LINEAR PROGRAMMING WITHOUT THE MATRIX - Semantic Scholar
LINEAR
PROGRAMMING
WITHOUT
THE
(Extended
Christos
ABSTRACT:
We study
the
decision-makers
who must
select
of a linear
program,
available sible
to each
solution
possible, able
of
that
When
and
we show imum
when
each
knows that
all
the
number
constraints
and a simple
“safe”
constrained
besides
performance tributed inst ante; When
the
constraint
tern
of the matrix)
ables
are partitioned
optimum ated
ratio
in
terms
but
several
program
maximize
as
one
subject
to
this prob-
The
matrix
is a novel
bers;
such
feasible
which
dis-
prise
on the current
for
this
and
the
parameter
which
we bound
from
of clique
and
above
coefficients, other
are
be-
diverse in
aij
needed
>
to
Finding
characterized
set
linear this
direct
commercial
title
of the
that
copying
25th
To copy
e 1993
the ACM
otherwise,
or distributed
copyright
appear,
notice
and notice
of the Association
is for
the
and the is given
requires
‘93-51931CA,
which
variables
the
In
of the
jth
re-
by
the
ith
are
We
solutions,
objective
function.
the worst-case distributed if all
application
shall
be
achieve
ratio
of A had areas,
to been
as well
the
in
developing
utility
available. as important
pro-
values
we wish
exact
case in
always
reasonable
of the total
a differbetween
that
In particular,
algorithm
general
interested
xi
a part
we shall
by
partitioned
heuristics
and
only
Usually the
DP,
[MMs].
variables
is decided
somehow
distributed
feasible
of the
each knowing
also consider
problem
eral
121
we shall
BOW,
analysis
values
as
paper
decision-
[PY,
in isolation.
agents.
this
In this
of distributed
variable
several
our
.50
the
example,
is as hard
problem).
agents, acting
for
case of “frac-
problems”
of competitive
that
is of course
special
information
for
putable . ..$l
the
function are one.
(see,
this
as one
each
but
mine
a fee
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for Computing
or to republish,
situations
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methods
is, we assume
duce
permission.
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are not made
and its date
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be assuming
1 Department of Computer Science and Engineering, University of California at San Diego. Research supported by the National Science Foundation. 2 AT&T Bell Laboratories, Murray Hill, NJ 07974.
that
sides,
of utility
that
problem
with
of the matrix,
provided
Notice
all objective
percentage
programming
are set by independent
granted
in which
normalized
a unit
algorithmic
it can be shown
using
That
material
that
0 is the
produce
generalized
Pa]
of this
we have
in
enter-
resources.
right-hand
the optimum
a well-studied
making
all or part
of a single
activity.
coloring;
we study
fee
scarse
num-
situations
an environment
a way
as all
real
common
activities
for
in such as well
the general
without
of nonnegative model
compete
words,
tional
copy
< 1
no loss of generality,
vari-
and
from
program
source
0 consists programs
be maximized,
with
linear
the
>
activities
that,
of the associ-
graph
is to
patthen
A linear
the utility
these
parameter.
zero-nonzero
cases
the
[Se, PS];
to
Ax
X>o
completely.
Permission
k’~ i=l
vari-
variable,
decision-makers,
special
linear
a fea-
comparing
of variants interesting
the
tc,the max-
in advance,
bet ween
Consider
are
optimum
best
(the
is a complicated
hypergraph,
low
the
bounds
is known
Yannakakisz
1. INTRODUCTION
of
Since
specializing
structure
Mihalis
there
ratio,
with
)
in each constraint,
is optimal.
different
to find
this
optimization,
perhaps
we show
true
appearing
of a heuristic algorithm,
is
the
is related
competitive
a set
decides
ratio
heuristic
the
goal
involving
worst-case
lem
criterion,
to
and
the variables
of the matrix
decision-maker
of variables
involves
for
parts
The
is as close
facing
values
Abstract
Papadimitrioul
problem
only
them.
H.
