Angewandte

Mathematik

und

Informatik

Universit

at

zu

K

oln

Report

No.

Tight

Approximations

for

Resource

Constrained

Scheduling

and

Bin

Packing

by

Anand

Srivastav

Peter

Stangier

Parts

of

this

paper

appeared

as

extended

abstract

in

J.

van

Leeuwen

-LRB-

ed

-RRB-

Proceedings

of

the

Second

Annual

European

Symposium

on

Algorithms

-LRB-

ESA

-RRB-

pages

-LCB-

Lecture

Notes

in

Computer

Science

Vol

Springer

Verlag

Anand

Srivastav

Institut

f?ur

Informatik

Lehrstuhl

Algorithmen

und

Komplexit?at

Humboldt

Universit?at

zu

Berlin

Unter

den

Linden

Berlin

Germany

srivasta@informatik.hu-berlin.de

Peter

Stangier

Institut

f?ur

Informatik

Universit?at

zu

K?oln

Pohligstr

K?oln

Germany

stangier@informatik.uni-koeln.de

Mathematics

Subject

Classification

Tight

Approximations

for

Resource

Constrained

Scheduling

and

Bin

Packing

Anand

Srivastav

Peter

Stangier

September

Abstract

We

consider

the

following

resource

constrained

scheduling

problem

Given

m

identical

processors

s

resources

with

upper

bounds

n

independent

tasks

of

unit

length

where

each

task

has

a

start

time

and

requires

one

processor

and

a

task-dependent

amount

of

every

resource

The

optimization

problem

is

to

schedule

all

tasks

at

discrete

times

in

IIN

minimizing

the

latest

completion

time

Cmax

subject

to

the

processor

resource

and

start-time

constraints

Multidimensional

bin

packing

is

a

special

case

of

this

problem

The

problem

is

NP-hard

even

under

much

simpler

assumptions

In

case

of

zero

start

times

R?ock

and

Schmidt

-LRB-

-RRB-

showed

an

-LRB-

m

-RRB-

factor

approximation

algorithm

and

de

la

Vega

and

Lueker

-LRB-

-RRB-

improving

a

classical

result

of

Garey

Graham

Johnson

and

Yao

-LRB-

-RRB-

gave

for

every

ffl

a

linear

time

algorithm

with

an

asymptotic

approximation

guarantee

of

s

ffl

The

contribution

of

this

paper

is

to

break

the

O

-LRB-

m

-RRB-

resp

O

-LRB-

s

-RRB-

barrier

even

in

the

case

of

zero

start

times

at

least

for

problems

where

the

number

of

processors

and

the

resource

bounds

are

in

-LRB-

log

jIj

-RRB-

jIj

being

the

input

size

of

the

problem

The

main

results

are

constant

factor

approximation

algorithms

for

such

problems

and

the

proof

of

the

optimality

of

the

achieved

approximation

under

the

hypothesis

P

NP

Introduction

Problem

Definition

and

Complexity

Resource

constrained

scheduling

with

start

times

is

the

following

problem

The

input

is

ffl

a

set

T

Tng

of

independent

tasks

Each

task

Tj

needs

one

time

unit

for

its

completion

and

can

not

be

scheduled

before

its

start

time

rj

rj

IIN

ffl

a

set

P

Pmg

of

identical

processors

Each

task

needs

one

processor

ffl

a

set

R

Rsg

of

limited

resources

This

means

that

at

any

time

all

resources

are

available

but

the

available

amount

of

each

resource

Ri

is

bounded

by

bi

IIN

ffl

For

i

s

j

n

let

Ri

-LRB-

j

-RRB-

-LSB-

-RSB-

be

rational

resource

requirements

indicating

that

every

task

Tj

needs

Ri

-LRB-

j

-RRB-

amount

of

resource

Ri

in

order

to

be

processed

The

combinatorial

optimization

problem

is

Definition

-LRB-

Resource

Constrained

Scheduling

-RRB-

Find

a

schedule

-LRB-

or

assignment

-RRB-

oe

T

IIN

of

minimal

time

length

subject

to

the

start

times

processor

and

resource

constraints

According

to

the

standard

notation

of

scheduling

problems

the

unweighted

version

of

our

problem

-LRB-

Ri

-LRB-

j

-RRB-

-RRB-

can

be

formalized

as

P

jres

rj

pj

This

notation

means

that

the

number

of

identical

processors

is

part

of

the

input

-LRB-

P

j

-RRB-

that

resources

are

involved

-LRB-

res

-RRB-

that

the

number

of

resources

and

the

amount

of

every

resource

are

part

of

the

input

too

-LRB-

res

-RRB-

that

every

task

needs

at

most

unit

of

a

resource

-LRB-

res

-RRB-

that

start

times

are

involved

-LRB-

rj

-RRB-

and

that

the

processing

time

of

all

tasks

is

equal

-LRB-

pj

-RRB-

and

that

the

optimization

problem

is

to

schedule

the

tasks

as

soon

as

possible

-LRB-

jCmax

-RRB-

The

problem

is

NP

hard

in

the

strong

sense

even

if

rj

for

all

j

n

s

and

m

-LSB-

-RSB-

An

interesting

special

case

of

resource

constrained

scheduling

is

the

following

generalized

version

of

the

multidimensional

bin

packing

problem

Definition

-LRB-

Bin

Packing

Problem

BIN

-LRB-

l

d

-RRB-

-RRB-

Let

d

l

n

be

non

negative

integers

Given

vectors

vn

-LSB-

-RSB-

d

pack

all

vectors

in

a

minimum

number

of

bins

such

that

in

each

bin

B

and

for

each

coordinate

i

i

d

P

vij

l.

Observe

that

BIN

-LRB-

d

-RRB-

is

the

multidimensional

bin

packing

problem

and

BIN

-LRB-

-RRB-

is

the

classical

bin

packing

problem

The

intention

behind

the

formulation

with

a

parameter

l

is

to

analyse

the

relationship

between

bin

sizes

and

polynomial-time

approximability

of

the

problem

Previous

Work

The

known

approximation

algorithms

for

the

problem

class

P

jres

rj

pj

are

due

to

Garey

Graham

Johnson

Yao

-LSB-

-RSB-

and

R?ock

and

Schmidt

-LSB-

-RSB-

Garey

et

al.

constructed

with

the

First-Fit-Decreasing

heuristic

a

schedule

of

length

CFFD

which

asymptotically

is

a

-LRB-

s

-RRB-

factor

approximation

i.e.

there

is

an

non

negative

integer

N

such

that

for

all

Copt

N

CFFD

Copt

-LRB-

s

-RRB-

de

la

Vega

and

Lueker

-LSB-

-RSB-

improved

this

result

presenting

for

every

ffl

a

linear-time

algorithm

with

asymptotic

approximation

factor

d

ffl

R?ock

and

Schmidt

showed

employing

the

polynomial-time

solvability

of

the

simpler

problem

rj

pj

with

processors

a

e

factor

approximation

algorithm

Thus

for

problems

with

small

optimal

schedules

or

many

resource

constraints

resp

processors

these

algorithms

have

a

weak

performance

Note

that

all

these

results

are

based

on

the

assumption

that

no

start-times

are

given

i.e.

