On Some Stochastic Load Balancing Problems Anupam Gupta Carnegie - - PowerPoint PPT Presentation

On Some Stochastic Load Balancing Problems

Anupam Gupta

Carnegie Mellon University

Joint work with Amit Kumar, IIT Delhi Viswanath Nagarajan, Michigan Xiangkun Shen, Michigan (appeared at SODA 2018)

1 / 26

Optimization under Uncertainty

Question: How to model/solve problems with uncertainty in input/actions?

◮ data not yet available, or obtaining exact data difficult/expensive ◮ actions have uncertainty in outcomes ◮ (my talk) we only have stochastic predictions 2 / 26

Optimization under Uncertainty

Question: How to model/solve problems with uncertainty in input/actions?

◮ data not yet available, or obtaining exact data difficult/expensive ◮ actions have uncertainty in outcomes ◮ (my talk) we only have stochastic predictions

Goal: get algos making (near)-optimal decisions given predictions.

2 / 26

Optimization under Uncertainty

Question: How to model/solve problems with uncertainty in input/actions?

◮ data not yet available, or obtaining exact data difficult/expensive ◮ actions have uncertainty in outcomes ◮ (my talk) we only have stochastic predictions

Goal: get algos making (near)-optimal decisions given predictions. worst-case analysis (vs. queueing perspective, cf. Mor’s talk)

◮ Relate to performance of best strategy on worst-case instance 2 / 26

Optimization under Uncertainty

Question: How to model/solve problems with uncertainty in input/actions?

◮ data not yet available, or obtaining exact data difficult/expensive ◮ actions have uncertainty in outcomes ◮ (my talk) we only have stochastic predictions

Goal: get algos making (near)-optimal decisions given predictions. worst-case analysis (vs. queueing perspective, cf. Mor’s talk)

◮ Relate to performance of best strategy on worst-case instance

given predictions (vs. all-adversarial model as in competitive analysis)

2 / 26

Approximation Algorithms for Stochastic Optimization

High Level Model:

◮ Inputs random variables X1, X2, . . . with known distributions ◮ (for today) assume discrete distributions, explicit access to them 3 / 26

Approximation Algorithms for Stochastic Optimization

High Level Model:

◮ Inputs random variables X1, X2, . . . with known distributions ◮ (for today) assume discrete distributions, explicit access to them ◮ outcomes not known a priori, revealed over time ◮ want to optimize, say, E[objective]. 3 / 26

Approximation Algorithms for Stochastic Optimization

High Level Model:

◮ Inputs random variables X1, X2, . . . with known distributions ◮ (for today) assume discrete distributions, explicit access to them ◮ outcomes not known a priori, revealed over time ◮ want to optimize, say, E[objective]. ◮ many different models ⋆ e.g., adaptive vs. non-adaptive ⋆ e.g., single-stage vs. multi-stage 3 / 26

Approximation Algorithms for Stochastic Optimization

High Level Model:

◮ Inputs random variables X1, X2, . . . with known distributions ◮ (for today) assume discrete distributions, explicit access to them ◮ outcomes not known a priori, revealed over time ◮ want to optimize, say, E[objective]. ◮ many different models ⋆ e.g., adaptive vs. non-adaptive ⋆ e.g., single-stage vs. multi-stage

Today: centralized load-balancing problem, minimizing E[makespan].

3 / 26

today’s problem

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(Classical) Load Balancing Problem

Schedule n jobs on m machines to minimize makespan. Graham list-scheduling from 1966.

A B C D

5 / 26

(Classical) Load Balancing Problem

Schedule n jobs on m machines to minimize makespan. Graham list-scheduling from 1966.

A B C D

5 / 26

(Classical) Load Balancing Problem

Schedule n jobs on m machines to minimize makespan. Graham list-scheduling from 1966.

A B C D

5 / 26

(Classical) Load Balancing Problem

Schedule n jobs on m machines to minimize makespan. Graham list-scheduling from 1966.

A B C D makespan

5 / 26

much work on the deterministic problem

Simplest Model: Identical machines:

◮ List Scheduling: 2-approx [Graham 66], PTAS [Hochbaum, Shmoys 87]. 6 / 26

much work on the deterministic problem

Simplest Model: Identical machines:

◮ List Scheduling: 2-approx [Graham 66], PTAS [Hochbaum, Shmoys 87].