MATRIX
achieved
optimum, There
of
to deterby comare
questions
sevin
other
fields,
that
make
this
problem
seem
the
to us worth-
organizational
signment
while:
make
Managing
a Network.
interconnects
thousands
community
of users.
bination path
video
oversee ing
with In
the
users
to capture speed
managers.
well
networks
(see,
Suppose
then
tion.
that
request
Presumably from
available
other
much
based
only
plus
nature
of the
of the
theory
must
ture
the
degradation
in
scal-
lack
of information
erating we have
proposed
competitive
of
work
as-
in
in this
its
rive
full at
on
informa-
world”
include
ficients
requirements
degraded
by the
2 addresses
Zt(.fl,
...,
that
several
schedulers
need
chines.
Each
task
can
(though
not
all),
and
machines;
but
machine ity.
Each
on its
information
How
of incomplete
schedule
only
be
split
of all
knows
which The
much
information?
This
2.1 (Theorem
that
and
based
may
Define a
affected
regime
r(l’’vl,
...,
extensive
set of tasks
fundamental are these
tasks
though
the
these
products
profitability may
the
tern
constraint
of the
which
in
may products
products
and change
constraint be fixed. compete
Theory
among resource
In
the
plete
That for
of the
with associated
resources.
~
VV of
is a function
(Wl,
. . . . W~),
the
available
quality
and quality
cost
of
of the
regime? an attractive ratio
under
answerl. the infor-
to be
min
tL(fl(w),.. .,MW)W
max
U(fl(lvl),
J$’
. . .,hwk),
m’
numerator
worst-case
of the
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the larger
of
utility
the
reversal
problem).
a monotonically regime, made Wi ‘s, the
in
the
optimum
the
denominator
achieved
the
information
of decisions
Al-
have
while
(notice
obviously
how
we
within closer
by due
The
ratio
the to
under
best the
T(W1,
this
have
decision
al-
maximization . . . . W~)
non-increasing arguably
com-
we
captures
information
to one r is bound
~
function the
1, of
quality
regime: to be.
The The
market
sparsity
is, it is known which
is,
requirements the
=
information,
nature
a very
managers?
dynamically structure,
matrix
has
manufactured.
Organization
to be partitioned
conditions, program,
—say,
question
IVi
is to form
is answered
enterprise
coef-
his/her
is the
competitive
~1,..,fk
the
A large
W~)
to
2).
Theory.
of the make
the
provides
valpro-
How
information
“state
linear
information
by the
be
gorithm
Organization
time,
. . . . W~)
values
sets
analysis
(Wl,
in our
decisions
the
network.
the worst-case
mation
on this
by this
question
at decision
Competitive
by
of its task
is lost
agents
The
are
n observed
a subset
the
ar-
decisions
i must
of these
capture
communication
of
problem must
world.
the
of
the
These
a set
are
and frame-
k agents
agent
Intuitively,
work
its capac-
of all fractions
efficiency
W).
regime,
a credible In past
To state
example,
only
utility
by
coop-
appropriate
that
wi’s
cap-
decision-making
of the
each
based The
fk,
decisions
several
tasks
fractions
sum
on ma-
executed
exceed
the
their
machines
between
tasks
machines,
decide
tasks
by several
no circumstances
machines.
be maximized. in Section
can
pertinent
it must
to which
it
fraction
under
scheduler
executed send
the
must
to
be executed
Suppose
the
revered
between [Ar],
Pa].
by
be
caused
providing
1, . . . . k.
Wn } (for
However,
~i(ll’i) values.
DP,
knowledge
problem
these
thus
suppose
is captured
aij ).
decision
How
[PY,
...,
gramming
the
resource.
Problem.
the
an algorith-
to concretely
be a most
~~, i =
partial
ues W={wl,
the
requirements,
may
regard
decisions
to the
Assignment
also
can
at
decisions
distributed
generality,
information
Distributed
work
of information
that
analysis
based
Section
are
communication
agents,
of the value
of the
system
in Section
A much
is how
economic
and
or competing
estimation
of service
same
ascan
partitions
of information. Theory
not
available
value
in Economic
question.