rj

for

all

tasks

Tj

T

For

example

R?ock

and

Schmidt

's

algorithm

can

not

be

used

when

start-times

are

given

because

the

problem

rj

pj

is

also

NP

complete

so

their

basis

solution

can

not

be

constructed

in

polynomial

time

In

-LSB-

-RSB-

we

showed

a

approximation

algorithm

for

resource

constrained

scheduling

problems

when

the

resource

bounds

and

the

number

of

processors

are

in

-LRB-

log

-LRB-

Cs

-RRB-

-RRB-

where

C

is

the

length

of

an

optimal

fractional

schedule

and

proved

that

even

for

this

class

a

polynomial-time

aeapproximation

algorithm

for

ae

can

not

exist

unless

P

NP

Since

this

non-approximability

result

was

derived

with

a

reduction

to

the

NP-complete

problem

of

deciding

if

a

schedule

of

length

does

exist

or

not

it

was

conjectured

that

for

other

instances

exept

this

pathological

one

better

approximations

might

be

possible

We

will

show

that

this

conjecture

is

true

in

a

comprehensive

sense

The

Results

Let

Copt

be

the

minimum

schedule

length

and

let

the

integer

C

denote

the

minimum

length

of

a

schedule

if

we

consider

the

LP

relaxation

where

fractional

assignments

of

the

tasks

to

scheduling

times

are

allowed

We

briefly

call

solutions

to

the

LP

relaxation

fractional

schedules

and

solutions

to

the

original

integer

problem

integral

schedules

As

the

one

main

result

we

will

present

a

polynomial-time

approximation

algorithm

for

the

problem

class

P

jres

rj

pj

including

its

rational

weighted

version

-LRB-

Ri

-LRB-

j

-RRB-

-RRB-

-LRB-

a

-RRB-

For

every

ffl

-LRB-

ffl

-RRB-

IIN

we

can

find

in

strongly

polynomial

time

an

integral

schedule

of

size

at

most

d

-LRB-

ffl

-RRB-

Copte

provided

that

all

resource

bounds

bi

are

at

least

-LRB-

ffl

-RRB-

log

-LRB-

-RRB-

and

the

number

of

processors

is

at

least

-LRB-

ffl

-RRB-

log

-LRB-

-RRB-

As

a

surprising

consequence

a

schedule

of

length

Copt

can

be

constructed

in

polynomialtime

whenever

bi

-LRB-

C

-RRB-

log

-LRB-

Cs

-RRB-

-RRB-

for

all

resource

bounds

bi

and

m

-LRB-

C

-RRB-

log

C

This

approximation

guarantee

is

independent

of

the

number

of

processors

or

resources

and

can

be

used

also

for

small

schedules

for

example

Cmax

or

The

results

can

be

extended

to

the

case

of

integer

resource

requirements

-LRB-

Ri

-LRB-

j

-RRB-

IIN

-RRB-

by

scaling

and

exploiting

the

fact

that

we

can

handle

rational

weights

thus

we

obtain

polynomial-time

approximation

algorithm

for

problems

of

the

form

P

jres

rj

pj

One

might

wonder

if

under

the

assumptions

bi

-LRB-

ffl

log

-LRB-

Cs

-RRB-

-RRB-

for

all

i

and

m

-LRB-

ffl

log

C

-RRB-

the

scheduling

problem

is

interesting

enough

In

other

words

can

it

be

that

such

problems

are

easily

solvable

in

polynomial

time

The

answer

is

that

under

the

hypothesis

P

NP

our

approximation

is

best

possible

-LRB-

b

-RRB-

For

the

simpler

problem

with

zero

start-times

and

C

it

is

NP

complete

to

decide

the

question

Is

Copt

C

even

if

ffl

a

fractional

schedule

of

size

C

is

known

ffl

an

integral

schedule

of

size

C

is

known

ffl

bi

-LRB-

log

-LRB-

Cs

-RRB-

-RRB-

for

all

resource

bounds

bi

ffl

m

-LRB-

log

C

-RRB-

In

conclusion

both

results

together

precisely

determine

the

border

of

approximability

of

resource

constrained

scheduling

and

sheed

light

on

its

complexity

An

interesting

special

case

of

resource

constrained

scheduling

is

the

multidimensional

bin

packing

problem

and

the

above

results

imply

-LRB-

c

-RRB-

Let

ffl

-LRB-

ffl

-RRB-

IIN

If

l

-LRB-

ffl

-RRB-

dlog

-LRB-

-RRB-

-RRB-

e

then

we

can

find

in

O

-LRB-

log

-LRB-

nd

-RRB-

-RRB-

time

a

bin

packing

with

L

bins

such

that

L

d

-LRB-

ffl

-RRB-

Lopte

But

even

if

l

-LRB-

log

-LRB-

nd

-RRB-

-RRB-

then

it

is

NP

complete

to

decide

whether

Lopt

LR

or

Lopt

LR

Furthermore

extending

the

method

of

logc

n-wise

independence

from

the

case

of

to

multivalued

random

variables

we

can

parallelize

our

algorithm

in

special

cases

Let

o

log

n

and

suppose

that

m

bi

-LRB-

s

-RRB-

for

all

i

Then

there

is

a

NC

algorithms

that

runs

on

O

-LRB-

-LRB-

ns

-RRB-

-RRB-

parallel

processors

and

finds

in

O

-LRB-

logn

-LRB-

ns

-RRB-

-RRB-

time

a

schedule

of

size

at

most

Observe

that

there

is

a

gap

of

a

n

o

factor

between

the

lower

bounds

of

the

resource

bounds

and

number

of

processors

in

the

sequential

and

in

the

parallel

algorithm

This

factor

comes

into

picture

due

to

the

estimation

of

higher

moments

required

by

the

method

of

logc

n-wise

independence

should

not

cause

any

confusion

C

is

always

an

integer

only

the

assignments

of

tasks

to

times

in

the

discrete

interval

Cg

are

fractional

numbers

that

C

is

at

most

the

sum

of

n

and

the

maximal

start

time

hence

the

factor

log

-LRB-

Cs

-RRB-

is

within

the

size

of

the

problem

input

C

see

-LSB-

-RSB-

We

leave

open

the

question

if

there

is

a

parallel

approximation

algorithm

for

problems

with

m

bi

-LRB-

log

-LRB-

Cs

-RRB-

-RRB-

The

paper

is

organized

as

follows

In

section

we

study

a

general

system

of

integer

inequalities

related

to

resource

constrained

scheduling

show

under

which

circumstances

an

integer

solution

can

be

constructed

via

derandomization

and

apply

the

results

to

resource

constrained

scheduling

In

section

it

is

proved

that

the

achieved

approximation

is

optimal

unless

P

NP

In

section

as

an

example

the

multidimensional

bin

packing

problem

is

discussed

and

in

section

we

give

for

some

special

cases

parallel

counterparts

of

our

scheduling

and

bin

packing

algorithms

based

on

an

extension

of

the

method

of

logc

n-wise

independence

to

multivalued

random

variables

-LRB-

appendix

section

-RRB-

Integer

Inequalities

Resource

constrained

scheduling

can

be

modelled

as

a

special

case

of

a

system

of

integral

inequalities

with

equality

constraints

Let

l

n

N

be

non

negative

integers

For

k

N

let

A

-LRB-

k

-RRB-

be

rational

l

n

matrices

with

a

-LRB-

k

-RRB-

ij

and

b

-LRB-

k

-RRB-

Ql

For

a

vector

yj

let

yjk

be

its

k-th

component

k

N

Let

y

-LRB-

k

-RRB-

be

the

vector

of

all

the

k-th

components

i.e.

y

-LRB-

k

-RRB-

-LRB-

ynk

-RRB-

Consider

the

following

system

of

linear

inequalities

Inequality

System

-LRB-

IS

-RRB-

A

-LRB-

k

-RRB-

y

-LRB-

k

-RRB-

b

-LRB-

k

-RRB-

N

n

yj

y

-LRB-

k

-RRB-

-LRB-

ynk

-RRB-

Resource

constrained

scheduling

fits

in

this

scheme

as

follows

consider

the

processors

as

a

resource

Rs

with

requirement

Rs

-LRB-

j

-RRB-

for

all

j

and

bound

m.

Let

R

-LRB-

Ri

-LRB-

j

-RRB-

-RRB-

ij

be

the

resource

constraint

matrix

set

A

-LRB-

k

-RRB-

R

and

b

-LRB-

k

-RRB-

-LRB-

b

m

-RRB-

for

all

k

N

Then

the

problem

of

finding

a

minimum

N

such

that

a

so

defined

inequality

system

-LRB-

IS

-RRB-

has

a

solution

is

equivalent

to

the

problem

of

finding

a

schedule

of

minimal

length

In

general

one

can

not

expect

to

solve

-LRB-

IS

-RRB-

in

polynomial

time

even

if

a

fractional

solution

exists

because

then

we

would

be

able

to

decide

the

solvability

of

resource

constrained

scheduling

The

key

observation

is

that

-LRB-

IS

-RRB-

can

be

solved

if

the

same

system

with

a

tighter

right

hand

side

say

-LRB-

ffl

-RRB-

-LRB-

k

-RRB-

instead

of

b

-LRB-

k

-RRB-

is

fractionally

solvable

Define

the

ffl-version

of

-LRB-

IS

-RRB-

for

ffl

as

ffl-Inequality

System

-LRB-

IS

-LRB-

ffl

-RRB-

-RRB-

A

-LRB-

k

-RRB-

y

-LRB-

k

-RRB-

-LRB-

ffl

-RRB-

-LRB-

k

-RRB-

N

n

yj

y

-LRB-

k

-RRB-

-LRB-

ynk

-RRB-

We

need

the

following

parameters

Let

be

an

integer

with

l

and

define

bffl

-LRB-

ffl

-RRB-

log

-LRB-

-RRB-

and

bffl

-LRB-

ffl

-RRB-

log

-LRB-

-LRB-

l

-RRB-

N

-RRB-

-LRB-

-RRB-

Theorem

-LRB-

Rounding

Theorem

-RRB-

Let

ffl

and

bffl

i

as

in

-LRB-

-RRB-

Suppose

that

b

-LRB-

k

-RRB-

i

bffl

for

all

i

and

b

-LRB-

k

-RRB-

i

bffl

for

all

i

Suppose

that

u

-LRB-

un

-RRB-

with

uj

-LSB-

-RSB-

N

is

a

fractional

solution

for

the

ffl-scaled

system

IS

-LRB-

ffl

-RRB-

Then

a

vector

x

-LRB-

xn

-RRB-

with

xj

satisfying

the

inequalities

of

system

IS

i.e.

A

-LRB-

k

-RRB-

x

-LRB-

k

-RRB-

b

-LRB-

k

-RRB-

and

can

be

constructed

in

O

-LRB-

N

log

-LRB-

N

ln

-RRB-

-RRB-

time

For

the

proof

we

need

the

following

derandomization

result

which

is

an

algorithmic

version

of

the

Angluin-Valiant

inequality

for

multi-valued

random

variables

Let

l

n

N

be

non-negative

integers

We

are

given

n

mutually

independent

random

variables

Xj

with

values

in

Ng

and

probabil

ity

distribution

P

rob

-LRB-

Xj

k

-RRB-

xjk

for

all

j

n

k

N

and

PNk

xjk

Suppose

that

the

xjk

are

rational

numbers

with

xjk

Let

Xjk

denote

the

random

variable

which

is

if

Xj

k

and

is

else

For

k

N

i

l

j

n

let

w

-LRB-

k

-RRB-

ij

be

rational

weights

with

w

-LRB-

k

-RRB-

ij

For

i

l

and

k

N

define

the

sums

ik

by

ik

n

X

j

w

-LRB-

k

-RRB-

ij

Xjk

-LRB-

-RRB-

Let

ik

be

rational

numbers

Denote

by

Eik

the

event

ik

IE

-LRB-

ik

-RRB-

-LRB-

ik

-RRB-

-LRB-

-RRB-

Let

-LRB-

Eik

-RRB-

be

a

collection

of

lN

such

events

Define

f

-LRB-

ik

-RRB-

exp

-LRB-

-LRB-

ik

-RRB-

-RRB-

By

the

Angluin-Valiant

inequality

-LRB-

-LSB-

-RSB-

Theorem

-RRB-

we

have

for

all

k

Proposition

IP

-LSB-

Ecik

-RSB-

f

-LRB-

ik

-RRB-

Thus

the

probability

that

there

is

some

-LRB-

i

k

-RRB-

so

that

Ecik

holds

is

at

most

Pmi

PNk

f

-LRB-

ik

-RRB-

Let

us

assume

that

this

is

bounded

away

from

one

i.e.