Most General: Unrelated machines:

◮ Jobs have different sizes on different machines. 6 / 26

much work on the deterministic problem

Simplest Model: Identical machines:

◮ List Scheduling: 2-approx [Graham 66], PTAS [Hochbaum, Shmoys 87].

Most General: Unrelated machines:

◮ Jobs have different sizes on different machines. ◮ 2-approx [Lenstra, Shmoys, Tardos 90], better for special cases. 6 / 26

Stochastic Load Balancing

Job j on machine i takes on size Xij (r.v. with known distribution) Today: these r.v.s are independent Find an assignment to minimize expected makespan: E  

m

max

i=1

• j∈Ji

Xij   .

? ? ? ? ?

7 / 26

Stochastic Load Balancing

Job j on machine i takes on size Xij (r.v. with known distribution) Today: these r.v.s are independent Find an assignment to minimize expected makespan: E  

m

max

i=1

• j∈Ji

Xij   .

? ? ? ? ?

7 / 26

Stochastic Load Balancing

Job j on machine i takes on size Xij (r.v. with known distribution) Today: these r.v.s are independent Find an assignment to minimize expected makespan: E  

m

max

i=1

• j∈Ji

Xij   .

? ? ? ? ? ? ? ? ? ?

7 / 26

Stochastic Load Balancing

Job j on machine i takes on size Xij (r.v. with known distribution) Today: these r.v.s are independent Find an assignment to minimize expected makespan: E  

m

max

i=1

• j∈Ji

Xij   .

? ? ? ? ?

7 / 26

Stochastic Load Balancing

Job j on machine i takes on size Xij (r.v. with known distribution) Today: these r.v.s are independent Find an assignment to minimize expected makespan: E  

m

max

i=1

• j∈Ji

Xij   .

minimize expected makespan

? ? ? ? ?

7 / 26

Related Work: Stochastic Job Sizes

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

◮ Poisson distributed job sizes: 2-approximation. ◮ Exponential distributed job sizes: PTAS. 8 / 26

Related Work: Stochastic Job Sizes

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

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

What about general unrelated case?

8 / 26

Main Result

Theorem

An O(1)-approx algo for minimizing E[makespan] on unrelated machines.

9 / 26

Roadmap and Assumptions

Identical machines case Ideas needed for unrelated machines (and sketch of proof)

10 / 26

Roadmap and Assumptions

Identical machines case Ideas needed for unrelated machines (and sketch of proof) By scaling, assume E[OPTmakespan] = 1.

10 / 26

Roadmap and Assumptions

Identical machines case Ideas needed for unrelated machines (and sketch of proof) By scaling, assume E[OPTmakespan] = 1. Assume each job is “small”: Pr[size > E[OPTmakespan]] = 0.

◮ Easy extension to general sizes 10 / 26

Deterministic Surrogate

Find deterministic quantity as a surrogate for each r.v. Do optimization over these deterministic quantities Surrogate = expected size?

11 / 26

Deterministic Surrogate

Find deterministic quantity as a surrogate for each r.v. Do optimization over these deterministic quantities Surrogate = expected size? (No!)

11 / 26

Deterministic Surrogate

Find deterministic quantity as a surrogate for each r.v. Do optimization over these deterministic quantities Surrogate = expected size? (No!)

Bad Example

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

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

1 √m m machines

E[mkspan] =

log m log log m 11 / 26

Deterministic Surrogate

Find deterministic quantity as a surrogate for each r.v. Do optimization over these deterministic quantities Surrogate = expected size? (No!)

Bad Example

Type 1: size 1 (deterministic). Type 2: size Bernoulli(0, 1) 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 11 / 26

Deterministic Surrogate

Find deterministic quantity as a surrogate for each r.v. Do optimization over these deterministic quantities Surrogate = expected size? (No!)

Bad Example

Type 1: size 1 (deterministic). Type 2: size Bernoulli(0, 1) 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 11 / 26

Deterministic Surrogate

Find deterministic quantity as a surrogate for each r.v. Do optimization over these deterministic quantities Surrogate = expected size? (No!)

