machines √m machines

E[mkspan] =

log m log log m

m − √m machines m − √m machines √m machines

E[mkspan] ≤ 2

√m √m

Effective size [Hui 88] [Elwalid, Mitra 93] [Kelly 96] [Kleinberg, Rabani, Tardos 00].

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 8 / 25

slide-23
SLIDE 23

Outline

1

Introduction Motivation Related Work Results

2

Techniques Effective Size Algorithm

3

Conclusion

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 9 / 25

slide-24
SLIDE 24

Related Work: Deterministic Job Sizes

Identical machines:

◮ List Scheduling: 2-approximation [Graham 66]. ◮ Sorted List Scheduling:

4 3-approximation [Graham 69].

◮ PTAS [Hochbaum, Shmoys 87].

Related machines:

◮ 2-approximation [Morrison 88] [Gonzalez, Ibarra, Sahni 77]. ◮ PTAS [Hochbaum, Shmoys 88].

Unrelated machines:

◮ 2-approximation [Lenstra, Shmoys, Tardos 90]. ◮ APX-hardness [Shmoys, Tardos 93]. ◮ 2 − 1

6 + ǫ-approximation for restricted assignment [Jansen, Rohwedder 17].

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 10 / 25

slide-25
SLIDE 25

Related Work: Stochastic Job Sizes

O(1)-approximation for identical machines [Kleinberg, Rabani, Tardos 00]. #P-hard to evaluate objective Better results for special classes of job size distributions [Goel, Indyk 99].

◮ Poisson distributed job sizes: 2-approximation. ◮ Exponential distributed job sizes: PTAS.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 11 / 25

slide-26
SLIDE 26

Related Work: Stochastic Job Sizes

O(1)-approximation for identical machines [Kleinberg, Rabani, Tardos 00]. #P-hard to evaluate objective Better results for special classes of job size distributions [Goel, Indyk 99].

◮ Poisson distributed job sizes: 2-approximation. ◮ Exponential distributed job sizes: PTAS.

Related knapsack and bin-packing problems [Deshpande, Li 11]. Other models for stochastic combinatorial optimization:

Two-stage [Shmoys, Swamy 04] [Gupta, Pal, Ravi, Sinha 04].. Adaptive [Dean Goemans Vondrak 08] [Guha Munagala 07] [Bhalgat Goel Khanna 11]..

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 11 / 25

slide-27
SLIDE 27

Related Work: Stochastic Job Sizes

O(1)-approximation for identical machines [Kleinberg, Rabani, Tardos 00]. #P-hard to evaluate objective Better results for special classes of job size distributions [Goel, Indyk 99].

◮ Poisson distributed job sizes: 2-approximation. ◮ Exponential distributed job sizes: PTAS.

Related knapsack and bin-packing problems [Deshpande, Li 11]. Other models for stochastic combinatorial optimization:

Two-stage [Shmoys, Swamy 04] [Gupta, Pal, Ravi, Sinha 04].. Adaptive [Dean Goemans Vondrak 08] [Guha Munagala 07] [Bhalgat Goel Khanna 11]..

Stochastic load-balancing has remained open: even related machines.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 11 / 25

slide-28
SLIDE 28

Outline

1

Introduction Motivation Related Work Results

2

Techniques Effective Size Algorithm

3

Conclusion

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 12 / 25

slide-29
SLIDE 29

Main Results

Theorem

There is an O(1)-approximation algorithm for minimizing expected makespan on unrelated machines.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 13 / 25

slide-30
SLIDE 30

Main Results

Theorem

There is an O(1)-approximation algorithm for minimizing expected makespan on unrelated machines.

Extension 1

There is an O(1)-approximation algorithm for budgeted expected makespan minimization on unrelated machines.

Extension 2

There is an O(

q log q)-approximation algorithm for the expected q-norm

minimization problem on unrelated machines.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 13 / 25

slide-31
SLIDE 31

Outline

1

Introduction Motivation Related Work Results

2

Techniques Effective Size Algorithm

3

Conclusion

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 14 / 25

slide-32
SLIDE 32

Effective Size

For random variable X and integer k ≥ 1

βk(X) := 1 log k · log E

  • e(log k)·X

. In identical machines use k = m (number of machines). Recall example: deterministic size-1 and Bernoulli w.p.

1 √m jobs.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 15 / 25

slide-33
SLIDE 33

Effective Size

For random variable X and integer k ≥ 1

βk(X) := 1 log k · log E

  • e(log k)·X

. In identical machines use k = m (number of machines). Recall example: deterministic size-1 and Bernoulli w.p.

1 √m jobs.

m − √m jobs: size (deterministic) 1 m jobs: size Bernoulli r.v. (0, 1) w.p.