m

X

i

N

X

k

f

-LRB-

ik

-RRB-

ffi

-LRB-

-RRB-

for

some

ffi

The

following

theorem

is

a

special

case

of

the

algorithmic

version

of

the

Angluin-Valiant

inequality

for

multivalued

random

variables

-LRB-

Theorem

in

-LSB-

-RSB-

-RRB-

Theorem

-LRB-

-LSB-

-RSB-

Let

ffi

and

Eik

be

as

above

satisfying

-LRB-

-RRB-

Then

IP

-LRB-

m

i

N

k

Eik

-RRB-

ffi

and

a

vector

x

Tmi

TNk

Eik

can

be

constructed

in

O

N

log

Nlnffi

-RSB-

time

We

are

ready

to

prove

Theorem

Proof

of

Theorem

Set

-LSB-

N

-RSB-

Ng

Let

Xn

be

mutually

independent

random

variables

with

values

in

-LSB-

N

-RSB-

defined

by

IP

-LSB-

Xj

k

-RSB-

ujk

k

-LSB-

N

-RSB-

For

j

k

let

Xjk

be

the

random

variable

which

is

if

Xj

k

and

else

Furthermore

for

each

k

-LSB-

N

-RSB-

let

X

-LRB-

k

-RRB-

be

the

vector

-LRB-

Xnk

-RRB-

For

i

l

define

sums

ik

by

ik

-LRB-

A

-LRB-

k

-RRB-

X

-LRB-

k

-RRB-

-RRB-

i

n

X

j

a

-LRB-

k

-RRB-

ij

Xjk

Let

Eik

be

the

event

ik

-LRB-

ffl

-RRB-

-LRB-

ffl

-RRB-

-LRB-

k

-RRB-

i

By

the

Angluin-Valiant

inequality

and

the

assumption

b

-LRB-

k

-RRB-

i

bffl

for

i

and

b

-LRB-

k

-RRB-

i

bffl

for

i

l

IP

-LSB-

Ecik

-RSB-

exp

-LRB-

k

-RRB-

i

-LRB-

ffl

-RRB-

where

or

l

resp

Hence

IP

-LSB-

-LSB-

ik

Ecik

-RSB-

With

ffi

we

can

invoke

the

algorithmic

Angluin-Valiant

inequality

for

multi-valued

random

variables

-LRB-

Theorem

-RRB-

and

can

construct

vectors

xn

with

xj

and

such

that

A

-LRB-

k

-RRB-

x

-LRB-

k

-RRB-

b

-LRB-

k

-RRB-

in

O

-LRB-

N

log

-LRB-

N

ln

-RRB-

-RRB-

time

In

order

to

show

the

claimed

approximation

guarantee

for

resource

constrained

scheduling

we

proceed

as

follows

In

the

first

step

we

solve

the

linear

programming

relaxation

of

the

integer

program

associated

to

resource

constrained

scheduling

Now

an

integer

schedule

can

be

generated

in

principle

with

our

rounding

theorem

-LRB-

Theorem

-RRB-

Since

the

rounding

theorem

can

be

applied

only

if

a

fractional

solution

within

the

smaller

resource

bound

vector

-LRB-

ffl

-RRB-

is

available

in

the

second

step

we

must

show

the

existence

of

such

a

solution

Fractional

Solutions

Let

rmax

maxj

n

rj

and

D

rmax

n.

Thus

C

Copt

D

The

following

integer

linear

program

is

equivalent

to

resource

constrained

scheduling

with

start

times

minS

P

j

Ri

-LRB-

j

-RRB-

xjz

bi

Ri

R

z

Dg

P

z

xjz

Tj

T

xjz

Tj

T

z

rj

and

Tj

T

z

S

xjz

The

fractional

optimal

schedule

C

can

be

found

in

a

standard

way

-LRB-

see

for

example

-LSB-

-RSB-

-RRB-

Start

with

an

overall

integer

deadline

eC

D

and

check

whether

the

LP

P

j

Ri

-LRB-

j

-RRB-

xjz

bi

Ri

R

z

Dg

P

z

xjz

Tj

T

xjz

Tj

T

z

rj

and

Tj

T

z

eC

xjz

-LSB-

-RSB-

has

a

solution

Using

binary

search

we

can

find

C

along

with

fractional

assignments

-LRB-

exjz

-RRB-

solving

at

most

log

D

such

LPs

W.l.o.g

we

can

assume

that

D

O

-LRB-

n

-RRB-

Hence

this

procedure

is

a

polynomialtime

algorithm

if

we

use

standard

polynomial-time

LP

algorithms

Our

goal

is

to

find

an

integer

solution

using

the

rounding

theorem

-LRB-

Theorem

-RRB-

But

at

this

moment

we

can

not

apply

it

because

the

rounding

theorem

requires

a

fractional

solution

within

the

tighter

resource

bound

ffl

while

C

and

the

assignments

-LRB-

xij

-RRB-

are

feasible

only

within

the

bound

b

It

should

be

intuitively

clear

that

given

a

fractional

solution

within

the

resource

bound

b

and

a

fractional

schedule

of

length

C

a

new

fractional

solution

within

ffl

might

be

constructable

enlarging

the

length

of

the

schedule

to

some

Cffl

C

But

how

can

we

choose

an

appropriate

Cffl

Let

ffl

with

-LRB-

ffl

-RRB-

IIN

Consider

the

time

interval

d

-LRB-

ffl

-RRB-

Ceg

Define

a

new

fractional

solution

as

follows

Call

fC

d

-LRB-

ffl

-RRB-

Ceg

the

fflcompressed

image

of

Cg

and

put

ffi

ffl

Set

I

d

-LRB-

ffl

-RRB-

Ceg

Cg

I

fC

C

bfflCcg

and

I

-LRB-

if

fflC

IIN

fd

-LRB-

ffl

-RRB-

Ceg

else

Then

I

-LSB-

I

-LSB-

I

For

integers

l

let

g

f

be

functions

defined

by

f

-LRB-

l

-RRB-

l

C

ffl

and

g

-LRB-

l

-RRB-

l

C

ffl

ffl

The

new

fractional

assignments

bxjl

are

bxjl

ffi

exj

l

for

l

f

-LRB-

l

-RRB-

P

t

g

-LRB-

l

-RRB-

-LRB-

ffi

-RRB-

exjt

for

l

I

CP

t

g

-LRB-

l

-RRB-

-LRB-

ffi

-RRB-

exjt

for

l

I

-LRB-

-RRB-

It

is

instructive

to

state

the

randomized

rounding

algorithm

behind

the

rounding

theorem

-LRB-

Theorem

-RRB-

for

resource

constrained

scheduling

explicitly

Here

is

the

algorithm

Algorithm

RANDOM-SCHEDULE

Schedule

the

tasks

at

times

selected

by

the

following

randomized

procedure

-LRB-

a

-RRB-

Cast

n

mutually

independent

dice

each

having

N

d

-LRB-

ffl

-RRB-

Ce

faces

where

the

z-th

face

of

the

j-th

die

corresponding

to

task

j

appears

with

probability

exjz

-LRB-

The

faces

stand

for

the

scheduling

times

-RRB-

-LRB-

b

-RRB-

Schedule

task

Tj

at

the

time

selected

in

-LRB-

a

-RRB-

The

following

examples

illustrate

this

algorithm

For

ffl

we

double

the

interval

and

get

C

C

Then

each

task

Tj

is

randomly

scheduled

at

time

z

with

probability

exjz

and

is

scheduled

at

time

z

C

with

the

same

probability

While

this

is

quite

clear

the

idea

of

ffl-compressed

image

becomes

more

transparent

looking

at

ffl

For

ffl

we

have

the

enlarged

interval

C

C

Ceg

Given

a

task

Tj

we

randomly

assign

it

to

a

time

z

Cg

with

probability

exjz

assign

it

to

z

C

with

probability

to

z

C

with

probability

and

so

on

up

to

the

assignment

to

z

Cc

Now

there

are

two

cases

If

is

even

we

assign

Tj

to

Cc

with

probability

exj

C

exj

C

and

since

Ce

Cc

we

are

done

If

is

odd

then

we

assign

Tj

to

Cc

with

probability

exj

C

exj

C

and

to

Cc

with

probability

exjC

The

following

figures

illustrate

this

schedule

enlargment

Given

a

fractional

schedule

as

indicated

in

figure

we

generate

the

schedule

as

shown

in

figure

Figure

A

fractional

schedule

Figure

The

new

fractional

schedule

The

following

lemma

shows

that

the

new

fractional

schedule

is

indeed

feasible

Lemma

The

vectors

bxj

-LRB-

bxjN

-RRB-

j

n

form

a

fractional

solution

for

resource

constrained

scheduling

with

-LRB-

ffl

-RRB-

processors

and

resource

bound

vector

-LRB-

ffl

-RRB-

Proof

Let

ffi

ffl

and

consider

an

arbitrary

resource

Ri

-LRB-

We

can

regard

the

processor

constraint

as

an

additional

resource

thus

the

following

arguments

also

apply

to

the

processor

constraint

-RRB-

The

resource

constraints

are

satisfied

because

for

l

we

have

n

X

j

Ri

-LRB-

j

-RRB-

bxjl

ffi

bi

for

l

I

-LRB-

using

f

-LRB-

l

-RRB-

-LRB-

g

-LRB-

l

-RRB-

-RRB-

-RRB-

n

X

j

Ri

-LRB-

j

-RRB-

bxjl

n

X

j

f

-LRB-

l

-RRB-

X

t

g

-LRB-

l

-RRB-

-LRB-

ffi

-RRB-

Ri

-LRB-

j

-RRB-

exjt

ffl

-LRB-

ffi

-RRB-

bi

ffi

bi

and

for

l

I

-LRB-

using

C

-LRB-

g

-LRB-

l

-RRB-

-RRB-

-RRB-

n

X

j

Ri

-LRB-

j

-RRB-

bxjl

n

X

j

C

X

t

g

-LRB-

l

-RRB-

-LRB-

ffi

-RRB-

Ri

-LRB-

j

-RRB-

exjt

ffl

-LRB-

ffi

-RRB-

bi

ffi

bi

Furthermore

for

all

tasks

Tj

X

bxjl

X

ffi

exjl

X

f

-LRB-

l

-RRB-

X

t

g

-LRB-

l

-RRB-

-LRB-

ffi

-RRB-

exjt

X

C

X

t

g

-LRB-

l

-RRB-

-LRB-

ffi

-RRB-

exjt

C

X

l

ffi

exjl

C

X

l

-LRB-

ffi

-RRB-

exjl

Using

the

rounding

theorem

-LRB-

Theorem

-RRB-

the

derandomized

version

of

the

algorithm

RAN

DOM

SCHEDULE

simply

is

Algorithm

SCHEDULE

Determine

by

the

rounding

theorem

for

each

task

its

scheduling

time

Theorem

For

every

ffl

with

-LRB-

ffl

-RRB-

IIN

the

algorithm

SCHEDULE

finds

a

feasible

integral

schedule

of

size

at

most

d

-LRB-

ffl

-RRB-

Ce

in

polynomial

time

provided

that

m

-LRB-

ffl

-RRB-

dlog

-LRB-

-RRB-

-RRB-

e

and

bi

-LRB-

ffl

-RRB-

dlog

-LRB-

-RRB-

-RRB-

e

for

all

i

s.