The
of the
project
problem
to high-
rate
would
process?
mic
by
managers
on
the
be ac-
information.
the resource
requesting
aiming
the
the resource
levels,
This
network
information
performance
distributed
various
this
latter
direction.
of Information.
). Because
updated
in this
of a larger
is known
one may
concerning
such
users
is the
the
the service,
resource
of any
[HLP]
problem
and fully
decisions
user
revenue
this
control
at their
of each
example,
of the
results
Value
compet-
relevant
concrete
guide the
results
of such
as part
possibly
programming
problems
for
nature
centralized
arrive
Linear
Our
ratio
The
priced
accommodate
decisions?
seen
manager
request,
to
optimization
distributed
sume
a network
of service
may or
informed competitive
voice
storage,
request
should so that
com-
on a specific
services,
of the user’s
rate
and
user
these
first
to
that
managers,
more
3.2 on the
a large
a complex
computers,
Each for
network
services
as bandwidth
event,
the servicing
down
the
etc. any
and
user requests
on several
a bid
high-speed
sites
such
services,
system.
ing
Each
cycles
companied the
of
of resources,
or tree,
and
A large
principles
of tasks
pat-
1 The relevance of competitive analysis to Economic Theory should come as no surprise. Essentially what we now call the competitive ratio was first proposed by the famous economist Leonard Savage in the 1950’s [Sa] under the term “regret ratio. ”
linear
in advance What
are
122
difference
of this
mation the
ratio
regimes
“extra
between
two
is a defensible
information”
comparable
estimate
agents.
infor-
of the
value
ter
of
colored
or communication.
the Outline. plete
We
study
information.
which
each agent
all constraints the
“safe”
achieves
where
d is the
in any row can
achieve
ilar
results
the
icients
of the
the
are
hold
bound
to
argument,
optimal
matrices,
have
On of
the
the
of nonzero umn the
straints),
and
show
of the
instance),
it
the
algor~hms, for
all
In Section pattern
that feasible
the
the
agent
very
use
no
use
the
the
In
colnot
this A
will
make
same
know
the
algo-
algorithm,
do not
to
and the
rise
algo-
ratio
is equal
to
respect
to
of the
is a partition
and
(let
matrix of the
in which
A is known variables
5).
any
0 for
0-1
matrix
satisfies the utility
algorlthm
D on ‘instance measures
on
the
not
on the
basis
with
of A D
%j.
A,
has
the
A$D(A) A,
Define
The
as the
vector
S 1
achieved
D:
of
a pos-
We
by the the
fol-
competitive
of D is
the
opt(A) = mAax — D(A)
distributed
ratio
d(D)
= myx
of D is maxDf
D’(A) ‘
zerowhere
~k advance,
among
has
need
submatrix
>
zj ‘(A’)
performance
constraint
appears
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to be ~~=1
ma-
algorithms
D is a distributed
this
Zj~(AJ)
For
no
which, Xj
us denote
a value
of the
our
that
in which
d = 2
is 1 for
entries
algorithms
D(A) model
per
of Theorem
that our
is, an algorithm
T(D) while
all
Suppose
matrix
on every
With
the
agents;
variable
CONSTRAINTS
that
(again,
of d).
is equal
3, and ratio
value
= (@A1),.
d >3. 3 we study
one
be O-1 (although
d variables
property:
distributed lowing
assume to
coefficient
let D(A)
to our
safe
is optimal
distributed
XD(A)
- a rather
d =
for
competi-
(a) Two
(c)
OWN
no use of this),
of the
following
infor-
giving
we
known
Aj ), produces
to
is optimal.
