grouping techniques for scheduling problemscgi.csc.liv.ac.uk/~ctag/seminars/tim-harnack.pdf · tim...

Grouping Techniques For Scheduling Problems

Tim Hartnack

Theory of ParallelismInstitute of Computer Science

Christian-Albrechts-University of Kiel

October 11, 2007

October 11, 2007 Tim Hartnack Grouping Techniques For Scheduling Problems 1 of 26

Introduction Overview

Overview

1 IntroductionOverview

2 Unrelated parallel machines with costsBasic ideasRounding and profiling jobsGrouping jobsDynamic programming

3 Outlook and discussion

Unrelated parallel machines with costs

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

n independent jobsm unrelated parallel machines

jobs without interruption

each machine: one job at a moment

job Jj on machine i requires pij ≥ 0

and incurs cij ≥ 0 costs, i = 1, · · · ,m, j = 1, · · · ,n

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

Problem

0 < ε < 1 fixed

m≥ 2 fixedGiven:

Objective function of unrelated parallel machines with costs

Objective function

T + µ

∑j=1

∑i=1

xijcij (1)

with xij =

{1, if job Jj runs on machine i0, else

T makespan, and µ ≥ 0

By multiplying each cost value by µ we may assume, w.l.o.g. that µ = 1

Objective function

T + µ

∑j=1

∑i=1

xijcij (1)

with xij =

Objective function

T + µ

∑j=1

∑i=1

xijcij (1)

with xij =

Objective function

T + µ

∑j=1

∑i=1

xijcij (1)

with xij =

Notation and scaling factors

Definition (scaling factor)

Define for each job Jj ∈J

1 dj = mini=1,··· ,m (pij + cij)2 D = ∑

nj=1 dj

Upper and lower bound of the objective function

LemmaFor the objective function, the following inequality holds: D≤ OPT ≤ m

Proof.

∑j=1

dj ≤m

∑i=1

∑j=1

x∗ijcij +m

∑i=1

∑j=1

x∗ijpij

≤ C∗+T∗ ≤ m(C∗+T∗) = m ·OPT

Let mj indicate a machine such that dj = pmj,j + cmj,j

Assign each job Jj to machine mj

The objective function is bounded by

∑j∈J

cmj,j + ∑j∈J

pmj,j = D

OPT ∈[D

By dividing all times and costs by Dm we get:

1≤ OPT ≤ m

∑j∈J

cmj,j + ∑j∈J

pmj,j = D

OPT ∈[D

1≤ OPT ≤ m

∑j∈J

cmj,j + ∑j∈J

pmj,j = D

OPT ∈[D

1≤ OPT ≤ m

∑j∈J

cmj,j + ∑j∈J

pmj,j = D

OPT ∈[D

1≤ OPT ≤ m

∑j∈J

cmj,j + ∑j∈J

pmj,j = D

OPT ∈[D

1≤ OPT ≤ m

Unrelated parallel machines with costs Basic ideas

Overview of the algorithm

1 Rounding and profiling of jobs creates profilesconstant number of profiles

2 Grouping of jobsconstant number of jobs

3 Schedule constant number of jobs with dynamic programming

Observation (Transformation)

We say that a transformation produces 1+O(ε) loss at the objective function

Unrelated parallel machines with costs Rounding and profiling jobs

Sets of machines

For every Jj define:

fast machines pij ≤ ε

cheap machines cij ≤ ε

slow machines pij ≥ mε

expensive machines cij ≥ djε

Sets of machines

Rounding Jobs

fast machine i of Jj : pij = 0

cheap machine i of Jj : cij = 0

slow machine i of Jj : pij = +∞

expensive machine i ∈ of Jj : cij = +∞

other machine i of Jj round pij, cij to the nearest lower value of ε

m dj (1+ ε)h,for some h ∈ N

ObservationFor each job Jj ∈J there is always a machine wich is neither expensive norslow

Rounding Jobs

Results of rounding

LemmaRounding produces 1+4ε loss

Proof.Start by considering rounding to zero the times and costs of jobs on fastand cheap machines, respectively

Let A be an optimal schedule of thisThe objective function value of A≤ OPT

we just reduced times and costs

F and C denote sets of jobs, which are processed on fast and cheapmachines according to AReplace times and costs of the transformed instance by the originals

∑Jj∈F

mdj + ∑

Jj∈C

mdj ≤ 2

∑j=1

mdj = 2ε

Results of rounding

∑Jj∈F

mdj + ∑

Jj∈C

mdj ≤ 2

∑j=1

mdj = 2ε

Results of rounding

∑Jj∈F

mdj + ∑

Jj∈C

mdj ≤ 2

∑j=1

mdj = 2ε

Results of rounding

∑Jj∈F

mdj + ∑

Jj∈C

mdj ≤ 2

∑j=1

mdj = 2ε

Results of rounding

∑Jj∈F

mdj + ∑

Jj∈C

mdj ≤ 2

∑j=1

mdj = 2ε

Results of rounding

∑Jj∈F

mdj + ∑

Jj∈C

mdj ≤ 2

∑j=1

mdj = 2ε

Results of rounding II

Proof.Show: there exists an approximate schedule where jobs are scheduledneither on slow nor on expensive machines