Proof

As

above

let

N

d

-LRB-

ffl

-RRB-

Ce

Let

A

-LRB-

Ri

-LRB-

j

-RRB-

-RRB-

ij

be

the

extended

-LRB-

s

-RRB-

n

resource

constraint

matrix

where

the

resource

Rs

represents

the

processor

requirements

and

thus

is

defined

by

Rs

-LRB-

j

-RRB-

for

all

j.

For

k

N

let

b

-LRB-

k

-RRB-

-LRB-

b

m

-RRB-

be

the

resource

bound

vector

The

theorem

is

equivalent

to

the

problem

of

finding

a

solution

to

the

following

inequality

system

Ay

-LRB-

k

-RRB-

b

-LRB-

k

-RRB-

N

n

yj

y

-LRB-

k

-RRB-

-LRB-

ynk

-RRB-

-LRB-

-RRB-

-LRB-

The

vector

yj

and

the

condition

stand

for

the

assignment

of

task

Tj

to

exactly

one

time

in

Ng

-RRB-

By

Lemma

the

assignment

vectors

bxn

with

bxj

-LSB-

-RSB-

N

and

PNk

bxjk

are

a

fractional

solution

to

the

ffl-scaled

system

Ay

-LRB-

k

-RRB-

-LRB-

ffl

-RRB-

-LRB-

k

-RRB-

N

n

yj

y

-LRB-

k

-RRB-

-LRB-

ynk

-RRB-

We

invoke

the

rounding

theorem

-LRB-

Theorem

-RRB-

with

s

the

assumptions

of

the

theorem

in

the

new

notation

read

as

b

-LRB-

k

-RRB-

i

-LRB-

ffl

-RRB-

dlog

-LRB-

-RRB-

-RRB-

e

and

b

-LRB-

k

-RRB-

-LRB-

ffl

-RRB-

dlog

-LRB-

-RRB-

-RRB-

e

thus

the

conditions

required

by

the

rounding

theorem

are

satisfied

With

the

rounding

theorem

we

can

find

in

deterministic

polynomial

time

a

solution

to

the

inequality

system

-LRB-

-RRB-

With

ffl

we

get

our

result

of

-LSB-

-RSB-

Corollary

If

m

-LRB-

-RRB-

-RRB-

e

and

bi

-LRB-

-LRB-

s

-RRB-

-RRB-

e

for

all

i

then

a

schedule

of

size

at

most

can

be

found

in

deterministic

polynomial

time

And

with

ffl

we

infer

the

as

it

will

be

shown

in

the

next

section

optimal

approximation

Corollary

If

m

-LRB-

C

-RRB-

dlog

-LRB-

-RRB-

-RRB-

e

and

bi

-LRB-

C

-RRB-

dlog

-LRB-

-LRB-

s

-RRB-

-RRB-

e

for

all

i

then

a

schedule

of

size

at

most

C

can

be

found

in

deterministic

polynomial

time

Note

that

in

case

of

-LCB-

weights

-LRB-

i.e.

Ri

-LRB-

j

-RRB-

-RRB-

we

have

a

strongly

polynomial

approximtion

algorithm

because

then

the

strongly

polynomial

LP

algorithm

of

Tardos

-LSB-

-RSB-

can

be

used

to

find

the

optimal

fractional

schedule

In

any

case

the

running

time

of

the

LP

algorithm

dominates

clearly

the

running

time

of

derandomization

Finally

suppose

that

the

resource

requirements

Ri

-LRB-

j

-RRB-

are

integers

with

bounds

bi

whereas

in

the

problems

above

we

assumed

Ri

-LRB-

j

-RRB-

Scaling

the

resource

bounds

reduces

the

problem

to

the

-LCB-

case

with

rational

resource

requirements

and

we

may

apply

the

results

above

Scaling

goes

as

follows

compute

for

every

resource

Ri

the

number

Rmax

-LRB-

i

-RRB-

Ri

-LRB-

j

-RRB-

set

Ri

-LRB-

j

-RRB-

Ri

-LRB-

j

-RRB-

Rmax

and

bi

Rmax

Identifying

the

processors

with

resource

Rs

we

have

Corollary

Consider

the

resource

constrained

scheduling

problem

with

integer

resource

requirments

i.e.

Ri

-LRB-

j

-RRB-

IIN

for

all

i

n

and

i

s

For

every

ffl

-LRB-

ffl

-RRB-

IIN

an

integral

schedule

of

size

at

most

d

-LRB-

ffl

-RRB-

Copte

can

be

found

in

polynomial

time

provided

that

-LRB-

ffl

-RRB-

dlog

-LRB-

-LRB-

s

-RRB-

-RRB-

e

for

i

s

Proof

The

proof

is

based

on

the

observation

that

if

a

schedule

for

the

scaled

problem

using

and

Ri

-LRB-

j

-RRB-

is

feasible

then

the

rescaled

schedule

is

feasible

for

the

original

problem

Now

construct

an

integer

schedule

for

the

scaled

problem

with

Theorem

Furthermore

the

choice

of

ffl

yields

a

approximation

for

problems

of

the

general

form

P

jres

pj

rjjCmax

and

setting

ffl

the

optimum

of

an

arbitrary

resource

constrained

scheduling

problem

can

be

approximated

up

to

one

time-unit

provided

the

bi

's

are

as

large

as

required

above

Note

that

for

any

constant

Rmax

this

gives

a

tight

approximation

since

then

bi

still

grows

logarithmically

in

the

input

size

Remark

For

scheduling

of

unrelated

parallel

machines

results

of

similar

flavour

have

been

achieved

by

Lenstra

Shmoys

and

Tardos

-LSB-

-RSB-

and

Lin

and

Vitter

-LSB-

-RSB-

Lenstra

Shmoys

and

Tardos

-LSB-

-RSB-

gave

a

approximation

algorithm

for

the

problem

of

scheduling

independent

jobs

with

different

processing

times

on

unrelated

processors

and

also

proved

that

there

is

no

aeapproximation

algorithm

for

ae

unless

P

NP

Lin

and

Vitter

-LSB-

-RSB-

considered

the

generalized

assignment

problem

and

the

problem

of

scheduling

of

unrelated

parallel

machines

For

the

generalized

assignment

problem

with

resource

constraint

vector

b

they

could

show

for

every

ffl

an

ffl

approximation

of

the

minimum

assignment

cost

which

if

feasible

within

the

enlarged

packing

constraint

-LRB-

-RRB-

b.

Non-Approximability

In

this

section

we

do

not

distinguish

between

the

processor

and

the

other

resource

constraints

but

consider

the

processor

requirement

as

an

additional

resource

constraint

Under

the

assumption

bi

-LRB-

log

-LRB-

Cs

-RRB-

-RRB-

we

have

constructed

an

integral

schedule

of

size

at

most

C

This

is

very

close

to

the

truth

because

now

Copt

is

either

C

or

C

We

will

show

for

all

fix

C

that

even

under

the

assumption

bi

-LRB-

log

-LRB-

Cs

-RRB-

-RRB-

it

is

NP

complete

to

decide

whether

Copt

C

or

Copt

C.

-LRB-

For

C

we

refer

to

-LSB-

-RSB-

-RRB-

In

the

remainder

of

this

section

we

consider

the

simpler

problem

with

zero

start

times

Then

C

n

and

the

NP

completeness

of

scheduling

problems

with

bi

-LRB-

log

-LRB-

ns

-RRB-

-RRB-

implies

the

NP

completeness

of

problems

with

bi

-LRB-

log

-LRB-

Cs

-RRB-

-RRB-

The

following

results

show

the

NP

completeness

of

resource

constrained

scheduling

under

different

conditions

In

Theorem

we

consider

bi

and

-LCB-

resource

requirements

Ri

-LRB-

j

-RRB-

Theorem

covers

the

case

of

bi

-LRB-

log

-LRB-

ns

-RRB-

-RRB-

-LRB-

and

still

Ri

-LRB-

j

-RRB-

-RRB-

In

Theorem

we

include

the

case

of

some

Ri

-LRB-

j

-RRB-

being

fractional

i.e

Ri

-LRB-

j

-RRB-

g

and

Theorem

covers

the

case

bi

-LRB-

log

-LRB-

Cs

-RRB-

-RRB-

In

particular

this

shows

that

resource

constrained

scheduling

is

NP

complete

even

if

we

ask

for

small

schedules

i.e.

Is

Copt

Theorem

Under

the

assumptions

that

there

exist

a

fractional

schedule

of

size

C

and

an

integral

schedule

of

size

C

and

bi

for

all

resource

bounds

it

is

NP

complete

to

decide

whether

or

not

there

exists

an

integral

schedule

of

size

C.

Proof

We

give

a

reduction

to

the

chromatic

index

problem

which

is

NP

complete

-LSB-

-RSB-

The

following

is

known

about

the

chromatic

index

-LRB-

G

-RRB-

of

a

graph

G.