3/2 this
upper
a complete
of the
cases:
very prob-
Algorithm
that
rows
itive
cenalso
same
the
are
than
optimization. of
YOUR
Section
trix
more
algorithms
algorithm
but
the
give
proof
for
of the
that
thus
constraint;
le-
ratio
or be in
small
show
three
an alternative
Safe
in the con-
applicable
ratio
per
KNOWING
2.1 The
Also,
it
raticl
distributed
variables
(providing
we
the
some
bound
some
characterization
in the following
true
subtlety
and
and
for
number
optimal
the
distributed
to homogeneous
3.4
coincide,
with
either
the
directed
number, The
and
on two
associated
3.3 we give
Section
two
bounds
solutions,
ratio,
in
bounds
(b)
compare
constrained
is equal d >4,
and
there
Finally, lower
based
hyperedge
with
there
ratio,
graph.
illustrating
adversary
is, it does
to the
involves
2 (i.e.
uniform
nonzero
examples
easy-to-compute
plus
lower
to
competitive
it
respect
d =
every
yield
we call
that
with
d/2
intricate
2.
appear
a heuristic is natural
coincide
ratio,
variable,
runs
our
algorithms
because
1 for
and
is, it
may
and
the
(that that
although
always
variant
d for
(that
it
a parameter
We
variables
comparing
and
problem
to
of the
distributed
mation
rithm
needs to its
directed
tive
lower
makes
inequality,
algorithm,
novel
only
of the
in Subsection
the
Besides
to
It
coloring
per
when an upper
clique
the
a sin-
show
of an
of colors
in
by
as constraint
algorithm
in each
agent),
matrix
so.
tralised other
our
number
complex
competitive directed
in which
appear
replaced
We
parameters
maximum
between;
and
Our
agent:
of the
competitive
of which been
rows
agent,
optimum
is more
per
per
hypergraph,
has
situation
variables,
the
variable
The
parame-
the
as hyperedges
elements
on the
a hypergraph
also
basic
considered
corresponding
uniform
at each
assume
gal
lem.
hypergraphs
is homogeneous
the names
rithm
when
The
d =2).
The
the
minimum
coeff-
of the
one.
two
values:
safe algorithm
know
coefficients
The
hard-to-compute
sim-
but
all
complex
as nodes
of a derivative
constraints,
bound
graph:
and
5) is this:
rank
of
is a very has
case of one
variables
a lower
are involved
(for
or
hand,
matrix
more
sets of neighborhoocls.
coefficients
of the
that
variable
zero
values,
coefficients’
variable,
if all
that
node.
and
the
set
are
above;
only
a graph-theoretic
identical
other
its
same gle
problem:
not
the variables
involves
!prove
discussed
knows
as its
establishing
different
same
involving
either
technique
also
the
of nodes,
of
algorithm
d. We
set
ratio that
agents,
In
(Theorem
any
maximum,
the
is the
con-
coefficients
of the
agent
even
be
resource clf this
of nonzero
problem
each
involving
bounds known
the
by
ratio
out that
no distributed
than
variants
constraints
same
lower
that
constraints
the constraints in
number
assignment
case in which
splits
all participants
smaller
two
We point
optimum
hypergraph
matrix,
result in
iand knows
of d to the centralized
We show
a ratio
distributed
which
maximum
of A. for
variable.
among
a ratio
of incomthe model
for a variable,
this
fairly
frameworks
2 we examine
algorithm
constraint
straint
basic
is responsible
involving
simple
each
two
In Section
The
of the
D’
algorithms
several
123
ranges which
over
all distributed
specialize
to
the
algorithms instance
—even A,
but
of
course
must
stances
produce
as well.
performance we
deal
ratio
with
problem
each
The
minatj>O
=.
Notice
the
following
distributed
algorithm
safe
those
>
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algorithm
in the jth that
all
differentiate
algorithms
and
Theorem
column,
define
:=
and
no
dj
variables
~ d,
for
For
the
the
distributed
upper
algorithm
bound,
safe algorithm,
ratio
has
better
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For
just
notice
that,
if
S
the
{l,2,...,
optimum
d},
by the of the
optimum
value
O to variables this
the
following
sets of indices
d-l,
of the
those
used
for
for
and
sunflower
in the
i; each
to
this
case).
machines,
obvious
ex-
are
m
involving
at
is appropriate fractions
to maximise
Similar
of Theorem
of the
There
to assign
in order
executed.
proof
the
each
wish
ith
provided
machine
agents
the
fraction
set of variables
omit
is, each
fractions
j,
machine),
problem
where the
(we
The
to the
task
prob-
this
n sets,
modeling
agent
each
(that
d tasks).
sum
the
techniques
1 yield
the
as
follow-
result:
Theorem
2:
assignment
problem
that
any
are
a single mts
distributed
Xicv would
~(i, be
constraint
the
value
of d – 1 variables
n}
1 to the
that the
algorithm
involving to
identities
in
algorithm
some
lY
d), there
and are
corresponding set
thus
V).