1 √m

effective size: 1 effective size: O(1)

m − √m machines m machines

effective size load: O(√m)

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 15 / 25

slide-34
SLIDE 34

Effective Size

For random variable X and integer k ≥ 1

βk(X) := 1 log k · log E

  • e(log k)·X

. In identical machines use k = m (number of machines). Recall example: deterministic size-1 and Bernoulli w.p.

1 √m jobs.

m − √m jobs: size (deterministic) 1 m jobs: size Bernoulli r.v. (0, 1) w.p.

1 √m

effective size: 1 effective size: O(1)

m − √m machines √m machines √m √m

effective size load: O(√m)

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 15 / 25

slide-35
SLIDE 35

Effective Size

For random variable X and integer k ≥ 1

βk(X) := 1 log k · log E

  • e(log k)·X

. In identical machines use k = m (number of machines). Recall example: deterministic size-1 and Bernoulli w.p.

1 √m jobs.

m − √m machines √m machines √m √m

effective size load: O(√m)

m − √m machines √m machines

effective size load: O(1)

m − √m machines √m machines

effective size load: O(1)

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 15 / 25

slide-36
SLIDE 36

Effective Size

For random variable X and integer k ≥ 1

βk(X) := 1 log k · log E

  • e(log k)·X

. In identical machines use k = m (number of machines). Recall example: deterministic size-1 and Bernoulli w.p.

1 √m jobs.

Roughly, on identical machines [Kleinberg, Rabani, Tardos 00] showed effective size load ≤ 1 ⇒ expected makespan = O(1). effective size load ≥ 1 ⇒ expected makespan = Ω(1).

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 15 / 25

slide-37
SLIDE 37

Effective Size

For random variable X and integer k ≥ 1

βk(X) := 1 log k · log E

  • e(log k)·X

. In identical machines use k = m (number of machines). Recall example: deterministic size-1 and Bernoulli w.p.

1 √m jobs.

Roughly, on identical machines [Kleinberg, Rabani, Tardos 00] showed effective size load ≤ 1 ⇒ expected makespan = O(1). effective size load ≥ 1 ⇒ expected makespan = Ω(1). How to set parameter k on unrelated machines?

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 15 / 25

slide-38
SLIDE 38

Outline

1

Introduction Motivation Related Work Results

2

Techniques Effective Size Algorithm

3

Conclusion

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 16 / 25

slide-39
SLIDE 39

Approach in Deterministic Setting

[Lentsra, Shmoys, Tardos 90]

1 Guess and scale so optimum is 1. 2 Solve linear program relaxation (polynomial time). 3 Round to an integral assignment.

LP relaxation

m

  • i=1

yij = 1, ∀j ∈ [n], zi −

n

  • j=1

pij · yij = 0, ∀i ∈ [m], zi ≤ 1, ∀i ∈ [m], yij, zi ≥ 0, ∀i, j.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 17 / 25

slide-40
SLIDE 40

Our Approach in Stochastic Setting

1 Identify valid linear inequalities using effective size. 2 Formulate an LP relaxation (exponential size). 3 Rounding algorithm.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 18 / 25

slide-41
SLIDE 41

Our Approach in Stochastic Setting

1 Identify valid linear inequalities using effective size. 2 Formulate an LP relaxation (exponential size). 3 Rounding algorithm.

Assume that optimum is 1.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 18 / 25

slide-42
SLIDE 42

Valid Inequalities

Truncated Tij := Xij · I(Xij≤1) Exceptional Eij := Xij · I(Xij>1)

guessed expected makespan

Truncated Exceptional

Lemma (New Valid Inequalities)

Consider any feasible solution that assigns jobs Ji to each machine i ∈ [m] with expected makespan E

  • maxm

i=1

  • j∈Ji Xij
  • ≤ 1. Then

m

  • i=1
  • j∈Ji

E[Eij] ≤ 2, and for all K ⊆ [m],

  • i∈K
  • j∈Ji

βk(Tij) ≤ b · k, where k = |K|. Here b = O(1).

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 19 / 25

slide-43
SLIDE 43

LP Relaxation

yij = whether/not job j assigned to machine i. zi(k) = load on machine i with effective size βk.

m

  • i=1

yij = 1, ∀j ∈ [n], zi(k) −

n

  • j=1

βk(Tij) · yij = 0, ∀i ∈ [m], k ∈ [m],

m

  • i=1

n

  • j=1

E[Eij] · yij ≤ 2,

  • i∈K

zi(k) ≤ b · k, ∀K ⊆ [m] with |K| = k, ∀k ∈ [m], yij, zi(k) ≥ 0, ∀i, j, k

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 20 / 25

slide-44
SLIDE 44

Algorithm Outline

1 Define truncated/exceptional random variables. 2 Solve LP via the ellipsoid method

Separation oracle: sort the zi(k) values for each k.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 21 / 25

slide-45
SLIDE 45

Algorithm Outline

1 Define truncated/exceptional random variables. 2 Solve LP via the ellipsoid method

Separation oracle: sort the zi(k) values for each k.