Let

-LRB-

G

-RRB-

be

the

maximal

vertex

degree

in

G

Then

by

Vizing

's

theorem

-LSB-

-RSB-

-LRB-

G

-RRB-

-LRB-

G

-RRB-

-LRB-

G

-RRB-

and

an

edge

coloring

with

-LRB-

G

-RRB-

colors

can

be

constructed

in

polynomial

time

But

it

is

NP

complete

to

decide

whether

there

exists

a

colouring

that

uses

-LRB-

G

-RRB-

colours

even

for

cubic

graphs

i.e.

-LRB-

G

-RRB-

-LSB-

-RSB-

Therefore

the

edge

colouring

problem

is

NP

complete

for

any

fixed

This

is

an

important

fact

which

will

be

used

in

the

proofs

of

Theorem

and

Theorem

Now

to

the

reduction

Let

G

-LRB-

V

E

-RRB-

be

a

graph

with

jV

j

jEj

and

deg

-LRB-

v

-RRB-

for

all

v

V

We

construct

a

resource

constrained

scheduling

problem

asssociated

to

G

as

follows

Introduce

for

every

edge

e

E

exactly

one

task

Te

and

consider

jEj

identical

processors

We

will

freely

call

the

edges

tasks

and

vice

versa

For

every

node

v

V

define

a

resource

Rv

with

bound

and

resource/task

requirements

Rv

-LRB-

e

-RRB-

ae

if

v

e

if

v

e

Claim

There

exists

a

colouring

that

uses

colours

if

and

only

if

there

is

a

feasible

integral

schedule

of

size

Suppose

there

is

an

integral

schedule

of

size

colour

each

edge

with

the

colour

equal

to

its

scheduling

time

Now

all

edges

incident

to

the

same

node

have

different

colours

If

this

is

not

true

then

there

would

be

a

node

v

V

covered

by

two

edges

e

E

of

the

same

colour

which

means

that

the

tasks

corresponding

to

the

edges

are

scheduled

at

the

same

time

z

So

xez

and

Rv

-LRB-

-RRB-

Rv

-LRB-

e

-RRB-

xez

contradicting

our

assumption

of

a

feasible

schedule

Suppose

there

is

an

edge

colouring

E

g

Then

schedule

task

Te

at

the

time

-LRB-

e

-RRB-

It

is

easily

verified

that

this

is

an

feasible

integral

schedule

Furthermore

there

is

a

fractional

schedule

of

size

C

Simply

set

xez

z

This

schedule

has

size

and

we

have

X

z

xez

Te

and

X

e

Rv

-LRB-

e

-RRB-

xez

bv

z

In

Corollary

we

assumed

that

bi

-LRB-

-LRB-

s

-RRB-

e

which

we

did

not

respect

in

the

reduction

above

In

the

next

two

theorems

we

show

how

this

assumption

can

be

included

Theorem

Under

the

assumption

that

there

exist

a

fractional

schedule

of

size

C

and

an

integral

schedule

of

size

C

bi

-LRB-

log

-LRB-

ns

-RRB-

-RRB-

for

all

resource

bounds

and

C

is

fixed

it

is

NP

complete

to

decide

whether

or

not

there

exists

an

integral

schedule

of

size

C.

Proof

We

follow

the

proof

of

Theorem

But

instead

of

respresenting

each

edge

of

the

graph

G

-LRB-

V

E

-RRB-

by

one

task

we

consider

tasks

where

K

dlog

e

and

jEj

For

each

e

E

let

us

introduce

red

tasks

T

-LRB-

e

-RRB-

T

-LRB-

e

-RRB-

and

-LRB-

-RRB-

blue

tasks

T

-LRB-

e

-RRB-

T

-LRB-

-RRB-

-LRB-

e

-RRB-

We

get

rid

of

the

processor

constraint

considering

identical

processors

hence

there

are

enough

processors

to

schedule

every

task

at

any

time

For

every

node

v

V

we

define

a

resource

Rv

with

bound

and

introduce

for

every

edge

e

E

a

resource

Re

also

having

bound

The

requirements

are

every

red

task

T

ri

-LRB-

e

-RRB-

needs

one

unit

of

resource

Rv

if

e

is

incident

with

v.

All

the

other

tasks

including

the

blue

tasks

T

bi

-LRB-

e

-RRB-

do

not

need

any

unit

of

the

resource

Rv

Every

red

or

blue

task

needs

one

unit

of

its

corresponding

edge

resource

Re

Hence

a

feasible

schedule

of

size

has

to

schedule

all

tasks

corresponding

to

an

edge

in

packings

of

size

at

most

The

crucial

observation

is

If

we

can

ensure

that

all

red

tasks

corresponding

to

the

same

edge

are

scheduled

at

the

same

time

then

we

can

define

a

scheduling

time

for

this

edge

and

can

conclude

arguing

as

in

the

proof

of

Theorem

where

we

showed

that

it

is

as

hard

to

find

a

schedule

of

size

as

to

determine

the

existence

of

an

edge

colouring

with

colors

Figure

shows

such

a

feasible

schedule

To

ensure

that

all

red

tasks

corresponding

to

the

same

edge

are

scheduled

at

the

same

time

we

introduce

a

new

resource

type

For

every

edge

e

every

red

task

T

ri

-LRB-

e

-RRB-

and

every

K-element

subset

S

of

blue

tasks

corresponding

to

e

define

a

resource

RTri

-LRB-

e

-RRB-

S

with

bound

K

whose

requirements

are

defined

as

follows

Each

task

in

S

-LSB-

fT

ri

-LRB-

e

-RRB-

g

needs

one

unit

of

RTri

-LRB-

e

-RRB-

S

and

all

the

other

tasks

do

not

need

any

unit

of

the

resource

RTri

-LRB-

e

-RRB-

S

Observe

that

the

number

of

such

resources

is

-LRB-

-RRB-

K

red

tasks

tasks

z

z

z

blue

tasks

blue

Figure

A

part

of

a

feasible

schedule

with

tasks

corresponding

to

one

edge

We

are

ready

to

show

Claim

The

edges

of

G

can

be

colored

with

colors

if

and

only

if

the

above

defined

scheduling

problem

has

a

schedule

of

length

Here

is

a

proof

Suppose

that

the

edges

of

G

can

be

colored

with

colors

Let

Z

be

the

time

set

Z

g.

For

each

e

E

schedule

all

red

tasks

T

ri

-LRB-

e

-RRB-

at

the

time

corresponding

to

the

color

of

e

say

ze

Z

and

schedule

the

blue

tasks

T

bi

-LRB-

e

-RRB-

in

packets

of

at

the

remaining

times

z

ze

z

Z

It

is

easily

verified

that

this

is

a

feasible

schedule

Suppose

we

are

given

a

schedule

of

length

Due

to

the

bound

K

for

the

resources

RTri

-LRB-

e

-RRB-

S

it

is

not

hard

to

show

that

in

such

a

schedule

all

red

tasks

T

ri

-LRB-

e

-RRB-

corresponding

to

the

same

edge

e

must

be

scheduled

at

the

same

time

This

scheduling

time

defines

a

unique

color

for

every

edge

and

the

proof

of

Theorem

shows

that

this

is

a

desired

coloring

We

are

done

if

we

can

show

bi

-LRB-

log

-LRB-

ns

-RRB-

-RRB-

Let

jV

j

and

let

us

assume

that

In

total

we

introduced

s

log

-LRB-

-RRB-

log

log

log

log

resources

and

we

have

n

log

tasks

It

is

straightforward

to

show

the

existence

of

a

constant

with

ns

hence

log

-LRB-

ns

-RRB-

log

K

which

together

with

the

assumption

that

C

and

consequently

is

fix

imply

bi

-LRB-

K

-RRB-

-LRB-

log

-LRB-

ns

-RRB-

-RRB-

It

remains

to

show

that

there

is

a

fractional

schedule

of

size

and

an

integral

schedule

of

size

Since

the

chromatic

index

of

G

is

we

can

define

an

integral

schedule

of

this

size

as

in

the

proof

of

Claim

Furthermore

it

is

easily

checked

that

the

setting

xTz

for

all

tasks

T

and

times

z

defines

a

fractional

schedule

of

size

Finally

if

we

allow

that

some

resource

requirements

are

fractional

numbers

we

have

Theorem

Under

the

assumptions

that

there

exist

a

fractional

schedule

of

size

C

and

an

integral

schedule

of

size

C

bi

-LRB-

-LRB-

s

-RRB-

-RRB-

e

for

all

resource

bounds

C

is

fixed

and

Ri

-LRB-

j

-RRB-

it

is

NP

complete

to

decide

whether

or

not

there

exists

an

integral

schedule

of

size

C.