+ ~i~V~A performance
xv
~i~v
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of instances
1.
the
distributed
ratio
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set
V
We
must
of
that
possible We
V) look
choices
have
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>
because
and
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b~und
of
Algorithm: :=
the
framework
optimization,
~
in-
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lem
di
other
to our
is a new
j:
Safe
Let
applicable
on all other
ratio
fashion.
Consider for
solutions
distributed
constrained
in which
distributed
feasible
The
...
signs
can
on the
force
If a safe
on
can
: j
Theorem
7. K(DIC).
maxc
Sketch:
An
Note
agents
and
constraint
Examples
Two
In this
in
where
the
C).
any
constraint
In
Figure
a triangle, triangle
forces
6 and
colorings
two
maximum
is two
(although
the
at
fills
correct Let
straint, erality
and (all
constraints
alzl
a cycle,
better
simple
by
suppose three
that
agents
as follows:
its
and
Obviously,
if the
solution
of this
is this:
first
this
~.
optimum
was achieved.
value
If the
s
Then, as fol-
Take
allowed
by
possibility
can be done
Z1 to the
prevails,
because
be the
other
con-
then
X2
B knows
the
of zl.
the
the
large
a total
that 2,
as a hyA,
order).
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order
constraint AIso,
number,
ratio
a.
1 and
agents
we know
of
than
size of a linear
Thus,
is two.
+ a2x2
of two
side
ratio.
examples
(see Figures
chromatic
competitive
on the
is represented
not
overall
answer
the
solution
~.
gap,
aszs
in two.
to agents number
in
Figure
has The
as defined
lustrates
by Theorems
is between
two
The
+ a3z3 al
1 be
< az, without
know First,
optimum ~
small
and
this). all
three
We
these
algorithm the
top
per
con-
constraints
in
Figure
complexity
hypergraph.
each
colorings
are not
bound
algorithm:
with
$ of the
answer
X3 takes constraint
intricate
problem
even
total
order
maximum
D IC
of Theorem true
2 is more
of the
The
moreover,
these
126
example the
The
10SS of genbreak
2.
the
two; The
bound
between
the
~,
side right-hand
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three. is this:
s
the
compute
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and
right-hand
splits
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partition
bound
colored
7, to be three.
7 that
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can
the
straint
their
+ azxz
agents
maximum
I--(DIC)
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the
of the
other
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it forms
in Theorem
K
split
constraint
alzl
three
and
of colors
ratio
structure
the
large
lows:
that
let
number
the
an intricate
with
and
and
a of variables
thus
exist
constraint
pergraph
x~l,
variables
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1.
❑
we present
quite
r
right-hand
Suppose
of D,
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may
subsection
behave
as well:
variant
assignment
3.3
ratio
two
The
competitive
the
there
the
of the
conknow
value
bound.
However,
and
the not
variable
is, the
this
coloring,
per
then
Figure
as-
system,
nodes
appropriate
that
is a legal
>>
C. The
achieves
that
ail
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in constraint
assume
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to zero,
bound
E C}l;
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con-
largest
algorithm
that
“inefficient”
of the
the
rapidly:
share”
too
proceed.
is an upper
l{K(j)
rithm
“fair
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We
distributed
to be close
coloring
pearing
its
the
and
decrease
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wasted
is a legal
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variables
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was
maximum,
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achieved.
side
be
a,..
to %1 less than
is not the
say
considers
the
coefficients
>>
straints
argument
achieves
is two-colorable.
globally
compatible,
and for
il-
such
is again However,
and the up-
7 is three. here is ~, achieved
~ of the constraint (1,4,
5); zs takes
by the following (1, 2, 3); *5 takes
~ of the
constraint
(6, 2); Z7 takes variables
~ of the
whose
constraints
constraints:
It
for
turns
this
that
is a little
within
part
Special
plus
two
known (not
extra
algorithm
$ of the optimum;
of the proof.