Bad Example

Type 1: size 1 (deterministic). Type 2: size Bernoulli(0, 1) 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

11 / 26

Deterministic Surrogate

Find deterministic quantity as a surrogate for each r.v. Do optimization over these deterministic quantities Surrogate = expected size? (No!)

Bad Example

Type 1: size 1 (deterministic). Type 2: size Bernoulli(0, 1) 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

11 / 26

Our Chief Weapon: Effective Size

12 / 26

Effective Size

[Hui 88] [Elwalid, Mitra 93] [Kelly 96]

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

.

13 / 26

Effective Size

[Hui 88] [Elwalid, Mitra 93] [Kelly 96]

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

= 1 log k · log E

• kX

.

13 / 26

Effective Size

[Hui 88] [Elwalid, Mitra 93] [Kelly 96]

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

= 1 log k · log E

• kX

. E.g., X = C w.p. 1, then βk(X) = C.

13 / 26

Effective Size

[Hui 88] [Elwalid, Mitra 93] [Kelly 96]

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

= 1 log k · log E

• kX

. E.g., X = C w.p. 1, then βk(X) = C. E.g., X = Bernoulli(1/

√ k), then βk(X) = log(1/

√ k·k1+(1−1/ √ k)·k0)

log k

≥ 1/2.

13 / 26

Effective Size

[Hui 88] [Elwalid, Mitra 93] [Kelly 96]

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

= 1 log k · log E

• kX

. E.g., X = C w.p. 1, then βk(X) = C. E.g., X = Bernoulli(1/

√ k), then βk(X) = log(1/

√ k·k1+(1−1/ √ k)·k0)

log k

≥ 1/2. Increasing function of k

13 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

.

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

1 √m m − √m machines m machines

effective size load: O(√m)

14 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

.

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)

14 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

.

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)

14 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

.

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)

14 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

.

15 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

. [KRT’00] For indentical machines, set k = m:

15 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

. [KRT’00] For indentical machines, set k = m:

◮ total βm summed over all jobs ≥ m

⇒ expected OPT = Ω(1).

15 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

. [KRT’00] For indentical machines, set k = m:

◮ total βm summed over all jobs ≥ m

⇒ expected OPT = Ω(1).

◮ βm for jobs on each machine ≤ 1

⇒ expected makespan = O(1).

15 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

. [KRT’00] For indentical machines, set k = m:

◮ total βm summed over all jobs ≥ m

⇒ expected OPT = Ω(1).

◮ βm for jobs on each machine ≤ 1

⇒ expected makespan = O(1).

⋆ Pr[one machine load ≥ 1 + c] ≤ (1/m)c. (next slide.) ⋆ union bound over all machines. 15 / 26

Effective Size

For any random variable X and parameter k > 1, define effective size βk(X) := 1 log k · log E

• elog k·X

. [KRT’00] For indentical machines, set k = m:

◮ total βm summed over all jobs ≥ m

⇒ expected OPT = Ω(1).

◮ βm for jobs on each machine ≤ 1

⇒ expected makespan = O(1).

⋆ Pr[one machine load ≥ 1 + c] ≤ (1/m)c. (next slide.) ⋆ union bound over all machines. ◮ Hence O(1)-approximation. 15 / 26

Lemma (Upper Bound)

For indep. r.v.s Y1, . . . , Yn, if

i βk(Yi) ≤ 1 then Pr[ i Yi ≥ 1 + c] ≤ 1 kc .

16 / 26

Lemma (Upper Bound)

For indep. r.v.s Y1, . . . , Yn, if

i βk(Yi) ≤ 1 then Pr[ i Yi ≥ 1 + c] ≤ 1 kc .

Pr[

• i

Yi ≥ 1 + c] = Pr[e(log k)

i Yi ≥ e(log k)(1+c)] ≤ E[e(log k) i Yi]

e(log k)(1+c) =

• i E[e(log k)Yi]

e(log k)(1+c)

16 / 26

Lemma (Upper Bound)

For indep. r.v.s Y1, . . . , Yn, if

i βk(Yi) ≤ 1 then Pr[ i Yi ≥ 1 + c] ≤ 1 kc .