Proof

Once

again

we

reduce

the

problem

the

the

chromatic

index

problem

and

argue

as

in

the

proof

of

theorem

Let

G

-LRB-

V

E

-RRB-

be

a

graph

Put

K

log

-LRB-

-RRB-

and

jEj

For

every

edge

e

E

we

consider

red

and

-LRB-

-RRB-

blue

tasks

T

-LRB-

e

-RRB-

T

-LRB-

e

-RRB-

and

T

-LRB-

e

-RRB-

T

-LRB-

-RRB-

K

-LRB-

e

-RRB-

Consider

K

identical

processors

For

every

node

v

V

let

Rv

be

a

resource

with

bound

Its

resource

requirement

is

Rv

-LRB-

j

-RRB-

if

Tj

T

rj

-LRB-

e

-RRB-

and

e

contains

v

else

We

follow

the

argumentation

of

Theorem

and

define

one

more

resource

type

to

enforce

that

all

red

task

corresponding

to

the

same

edge

are

scheduled

at

the

same

time

Furthermore

we

have

to

introduce

some

resource

requirements

of

value

First

we

forbid

that

more

than

tasks

corresponding

to

the

same

edge

e

can

be

scheduled

at

the

same

time

Let

Re

be

a

resource

with

bound

and

resource

requirement

Re

-LRB-

j

-RRB-

ae

if

Tj

T

ri

-LRB-

e

-RRB-

or

Tj

T

bi

-LRB-

e

-RRB-

else

Let

T

denote

the

set

of

all

tasks

Tr

the

blue

tasks

Tb

the

blue

task

Tr

-LRB-

e

-RRB-

the

red

tasks

corresponding

to

an

edge

e

and

Tb

-LRB-

e

-RRB-

the

red

tasks

corresponding

to

an

edge

e

The

resource

with

fractional

requirements

is

For

every

red

task

T

ri

-LRB-

e

-RRB-

choose

exactly

one

other

red

task

g

-LRB-

T

ri

-LRB-

e

-RRB-

-RRB-

corresponding

to

e

as

follows

g

-LRB-

T

ri

-LRB-

e

-RRB-

-RRB-

T

ri

-LRB-

e

-RRB-

for

i

and

g

-LRB-

T

-LRB-

e

-RRB-

-RRB-

T

-LRB-

e

-RRB-

Let

us

call

g

-LRB-

T

ri

-LRB-

e

-RRB-

-RRB-

the

buddy

of

T

ri

-LRB-

e

-RRB-

For

every

red

task

T

ri

-LRB-

e

-RRB-

let

Ri

e

be

a

resource

of

bound

K

The

requirements

are

Ri

e

-LRB-

j

-RRB-

if

T

ri

-LRB-

e

-RRB-

T

rj

-LRB-

e

-RRB-

if

Tj

is

a

red

task

corresponding

to

the

edge

e

but

is

not

the

buddy

of

T

ri

-LRB-

e

-RRB-

else

Claim

G

has

an

edge

coloring

with

colors

if

and

only

if

the

above

defined

scheduling

problem

has

a

schedule

of

length

Suppose

that

the

edges

of

G

can

be

colored

with

colors

Let

Z

be

the

time

set

Z

g.

For

each

e

E

schedule

all

red

tasks

T

ri

-LRB-

e

-RRB-

at

the

time

corresponding

to

the

color

of

e

say

ze

Z

and

schedule

the

blue

tasks

T

bi

-LRB-

e

-RRB-

in

packets

of

at

the

remaining

times

z

ze

z

Z.

-LRB-

See

figure

-RRB-

blue

tasks

z

red

tasks

z

Figure

A

way

to

schedule

the

red

and

blue

tasks

The

resource

requirments

are

satisfied

set

xjz

if

the

task

Tj

Tj

T

is

scheduled

at

time

z

and

else

For

the

resources

of

type

Re

we

have

X

Re

-LRB-

j

-RRB-

xjz

Now

fix

an

arbitrary

e

E.

For

every

resource

Ri

e

corresponding

to

a

red

task

T

-LRB-

e

-RRB-

we

have

for

the

scheduling

time

ze

X

Ri

e

-LRB-

j

-RRB-

xjze

X

-LRB-

e

-RRB-

fg

-LRB-

-LRB-

e

-RRB-

-RRB-

g

Ri

e

-LRB-

j

-RRB-

xjze

-LRB-

-RRB-

K

and

for

all

other

z

ze

z

Z

X

Ri

e

-LRB-

j

-RRB-

xjz

X

-LRB-

e

-RRB-

Ri

e

-LRB-

j

-RRB-

xjz

-LRB-

-RRB-

K

This

proves

the

first

part

of

the

claim

Suppose

that

we

have

a

feasible

schedule

of

length

We

show

that

it

is

impossible

to

schedule

the

red

tasks

corresponding

to

the

same

edge

at

different

times

Then

we

can

again

argue

as

in

the

proof

of

Theorem

taking

the

scheduling

time

of

the

red

tasks

corresponding

to

an

edge

as

its

color

and

are

done

Assume

for

a

moment

that

there

is

an

edge

E

and

a

time

in

a

schedule

of

size

at

which

only

I

K

red

tasks

corresponding

to

Then

there

must

be

exactly

I

blue

tasks

being

scheduled

at

time

in

order

not

to

violate

the

constraint

of

resource

Re

at

any

time

Since

there

are

strictly

less

than

K

red

tasks

in

Tr

-LRB-

e

-RRB-

scheduled

at

time

there

is

a

task

T

-LRB-

e

-RRB-

whose

buddy

is

not

scheduled

at

time

In

fact

there

are

at

least

two

such

tasks

Then

we

have

for

resource

at

time

X

-LRB-

j

-RRB-

X

-LRB-

e

-RRB-

-LSB-

Tb

-LRB-

e

-RRB-

-LRB-

j

-RRB-

-LRB-

-LRB-

I

-RRB-

-LRB-

I

-RRB-

-RRB-

K

K

and

the

schedule

requires

more

than

K

units

of

resource

in

contradiction

to

the

feasibility

assumption

Hence

the

red

tasks

corresponding

to

the

edge

must

be

scheduled

at

the

same

time

We

must

show

bi

-LRB-

-LRB-

s

-RRB-

-RRB-

e

for

all

resource

bounds

We

introduced

n

tasks

and

s

K

resources

Using

K

log

-LRB-

-RRB-

the

estimate

s

holds

and

with

C

we

continue

-LRB-

-LRB-

s

-RRB-

-RRB-

e

log

-LRB-

-RRB-

log

-LRB-

-RRB-

K

bi

Finally

one

may

check

as

in

the

proof

of

Theorem

that

there

is

a

fractional

schedule

of

size

and

an

integral

schedule

of

size

With

K

c

for

a

constant

c

satisfying

c

-LRB-

-RRB-

dlog

-LRB-

-LRB-

s

-RRB-

-RRB-

e

the

proof

of

Theorem

implies

Theorem

Under

the

assumptions

that

there

is

a

fractional

schedule

of

size

C

and

an

integral

schedule

of

size

C

bi

-LRB-

C

-RRB-

dlog

-LRB-

-LRB-

s

-RRB-

-RRB-

e

for

all

resource

bounds

C

is

fixed

and

Ri

-LRB-

j

-RRB-

it

is

NP

complete

to

decide

whether

or

not

there

exists

an

integral

schedule

of

size

C.

Multidimensional

Bin

Packing

Consider

the

general

multidimensional

bin

packing

problem

BIN

-LRB-

l

d

-RRB-

as

defined

in

the

introduction

Since

BIN

-LRB-

l

d

-RRB-

is

nothing

else

than

the

resource

constrained

scheduling

problem

with

d

resources

resource

bounds

bi

l

for

all

i

and

zero

start

times

all

the

approximation

and

nonapproximability

results

proved

for

scheduling

are

valid

for

BIN

-LRB-

l

d

-RRB-

too

To

get

a

feeling

for

good

approximations

we

briefly

recall

the

known

asymptotic

approximation

results

Let

Lopt

be

the

optimal

bin

number

The

First-Fit

heuristic

gives

a

approximation

for

BIN

-LRB-

-RRB-

and

using

this

result

one

can

construct

a

solution

for

BIN

-LRB-

d

-RRB-

within

in

polynomial

time

-LSB-

-RSB-

A

similar

argumentation

shows

a

-LRB-

-RRB-

d

factor

approximation

for

BIN

-LRB-

l

d

-RRB-

Thus

good

approximations

must

beat

these

factors

The

results

of

Garey

Graham

Johnson

and

Yao

-LSB-

-RSB-

for

resource

constrained

scheduling

-LRB-

which

we

have

already

discussed

in

the

introduction

-RRB-

applied

to

bin

packing

show

the

existence

of

an

integer

such

that

LFFD

-LRB-

d

-RRB-

Lopt

for

all

instances

with

Lopt

By

de

la

Vega

and

Luecker

-LSB-

-RSB-

for

all

ffl

there

is

a

linear

time

algorithm

Affl

and

an

integer

Nffl

such

that

LAffl

-LRB-

d

ffl

-RRB-

Lopt

for

all

instances

with

Lopt

Nffl

Our

results

imply

for

BIN

-LRB-

l

d

-RRB-

an

approximation

within

a

factor

independent

of

d

and

l

Let

LR

be

the

minimum

number

of

bins

if

fractional

packing

is

allowed

i.e.