The
lower
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izations
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them,
show
bounds and
now
of Theorems
from
that
the
all
three
cases yielding
(the
proofs
are
6 and
correct
there
the
coincide
if the uncertainty
this
but
specializations
1:
competitive
When
ratio
in
agents
there
is two
which
only
appear,
in which pears.
6 and
the
intuitive
to
some
are
two
if and there
Corollary
2:
variables,
competitive
then
constraints
the ratio
the
competitive
component
to the
is two.
is a con-
with
one
of
[BOW]
that
and
utility
ratio
with
same
agent.
all
When
each
then
number
more
ap-
❑ than
two
is one if and (Multigraph
variables
the
variable
ratio
ng,
pp.
X.
Deng
Proc. 1992.
side
[HLP]
competitive
variables
T.
We studied
and
J. M.
problem
of a linear
this
understood.
OPEN
to a dif[L]
maxi-
appearing
L.
IFIP
We
analysed
(Section
2), we do not
know
pattern)
of the
of the nonzero help
(Section
the
On
basic
other:
In
in
M.
A.
pp.
the structure
which
Moscow,
first
we might
(and
ratio).
informais [Ps]
one
of the matrix
but
there
223-234,
side,
problems it would
left
open
[PY]
H.
Proc.
“Joint
for
ATS-
ACM
SIG-
and
Exer-
1992. Problems
A.
ACM
McGeoch,
D.
D.
On-line
Symp.
322-333,
Sleator,
Problems,”
on Theory
of Comput-
1988. “The
Value
at the World
August
1992.
Papadimitriou,
Optimization:
C.
H.
Value
one
Making,”
not [Sa]
by this
be interesting
Pacifici
1979. L.
of Information,”
Congress Paper
to
of Economics, appear
in
the
L.
J.
Princeton
for
127
K.
Steiglitz
Algorithms
Combinato-
and
Complexity,
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Papadimitriou,
M.
Yannakakis
“On
of
in
Distributed
Decision
Information Proc.
of Distributed
are many
technical
G.
Control
Nodes,”
Prentice-Hall,
coefficients. the
C. rial
know
second
Lazar,
September
proceedings.
as we saw, this
In the
“Distributed Information”,
Madrid,
Algorithmsfor
talk
of Computi-
Incomplete
A.
C. H. Papadimitriou invited
(zero-nonzero
though
coefficients
[Pa]
the vari-
on Theory
Combinatorial
20th
“Target Variables,”
Papadimitriou
North-Holland,
pp.
Winkler
Random
Admission
S. Manasse,
ing,
frameworks, the
the structure
matrix
competitive
3) we know
actual Clearly,
work.
constraint
problem
two
each
for
of partial
is an important
to
the value
values
on the basis
orthogonal
the
trade-offs
to exchange
of Information,
P.
Congress,
and
Lovasz
Ott,
with
Switching
cises,
PROBLEMS
of selecting
program
We believe
not
a
leads
1992.
G. H.
Hayman,
COMM,
[MMS]
AND
the
essentially
did
is
so that
1984.
Symp.
691-698,
12th
Proc.
little
agents
Economics
J.
ACM
Scheduling
•l
DISCUSSION
tion.
instance
are the
the
Programmed
Decision-making
only
is the
The
with 24th
of the
at each
the
is assigned
competitive
of nonequivalent
constraint.
ables
in Theo-
for
achieved?
Press,
Brightwell,
Based
3: agent,
G.
Proc.
agent
is one.
Otherwise
Arrow Univ.
to a constraint
has
of the
Kenneth Harvard
“Competitive 4.