pij,cij := +∞

Let A be an optimal schedule, T∗ Makespan C∗ total costsS and E sets,

containing jobs, running on slow and expensive machines

Assign Jj ∈ S∪E mj

This may increase the objective funtion value by at most

∑Jj∈S∪E

dj ≤ε

m ∑Jj∈S

pA(j),j + ε ∑Jj∈E

cA(j),j ≤ εT∗+ εC∗

since pA(j),j ≥ mε

dj for Jj ∈ S and cA(j),j ≥djε

for Jj ∈ E

pij,cij := +∞

∑Jj∈S∪E

dj ≤ε

m ∑Jj∈S

for Jj ∈ E

pij,cij := +∞

∑Jj∈S∪E

dj ≤ε

m ∑Jj∈S

for Jj ∈ E

pij,cij := +∞

∑Jj∈S∪E

dj ≤ε

m ∑Jj∈S

for Jj ∈ E

pij,cij := +∞

∑Jj∈S∪E

dj ≤ε

m ∑Jj∈S

for Jj ∈ E

pij,cij := +∞

∑Jj∈S∪E

dj ≤ε

m ∑Jj∈S

for Jj ∈ E

pij,cij := +∞

∑Jj∈S∪E

dj ≤ε

m ∑Jj∈S

for Jj ∈ E

Summary & Outlook

up to nowAll jobs rounded

nextCreate profiles of jobs

Summary & Outlook

Profiles for jobs

Definition (Execution profile)The execution profile of a job Jj is a m-tuple⟨

Π1,j, · · · ,Πm,j⟩,

so that pij = ε

m dj (1+ ε)Πi,j

Definition (Cost profile)The cost profile of a job Jj is a m-tuple⟨

Γ1,j, · · · ,Γm,j⟩,

so that cij = ε

m dj (1+ ε)Γi,j

Profiles for jobs

Definition (Execution profile)The execution profile of a job Jj is a m-tuple⟨

Π1,j, · · · ,Πm,j⟩,

so that pij = ε

m dj (1+ ε)Πi,j

Definition (Cost profile)The cost profile of a job Jj is a m-tuple⟨

Γ1,j, · · · ,Γm,j⟩,

so that cij = ε

m dj (1+ ε)Γi,j

Special cases in the profile

For pij = +∞ put Πi,j := +∞

For pij = 0 put Πi,j :=−∞

For cij = +∞ put Γi,j := +∞

For cij = 0 put Γi,j :=−∞

ObservationTwo jobs have the same profile, if they have the same execution profile as wellas the same cost profile

Number of profiles

LemmaThe number of different profiles is at most

3+2log1+ε

Unrelated parallel machines with costs Grouping jobs

Summary & Outlook

up to nowAll jobs roundedEvery job has a profileNumber of profiles is constant

next: Group jobs =⇒ Number of jobs constant

Summary & Outlook

Grouping Jobs

1 Make a partition of the jobs

L = {Jj : dj >ε

andS = {Jj : dj ≤

2 L set of big jobs3 S set of small jobs4 Partition S in Si, i = 1, · · · , l based on the profile

Ja,Jb ∈ Si with da,db ≤εm2

Create Jcfrom Ja and Jb

Continue this step until there is only one job Jj ∈ Si with dj ≤εm2 left

5 use the above grouping on all Si of S

Grouping Jobs

L = {Jj : dj >ε

andS = {Jj : dj ≤

Grouping Jobs

L = {Jj : dj >ε

andS = {Jj : dj ≤

Grouping Jobs

L = {Jj : dj >ε

andS = {Jj : dj ≤

Grouping Jobs

L = {Jj : dj >ε

andS = {Jj : dj ≤

Grouping Jobs

L = {Jj : dj >ε

andS = {Jj : dj ≤

Grouping Jobs

L = {Jj : dj >ε

andS = {Jj : dj ≤

Grouping Jobs

L = {Jj : dj >ε

andS = {Jj : dj ≤

Results of grouping

LemmaWith a loss of 1+ ε the number of jobs can be reduced tok := min{n,

(log m

)O(m)}

Proof.After the grouping there are at most l jobs, one from each subset Si, withdj ≤

Therefore the number of jobs is bounded to:

m+ l≤ 2m2

m+ l =

Proof of loss will be omitted

Results of grouping

(log m

)O(m)}

m+ l≤ 2m2

m+ l =

Results of grouping

(log m

)O(m)}

m+ l≤ 2m2

m+ l =

Unrelated parallel machines with costs Dynamic programming

Summary & Outlook

up to nowAll jobs roundedEvery job has a profileNumber of profiles constantGrouping =⇒ Number of jobs constant

next: Create a schedule with dynamic programming

Summary & Outlook

Dynamic Programming

1 J1, · · · ,Jk jobs of the transformed instance2 A schedule configuration s = (t1, · · · , tm,c) is a (m+1)-tuple

ti completion time of machine ic total cost

3 Vj a set of these tuples (f.a. j = 1, · · · ,n)for every i = 1, · · · ,m there is a tuple v ∈ Vj whose entries are 0

Exception: the ith component, which is pij,the (m+1)th component, which is cij

4 T (j,s) denote the truth value of: There is a schedule for J1, · · · ,Jj, forwhich s is the corresponding configuration

Calculate all T (j,s):

T (1,v) =

{true, if v ∈ Vj

false, if v /∈ Vj

T (j,s) =∨