LR

d

max

i

<=d

l

n

X

j

vije

Theorem

implies

Theorem

Let

ffl

with

-LRB-

ffl

-RRB-

IIN

If

l

-LRB-

ffl

-RRB-

dlog

-LRB-

-RRB-

-RRB-

e

then

we

can

find

in

O

-LRB-

log

-LRB-

nd

-RRB-

-RRB-

time

a

bin

packing

with

L

bins

such

that

L

d

-LRB-

ffl

-RRB-

Lopte

Proof

Since

we

get

the

optimal

fractional

solution

for

free

we

can

apply

Theorem

without

using

LP-algorithms

So

only

the

time

for

derandomization

counts

For

ffl

Theorem

implies

Corollary

If

l

-LRB-

-RRB-

e

then

we

can

find

in

O

-LRB-

log

-LRB-

nd

-RRB-

-RRB-

time

a

bin

packing

with

L

bins

such

that

L

and

for

ffl

L

R

we

get

Corollary

If

l

-LRB-

LR

-RRB-

dlog

-LRB-

-RRB-

-RRB-

e

then

we

can

find

in

O

-LRB-

log

nd

-RRB-

time

a

bin

packing

with

L

bins

such

that

L

Lopt

Since

we

allow

really

large

bin

sizes

one

might

wonder

if

the

class

of

problems

is

interesting

enough

The

following

negative

result

which

is

follows

from

the

proof

of

Theorem

gives

the

answer

Theorem

Even

if

l

-LRB-

log

-LRB-

nd

-RRB-

-RRB-

then

it

is

NP

complete

to

decide

whether

Lopt

LR

or

Lopt

LR

Parallel

Scheduling

and

Bin

Packing

There

is

no

obvious

way

to

achieve

the

approximation

guarantee

of

Theorem

in

NC

In

this

section

we

will

show

that

at

least

in

some

special

cases

there

is

a

NC

approximation

algorithm

The

algorithm

is

based

on

the

method

of

logc

n-wise

independence

The

important

steps

are

Fractional

Scheduling

in

Parallel

We

wish

to

apply

randomized

rounding

using

logc

-LRB-

n

-RRB-

wise

independence

and

therefore

first

have

to

generate

an

appropriate

probability

distribution

in

NC

Sequentially

this

is

easy

solve

the

linear

programming

relaxation

of

the

integer

programming

formulation

of

our

scheduling

problem

and

the

fractional

assignments

of

tasks

to

times

will

define

the

right

distribution

Unfortunately

linear

programming

is

P

complete

But

fortunately

due

to

the

fact

that

the

start

times

are

zero

we

have

-LRB-

as

for

the

bin

packing

problem

-RRB-

a

formula

for

the

optimal

fractional

schedule

Schedule

Enlargement

Having

found

the

fractional

optimal

schedule

of

length

say

C

we

enlarge

the

makespan

to

and

define

fractional

assignments

as

in

Lemma

Rounding

in

NC

is

performed

with

the

parallel

conditional

probability

method

for

multivalued

random

variables

-LRB-

Theorem

-RRB-

First

we

generate

a

fractional

optimal

schedule

in

NC

Let

xjz

be

a

-LCB-

variable

which

is

if

Tj

is

scheduled

at

time

t

and

is

else

-LRB-

z

ng

-RRB-

Lemma

Define

max

i

s

bi

n

X

j

Ri

-LRB-

j

-RRB-

that

it

suffice

to

deal

with

the

time

intervall

-LSB-

n

-RSB-

since

any

one-to-one

assignment

of

the

n

tasks

to

the

n

times

is

feasible

and

C

Then

C

is

the

length

of

an

optimal

fractional

schedule

and

the

corresponding

task-time

assignments

exjt

are

exjt

ae

Tj

T

t

C

else

Proof

Straightforward

Proposition

There

is

a

NC-algorithm

computing

an

optimal

fractional

schedule

on

O

-LRB-

n

s

-RRB-

EREW

PRAM-processors

in

O

-LRB-

log

s

log

log

s

log

n

-RRB-

time

Proof

Using

O

-LRB-

n

-RRB-

EREW

PRAM-processors

for

each

resource

Ri

we

can

compute

Pnj

Ri

-LRB-

j

-RRB-

in

O

-LRB-

logn

-RRB-

time

With

standard

sorting

algorithms

-LRB-

-LSB-

-RSB-

Remark

and

Corollary

-RRB-

we

can

find

max

i

s

bi

n

X

j

Ri

-LRB-

j

-RRB-

in

O

-LRB-

log

s

log

log

s

-RRB-

time

on

O

-LRB-

s

-RRB-

EREW

PRAM-processors

The

next

step

is

the

enlargement

of

the

fractional

makespan

to

as

in

Lemma

The

fractional

assignments

are

defined

as

follows

For

every

time

Cg

introduce

a

second

time

C

and

set

exjz

for

j

n

z

Clearly

this

new

fractional

schedule

has

size

and

by

Lemma

we

have

for

every

z

and

every

resource

Ri

R

n

X

j

Ri

-LRB-

j

-RRB-

exjz

bi

-LRB-

-RRB-

The

main

result

of

this

section

is

Theorem

Let

o

log

n

and

suppose

that

bi

-LRB-

s

-RRB-

for

all

i

Then

there

is

a

NC-algorithms

that

runs

on

O

-LRB-

-LRB-

ns

-RRB-

-RRB-

parallel

processors

and

finds

in

O

-LRB-

logn

-LRB-

ns

-RRB-

-RRB-

time

a

schedule

of

size

at

most

Proof

For

the

moment

let

k

be

an

arbitrary

integer

We

will

fix

k

in

the

proof

In

order

to

apply

Theorem

-LRB-

the

NC

version

of

the

conditional

probability

method

for

multivalued

random

variables

-RRB-

put

M

s

cij

Ri

-LRB-

j

-RRB-

and

d

Then

N

dMnk

Define

k-wise

independent

uniformly

distributed

random

variables

Xn

with

values

in

dg

as

in

Theorem

and

with

the

notation

there

we

define

for

a

resource

Ri

and

a

time

z

dg

the

function

fiz

by

fiz

-LRB-

Xn

-RRB-

n

X

j

Ri

-LRB-

j

-RRB-

-LRB-

Xjz

d

-RRB-

Theorem

the

Xj

's

take

on

values

in

the

set

d

But

this

is

only

a

convention

and

F

-LRB-

Xn

-RRB-

X

iz

fiz

-LRB-

Xn

-RRB-

By

Theorem

we

can

construct

xn

where

xj

dg

for

all

j

such

that

F

-LRB-

xn

-RRB-

IE

-LRB-

F

-LRB-

Xn

-RRB-

-RRB-

-LRB-

-RRB-

holds

using

O

-LRB-

N

-RRB-

O

-LRB-

dMnk

-RRB-

O

-LRB-

-LRB-

ns

-RRB-

-RRB-

parallel

processors

in

O

-LRB-

k

log

n

log

N

-RRB-

O

-LRB-

log

n

-LRB-

ns

-RRB-

-RRB-

time

To

complete

the

proof

we

must

show

that

the

assignment

of

task

Tj

to

time

xj

for

all

j

indeed

is

a

feasible

schedule

By

definition

of

the

function

F

we

have

for

all

Ri

and

z

dg

n

X

j

Ri

-LRB-

j

-RRB-

-LRB-

xjz

d

-RRB-

-LRB-

F

-LRB-

xn

-RRB-

-RRB-

-LRB-

IE

-LRB-

F

-LRB-

Xn

-RRB-

-RRB-

-LRB-

-RRB-

Utilizing

the

fact

that

the

random

variables

Xjz

are

binomially

distributed

and

putting

k

log

-LRB-

s

-RRB-

o

log

n

e

the

proof

of

Corollary

in

-LSB-

-RSB-

shows

-LRB-

IE

-LRB-

F

-LRB-

Xn

-RRB-

-RRB-

n

oplog

-LRB-

s

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

imply

for

all

Ri

and

z

dg

n

X

j

Ri

-LRB-

j

-RRB-

xjz

n

X

j

Ri

-LRB-

j

-RRB-

-LRB-

xjz

d

-RRB-

n

X

j

Ri

-LRB-

j

-RRB-

d

n

-LRB-

s

-RRB-

bi

bi

and

the

theorem

is

proved

For

multidimensional

bin

packing

we

get

Corollary

Let

o

log

n

and

suppose

that

l

o

plog

-LRB-

-RRB-

Then

there

is

a

NC

algorithms

for

the

problem

BIN

-LRB-

l

d

-RRB-

that

runs

on

O

-LRB-

-LRB-

nd

-RRB-

-RRB-

parallel

processors

and

finds

in

O

-LRB-

log

n

-LRB-

nd

-RRB-

-RRB-

time

a

packing

with

L

bins

such

that

L

Remark

The

reason

why

we

have

to

assume

bi

-LRB-

n

oplogns

-RRB-

is

due

to

the

estimation

of

the

k-th

moments

Improvements

of

this

method

with

the

goal

to

show

a

-LRB-

or

even

better

-RRB-

NC-algorithm

under

the

weaker

assumption

bi

-LRB-

logns

-RRB-

would

be

interesting

This

would

match

the

presently

best

sequential

approximation

guarantees

Conclusion

ffl

We

proved

that

there

are

polynomial-time

algorithms

approximating

the

optimum

of

a

resource

constrained

scheduling

problem

where

the

makespan

is

to

be

minimized

On

the

other

hand

we

showed

that

it

is

NP

complete

to

guarantee

better

approximations

We

showed

that

the

problem

of

finding

the

chromatic

index

of

a

graph

is

a

special

case

of

a

resource

constrained

scheduling

problem

ffl

We

hope

that

our

rounding

technique

can

be

applied

to

other

problems

of

similar

flavour

Acknowledgement

The

first

author

thanks

Professor

L?aszl?o

Lov?asz

for

helpful

discussions

during

a

research

stay

at

Yale

University

in

Spring

where

a

part

of

this

paper

has

been

written

We

would

also

like

to

thank

Andreas

Kr?amer

for

helpful

discussions

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Introduction

to

Parallel

Algorithms

Addison-Wesley

Publishing

Company

Reading

Massachusetts

-LSB-

-RSB-

K.

L.

Krause

V.

Y.

Shen

H

D.

Schwetmann

Analysis

of

several

job-scheduling

algorithms

for

a

model

of

multiprogramming

computer

systems

JACM

-LRB-

-RRB-

-LCB-

Erratum

JACM

-LRB-

-RRB-

p.

-LSB-

-RSB-

J.

K.

Lenstra

D.

B.

Shmoys

E.

Tardos

Approximating

algorithms

for

scheduling

unrelated

parallel

machines

Math

Programming

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

J.-H

Lin

J.

S.

Vitter

ffl-approximations

with

minimum

packing

constraint

violation

Proceedings

Annual

ACM

Symposium

on

the

Theory

of

Computation

-LRB-

-RRB-

Victoria

B.C.

Canada

-LSB-

-RSB-

F.J.

MacWilliams

N.J.A.

Sloane

The

theory

of

error

correcting

codes

North

Holland

Amsterdam

-LRB-

-RRB-

-LSB-

-RSB-

C.

McDiarmid

On

the

method

of

bounded

differences

Surveys

in

Combinatorics

J.

Siemons

Ed.

London

Math

Soc

Lectures

Notes

Series

Cambridge

University

Press

Cambridge

England

-LSB-

-RSB-

R.

Motwani

J.

Naor

M.

Naor

The

probabilistic

method

yields

deterministic

parallel

algorithms

Proceedings

IEEE

Conference

on

Foandation

of

Computer

Science

-LRB-

FOCS

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

H.

R?ock

G.

Schmidt

Machine

aggregation

heuristics

in

shop

scheduling

Math

Oper

Res

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

J.

Spencer

Ten

lectures

on

the

probabilistic

method

SIAM

Philadelphia

-LRB-

-RRB-

-LSB-

-RSB-

A.

Srivastav

P.

Stangier

Algorithmic

Chernoff-Hoeffding

inequalties

in

integer

programming

to

appear

in

Random

Structures

Algorithms

January

-LRB-

preliminary

version

in

Du

Zhang

-LRB-

eds

-RRB-

Proccedings

of

the

Annual

Inernational

Symposium

on

Algorithms

and

Computation

-LRB-

ISAAC

-RRB-

pages

Lecture

Notes

in

Computer

Science

Vol

Springer

Verlag

-RRB-

-LSB-

-RSB-

E.