3 If the LP is infeasible, optimal expected makespan is more than 1. 4 If the LP is feasible, run the rounding algorithm.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 21 / 25

slide-46
SLIDE 46

Rounding Overview

Round the fractional solution to satisfy carefully chosen subset of the constraints: hard to satisfy all.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 22 / 25

slide-47
SLIDE 47

Rounding Overview

Round the fractional solution to satisfy carefully chosen subset of the constraints: hard to satisfy all. Construct an instance of Generalized Assignment Problem (GAP) based on the fractional solution. Utilize the algorithm from [Shmoys, Tardos 93].

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 22 / 25

slide-48
SLIDE 48

Generalized Assignment Problem

n jobs and m machines. The size of job j on machine i: pij. The cost of assigning job j to machine i: cij. Makespan bound b. Find an assignment to minimize the total cost subject to the makespan being at most b.

Theorem [Shmoys, Tardos 93]

If the natural LP relaxation for GAP has optimal value C ⋆, then the algorithm finds in polynomial-time an assignment with cost at most C ⋆ and makespan at most b + maxij pij.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 23 / 25

slide-49
SLIDE 49

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β5 ≤ 5b pi,j ← β2(Ti,j)

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-50
SLIDE 50

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β5 ≤ 5b pi,j ← β2(Ti,j)

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-51
SLIDE 51

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β5 ≤ 5b β5 ≤ b

5

pi,j ← β2(Ti,j)

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-52
SLIDE 52

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β5 ≤ 5b pi,j ← β5(Ti,j)

5

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-53
SLIDE 53

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β4 ≤ 4b

5

pi,j ← β2(Ti,j)

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-54
SLIDE 54

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β4 ≤ 4b β4 ≤ b

5 4 4

pi,j ← β2(Ti,j)

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-55
SLIDE 55

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β4 ≤ 4b pi,j ← β4(Ti,j)

5 4 4

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-56
SLIDE 56

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β2 ≤ 2b

5 4 4

pi,j ← β2(Ti,j)

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-57
SLIDE 57

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β2 ≤ 2b

5 4 4

β2 ≤ b

2 2

pi,j ← β2(Ti,j)

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-58
SLIDE 58

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j] total β2 ≤ 2b

5 4 4 2 2

pi,j ← β2(Ti,j)

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-59
SLIDE 59

Rounding Algorithm: Constructing GAP Instance

Costs: cij ← E[Eij]. Processing times: set by iterative method. machines jobs ci,j ← E[Ei,j]

5 4 4 2 2

pi,j ← β2(Ti,j) total β2 ≤ 2b

  • i∈K

βk(Tij) · yij ≤ b · k, ∀K ⊆ [m] : |K| = k, ∀k = 1 · · · m

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 24 / 25

slide-60
SLIDE 60

Conclusion

A constant-factor approximation for stochastic load balancing problem on unrelated machines.

◮ O(1)-approximation for budgeted makespan minimization problem. ◮ O(

q log q)-approximation for q-norm minimization problem.

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 25 / 25

slide-61
SLIDE 61

Conclusion

A constant-factor approximation for stochastic load balancing problem on unrelated machines.

◮ O(1)-approximation for budgeted makespan minimization problem. ◮ O(

q log q)-approximation for q-norm minimization problem.

Exciting recent result by [Molinaro ’19] O(1)-approximation for every q-norm!

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 25 / 25

slide-62
SLIDE 62

Conclusion

A constant-factor approximation for stochastic load balancing problem on unrelated machines.

◮ O(1)-approximation for budgeted makespan minimization problem. ◮ O(

q log q)-approximation for q-norm minimization problem.

Exciting recent result by [Molinaro ’19] O(1)-approximation for every q-norm! Smaller constant factor (close to 2?) for even identical machines? Generalize to other combinatorial optimization problems?

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 25 / 25

slide-63
SLIDE 63

Conclusion

A constant-factor approximation for stochastic load balancing problem on unrelated machines.

◮ O(1)-approximation for budgeted makespan minimization problem. ◮ O(

q log q)-approximation for q-norm minimization problem.

Exciting recent result by [Molinaro ’19] O(1)-approximation for every q-norm! Smaller constant factor (close to 2?) for even identical machines? Generalize to other combinatorial optimization problems?

Thank you!

  • V. Nagarajan (UM)

Stochastic Load Balancing Aussois, 2019 25 / 25