Pr[

• i

Yi ≥ 1 + c] = Pr[e(log k)

i Yi ≥ e(log k)(1+c)] ≤ E[e(log k) i Yi]

e(log k)(1+c) =

• i E[e(log k)Yi]

e(log k)(1+c) Taking logarithms, log Pr[

• i

Yi ≥ 1 + c] ≤ (log k) ·

i

βp(Yi) − (1 + c)

• ≤ (log k) · (−c).

16 / 26

Effective size for Unrelated Machines Setting

17 / 26

Effective Size: Unrelated Machines

Must use different k for different kinds of jobs. (Fixed k fails.)

18 / 26

Effective Size: Unrelated Machines

Must use different k for different kinds of jobs. (Fixed k fails.) All jobs have size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

Type 1: can only be assigned to the first machine. Type 2: can be assigned to any machine.

18 / 26

Effective Size: Unrelated Machines

Must use different k for different kinds of jobs. (Fixed k fails.) All jobs have size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

Type 1: can only be assigned to the first machine. Type 2: can be assigned to any machine.

√m jobs: effective size θ m jobs: effective size θ

m machines effective size load: √mθ

E[mkspan] =

log m log log m

18 / 26

Effective Size: Unrelated Machines

Must use different k for different kinds of jobs. (Fixed k fails.) All jobs have size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

Type 1: can only be assigned to the first machine. Type 2: can be assigned to any machine.

√m jobs: effective size θ m jobs: effective size θ

• ne machine

√m √m √m m − √m − 1 machines effective size load: √mθ √m machines

E[mkspan] =

log m log log m

18 / 26

Effective Size: Unrelated Machines

Must use different k for different kinds of jobs. (Fixed k fails.) All jobs have size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

Type 1: can only be assigned to the first machine. Type 2: can be assigned to any machine.

√m jobs: effective size θ m jobs: effective size θ

• ne machine

√m √m √m m − √m − 1 machines effective size load: √mθ √m machines

E[mkspan] =

log m log log m

18 / 26

Effective Size: Unrelated Machines

Must use different k for different kinds of jobs. (Fixed k fails.) All jobs have size∼ (0, 1) Bernoulli r.v. with p =

1 √m.

Type 1: can only be assigned to the first machine. Type 2: can be assigned to any machine.

• ne machine

√m √m √m m − √m − 1 machines effective size load: √mθ √m machines

E[mkspan] =

log m log log m

• ne machine

m − 1 machines √m

E[mkspan] = 1 + 1

e

18 / 26

Our Solution for Unrelated Machines

19 / 26

Our Approach

1

valid inequalities using effective size.

2

(large) LP relaxation.

3

Rounding algorithm.

20 / 26

Valid Inequalities

Lemma (New Valid Inequalities)

If assignment satisfies E

• maxm

i=1

• j∈Ji Xij
• ≤ 1, then
• i∈K
• j∈Ji

β|K|(Xij) ≤ O(|K|) ∀K ⊆ [m].

21 / 26

LP Relaxation (cont’d)

Solve the LP using separation oracle = sorting.

22 / 26

LP Relaxation (cont’d)

Solve the LP using separation oracle = sorting.

◮ LP infeasible =

⇒ optimal expected makespan > 1.

⋆ Contrapositive to above lemma. ◮ LP feasible =

⇒ round fractional solution to satisfy subset of constraints

⋆ Suffice to bound E[makespan], show next. 22 / 26

LP Relaxation (cont’d)

Solve the LP using separation oracle = sorting.

◮ LP infeasible =

⇒ optimal expected makespan > 1.

⋆ Contrapositive to above lemma. ◮ LP feasible =

⇒ round fractional solution to satisfy subset of constraints

⋆ Suffice to bound E[makespan], show next.

Construct instance of unrelated machines problem Use rounding from [Lenstra, Shmoys, Tardos 92], [Shmoys, Tardos 93].

22 / 26

Valid Inequalities

Lemma (New Valid Inequalities)

If assignment satisfies E

• maxm

i=1

• j∈Ji Xij
• ≤ 1, then
• i∈K
• j∈Ji

β|K|(Xij) ≤ O(|K|) ∀K ⊆ [m].