Tardos

A

strongly

polynomial

algorithm

to

solve

combinatorial

linear

programs

Oper

Res

-LRB-

-RRB-

-LSB-

-RSB-

V.

G.

Vizing

On

an

estimate

of

the

chromatic

class

of

a

p-graph

-LRB-

Russian

-RRB-

Diskret

Analiz

-LRB-

-RRB-

Appendix

Multivalued

Random

Variables

and

logc

n-wise

Independence

In

this

section

we

extend

the

well-known

derandomization

technique

of

logc

n-wise

due

to

Berger

Rompel

-LSB-

-RSB-

and

Motwani

Naor

Naor

-LSB-

-RSB-

from

the

case

of

uniformly

distributed

to

uniformly

distributed

multivalued

random

variables

Let

n

for

some

IIN

A

representation

of

GF

-LRB-

-RRB-

as

a

algebra

over

GF

-LRB-

-RRB-

can

be

explicitly

constructed

using

irreducible

polynomials

for

example

the

polynomials

given

in

-LSB-

-RSB-

Theorem

Let

bn

be

the

n

non-zero

elements

of

GF

-LRB-

-RRB-

in

such

an

irreducible

representation

and

let

A

-LRB-

aij

-RRB-

the

following

n

dk

e

matrix

over

GF

-LRB-

-RRB-

A

bk

bk

bk

bn

bk

n

A

can

be

viewed

as

a

n

matrix

over

GF

-LRB-

-RRB-

with

dk

edlog

-LRB-

n

-RRB-

e

O

-LRB-

k

log

n

-RRB-

The

matrix

A

is

well-known

in

coding

theory

It

is

the

parity

check

matrix

of

binary

BCH

codes

and

every

set

of

k

row

vectors

of

A

is

linear

independent

over

GF

-LRB-

-RRB-

-LSB-

-RSB-

Alon

Itai

and

Babai

-LSB-

-RSB-

showed

that

k-wise

independent

random

variables

can

be

constructed

from

mutually

independent

random

variables

using

a

BCH-matrix

The

extension

to

multivalued

random

variables

goes

as

follows

Let

Yl

be

independent

and

uniformly

distributed

random

variables

with

values

in

d

d

IIN

Let

Y

be

the

vector

Y

-LRB-

Yl

-RRB-

Define

valued

random

variables

Xn

by

Xi

-LRB-

BY

-RRB-

i

mod

d

for

all

i

n.

With

X

-LRB-

Xl

-RRB-

we

briefly

write

X

BY

mod

d.

Theorem

The

random

variables

Xn

are

k-wise

independent

and

uniformly

distributed

Proof

We

modify

the

proof

of

Theorem

-LSB-

-RSB-

Choose

J

ng

with

jJ

j

k.

Let

x

k

be

an

arbitrarily

choosen

but

fixed

vector

Set

XJ

-LRB-

Xj

-RRB-

We

must

show

IP

-LSB-

XJ

x

-RSB-

d?k

-LRB-

-RRB-

Let

AJ

be

the

submatrix

of

A

with

row

indices

from

J

and

set

BJ

-LRB-

A

-RRB-

J

BJ

is

a

l

k

matrix

over

GF

-LRB-

-RRB-

As

in

the

proof

of

Theorem

we

extend

BJ

to

an

invertible

l

l

matrix

C

over

GF

-LRB-

-RRB-

Define

x

l

xi

for

i

kg

-LRB-

-RRB-

then

j?xj

dl?k

and

we

have

IP

-LSB-

XJ

x

-RSB-

IP

-LSB-

BY

mod

d

x

-RSB-

IP

-LSB-

CY

mod

d

x

-RSB-

X

IP

-LSB-

CY

mod

d

-RSB-

X

IP

-LSB-

Y

C

mod

d

-RSB-

X

d?l

-LRB-

the

Yi

's

are

independent

-RRB-

j?xj

dl

dl?k

dl

d?k

For

the

following

class

of

functions

F

arising

in

the

analysis

of

resource

constrained

scheduling

a

NC

algorithm

finding

points

below

or

above

the

expectation

IE

-LRB-

F

-LRB-

Xn

-RRB-

-RRB-

can

be

derived

We

keep

the

notation

i.e.

let

k

d

be

as

above

Let

vn

be

vectors

in

and

let

vjz

denote

the

z-th

component

of

vj

For

xj

d

let

xj

be

the

vector

that

has

a

in

its

xj-th

coordinate

and

elsewhere

Let

-LRB-

cij

-RRB-

be

M

n

matrix

with

rational

cij

For

i

M

and

z

define

the

functions

giz

-LRB-

xn

-RRB-

n

X

j

cij

-LRB-

xjz

d

-RRB-

-LRB-

-RRB-

and

F

-LRB-

xn

-RRB-

X

iz

giz

-LRB-

xn

-RRB-

-LRB-

-RRB-

Theorem

Let

d

k

M

n

IIN

and

N

dMnk

Let

Xn

be

k-wise

independent

random

variables

with

values

in

d

defined

as

in

Theorem

and

let

F

as

in

-LRB-

-RRB-

Then

with

O

-LRB-

N

-RRB-

parallel

processors

we

can

construct

xn

and

yn

in

O

-LRB-

k

log

n

log

N

-RRB-

time

such

that

-LRB-

i

-RRB-

F

-LRB-

xn

-RRB-

IE

-LRB-

F

-LRB-

Xn

-RRB-

-RRB-

-LRB-

ii

-RRB-

F

-LRB-

yn

-RRB-

IE

-LRB-

F

-LRB-

Xn

-RRB-

-RRB-

Proof

It

suffices

to

prove

-LRB-

i

-RRB-

Let

I

be

the

set

of

all

k

tupels

-LRB-

k

-RRB-

with

j

ng

Set

g

-LRB-

iz

-RRB-

j

-LRB-

xn

-RRB-

cij

-LRB-

xjz

d

-RRB-

and

define

for

a

k-tupel

I

g

-LRB-

iz

-RRB-

-LRB-

xn

-RRB-

kY

j

ciffj

-LRB-

x

j

z

d

-RRB-

Then

F

-LRB-

xn

-RRB-

X

iz

X

g

-LRB-

iz

-RRB-

-LRB-

xn

-RRB-

The

random

variables

Xn

by

definition

have

the

form

Xi

-LRB-

BY

-RRB-

i

mod

d

where

the

Yi

's

are

uniformly

distributed

valued

random

variables

B

AT

and

A

is

the

n

parity

check

matrix

of

BCH

codes

defined

in

Theorem

Therefore

we

may

restrict

us

to

the

assignments

of

the

Yi

's

The

conditional

probability

method

then

goes

as

follows

Suppose

that

for

some

t

we

have

computed

the

values

Yt

yt

then

choose

for

Yt

the

value

yt

that

maximizes

the

function

w

IE

-LRB-

F

-LRB-

Xn

-RRB-

j

yt

Yt

w

-RRB-

-LRB-

-RRB-

After

steps

we

have

Y

y

for

some

y

and

the

vector

x

-LRB-

xn

-RRB-

with

xj

-LRB-

By

-RRB-

j

mod

d

is

the

desired

solution

Let

Yt

-LRB-

Yt

-RRB-

and

yt

-LRB-

yt

-RRB-

We

are

done

if

we

can

compute

the

conditional

expectations

IE

-LRB-

F

-LRB-

Xn

-RRB-

j

Yt

yt

-RRB-

within

the

claimed

time

By

linearity

of

expectation

it

is

sufficient

to

compute

for

each

tripel

-LRB-

i

z

-RRB-

i

m

z

I

the

conditional

expectations

IE

-LRB-

g

-LRB-

iz

-RRB-

-LRB-

Xn

-RRB-

j

Yt

yt

-RRB-

-LRB-

-RRB-

Because

the

Xj

's

are

k-wise

independent

the

problem

reduces

to

the

computation

of

terms

fo

the

form

IE

-LRB-

g

-LRB-

iz

-RRB-

j

-LRB-

Xn

-RRB-

j

Yt

yt

-RRB-

for

j

ng

Now

IE

-LRB-

g

-LRB-

iz

-RRB-

j

-LRB-

Xn

-RRB-

j

Yt

yt

-RRB-

-RRB-

-LRB-

d

-RRB-

IP

-LSB-

Xj

z

j

Yt

yt

-RRB-

-RSB-

d

IP

-LSB-

Xj

z

j

Yt

yt

-RRB-

-RSB-

It

is

straightforward

to

verify

that

IP

-LSB-

Xj

z

j

Yt

yt

-RRB-

-RSB-

can

be

computed

in

constant

time

because

the

only

possible

values

it

attains

are

or

The

same

holds

for

IP

-LSB-

Xj

z

j

Yt

yt

-RRB-

-RSB-

In

other

words

for

fixed

t

i

z

and

given

yt

we

can

compute

IE

-LRB-

g

-LRB-

iz

-RRB-

-LRB-

Xn

-RRB-

j

Yt

yt

-RRB-

in

O

-LRB-

k

-RRB-

time

Thus

for

every

possible

value

yt

we

can

compute

IE

-LRB-

F

-LRB-

Xn

-RRB-

j

Yt

yt

-RRB-

in

O

-LRB-

logN

-RRB-

time

using

O

-LRB-

N

-RRB-

parallel

processors

That

value

of

yt

which

maximizes

-LRB-

-RRB-

can

be

computed

finding

the

maximum

of

the

d

conditional

expectations

with

standard

sorting

algorithms

-LRB-

see

-LSB-

-RSB-

Remark

and

Corollary

-RRB-

in

O

-LRB-

log

d

log

log

d

-RRB-

time

The

total

running

time

over

all

steps

then

is

O

-LRB-

-LRB-

log

N

log

d

log

log

d

-RRB-

-RRB-

O

-LRB-

k

log

n

log

N

-RRB-

and

the

theorem

is

proved

approximation algorithms
chromatic index
derandomization
multidimensional bin packing
resource constrained scheduling