23 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

machines jobs total β5 value ≤ 5 pi,j ← β2(Ti,j)

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β5 value ≤ 5 pi,j ← β2(Ti,j)

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β5 value ≤ 5 β5 ≤ 1

5

pi,j ← β2(Ti,j)

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β5 value ≤ 5 pi,j ← β5(Ti,j)

5

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β4 value ≤ 4

5

pi,j ← β2(Ti,j)

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β4 value ≤ 4 β4 ≤ 1

5 4 4

pi,j ← β2(Ti,j)

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β4 value ≤ 4 pi,j ← β4(Ti,j)

5 4 4

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β2 value ≤ 2

5 4 4

pi,j ← β2(Ti,j)

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β2 value ≤ 2

5 4 4

β2 ≤ 1

2 2

pi,j ← β2(Ti,j)

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

total β2 value ≤ 2

5 4 4 2 2

pi,j ← β2(Ti,j)

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

24 / 26

“Pre-Rounding” Algorithm

1

remaining machines L ← [m], ℓ = |L|.

2

while ℓ > 0:

1

“class ℓ machines: L′ ← {i ∈ L : zi(ℓ) ≤ 1}.

2

pij ← βℓ(Xij) for machines i ∈ L′, all jobs.

3

L ← L \ L′ and ℓ = |L|.

5 4 4 2 2

pi,j ← β2(Ti,j) total β5 value ≤ 5

• i∈K

β|K|(Xij) · yij ≤ |K|, ∀K ⊆ [m]

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Analysis Sketch

LP gives fractional sol. for unrelated machines (target load B = 1). Round it: 2-approx rounded assignment {Ji}m

i=1 satisfies:

• j∈Ji

βℓ(i)(Xij) ≤ 2, ∀i ∈ [m]

25 / 26

Analysis Sketch

LP gives fractional sol. for unrelated machines (target load B = 1). Round it: 2-approx rounded assignment {Ji}m

i=1 satisfies:

• j∈Ji

βℓ(i)(Xij) ≤ 2, ∀i ∈ [m]

Pr[machine i load ≥ 2 + c] ≤

1 ℓ(i)c .

25 / 26

Analysis Sketch

LP gives fractional sol. for unrelated machines (target load B = 1). Round it: 2-approx rounded assignment {Ji}m

i=1 satisfies:

• j∈Ji

βℓ(i)(Xij) ≤ 2, ∀i ∈ [m]

Pr[machine i load ≥ 2 + c] ≤

1 ℓ(i)c .

Union bound over machines gives Pr[maxload > 2 + c] ≤

• i

1 ℓ(i)c ≤

• i

1 ic ≪ 1. Can also bound E[maxload].

25 / 26

wrapup

O(1)-approx for stochastic load balancing on unrelated machines.

◮ Main tool: effective size 26 / 26

wrapup

O(1)-approx for stochastic load balancing on unrelated machines.

◮ Main tool: effective size

Extensions (in the paper):

◮ O(1)-approx for budgeted makespan minimization. ◮ O(

q log q )-approx for q-norm minimization.

⋆ Now O(1)-approx by M. Molinaro (forthcoming)

Tighter analysis (2?) even for identical machines?

26 / 26

wrapup

O(1)-approx for stochastic load balancing on unrelated machines.

◮ Main tool: effective size

Extensions (in the paper):

◮ O(1)-approx for budgeted makespan minimization. ◮ O(

q log q )-approx for q-norm minimization.

⋆ Now O(1)-approx by M. Molinaro (forthcoming)

Tighter analysis (2?) even for identical machines? Incorporate routing aspects?

26 / 26

wrapup

O(1)-approx for stochastic load balancing on unrelated machines.

◮ Main tool: effective size

Extensions (in the paper):

◮ O(1)-approx for budgeted makespan minimization. ◮ O(

q log q )-approx for q-norm minimization.

⋆ Now O(1)-approx by M. Molinaro (forthcoming)

Tighter analysis (2?) even for identical machines? Incorporate routing aspects? Data center issues?

26 / 26

wrapup

O(1)-approx for stochastic load balancing on unrelated machines.

◮ Main tool: effective size

Extensions (in the paper):

◮ O(1)-approx for budgeted makespan minimization. ◮ O(

q log q )-approx for q-norm minimization.

⋆ Now O(1)-approx by M. Molinaro (forthcoming)

Tighter analysis (2?) even for identical machines? Incorporate routing aspects? Data center issues? Thank you!

26 / 26