What
among
the
bounds
for each resource
ratio.
or
“organiza-
decisions
competes
What
complete,
of error,
what
consequence
partitioning
competitive
other
El
Corollary
mum
no constraint
is bipartite,
assigned
any
When
connected
ferent
margin
by our
obvious
extensions
is not
are suggested
communication
Shooting
is a path with
[Ar]
the
[DP]
ratio
some
is it
6
then
if there
associated
associated the
agents,
only
variables
but
a variable Otherwise
every
coefficients
within
of agents
a better
to the
and results:
Also,
fact
information
of Theorems
the
why
Furthermore,
related model
distribution?
One
number
between
character-
known
example,
NP-hard?).
basic
about
7?
3 (for
com-
of a hyper-
REFERENCES
Corollary
the
principles”
rem
ratio
problems
a known
tion
computational
Is it
of our
are
have
small
7):
straint
they
even
precise competitive
in Section
open
that
7 can
for
following
are many
and interpretation
competitive
the the
problem?
B).
bound
down
as explained
a decidable
to C), and to
pin
of computing
graph
~ of
known
to
plexity
Cases
seen that
important
and
is not
instance
program
remaining
is a distributed
program
between
ratio.
the
~ of (4,7)
remaining
easier.
We have
fairly
The
a linear
to all three,
constraint
involved
Three
differ
(4, 7).
to solve
X2 from
the
there
linear
is the most
3.4
on
Z4 from
out
solves
one
(6, 2) (this
one
now
are known
The
constraint the
constraint
z ~, X2, X4 have
10th
ACM
Computing,
Savage Univ.
The
Symp.
on Principles
pp.
61-64,
Foundations
of
Press,
1957.
the
1991. Statistics,
[Se] A.
Schrijver
Theory
gramming, H.
[Si]
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of Linear
Wiley,
Simon
and
Ni
Administrative
of Decision-Making Organization,
maximum
Pro-
Integer
1986. Behavior;
Processes
3rd
ed.,
in
Free,
sketch
We need (1)
For
for
every
the proof
to show every
two
1976.
3 for the cased=
3.
each
Ui sum
given
value
(hypergraph)
A
value
of rank
algorithm
S(A)
D,
1, ..., k, the
will
following
show
For
every
constant
C (and
therefore
instance
A and
C’(A)
the
uniform
in particular
and
(We
algorithm
also for S) there
a homogeneous
cD(A).
feasible
will
algorithm
show
exists
D
something
in
the
following
discussion
cardinality Thus,
exactly
the
node
We will
safe
and
number
of nodes.
that
contain rank
3 then
the
algorithm variable value
view D
and D(i,
Lemma rithm.
is an ordinary
for
Al.
Let
If D(i,
Gl)
distinct
nodes
contain
any
Statement
D
(1)
G
1/2
then
nodes
above
Proof:
and
of i)
the
G2)
graphs
with
the
follows
to
a
the
for
two
n
A2.
nodes.
D(A)
Any
the
For
nodes
with
node
view.
we can for
value
order
are nodes
the
homogeneous
each
i, let
choose all
views
larger
i. N1
than
k.
otherwise distinct By
to
without
1 through
be its 1/2
the
view
and
they
nodes Al, N~
are
maximum
distinct.
holds
distributed
for
for
any
algorithm.
B such
that,
= E(D(T(A))),
all
random
for
where
permutations
of the
sum
of the node
B gives
expected this
of the
=
N,
where
A, and
For
algorithm
the i and
B
expectation the labelling
e of A.
of e is equal
values
given to
for
than
1 for
and
a hyperedge to the nodes
is equal
values
is no more
B(A)
B as follows.
(view)
J(N))),
that
sum
sum
algorithm
choices
of expectation,
D.
by the
since
every
D.
follows
D
By
lin-
is feasible,
labelling,
also
The
expectation
and
B is also feasible.
13( D(m(A)))
The
from
thus
equal-
linearity
of
❑ A4.
nodes
value
that
be their
Let
belong
views.
y such
If
view
of y in NV.
them
in
Lemma
degree
in
and
in
A.
A be a hypergraph,
let
to
and
a common
Then
node
that
that
the
NU
edge,
has
view
a node
u, v be two let
x and
of x in NU is the
N., N.
Nv has
same
degrees.
B
that Then
be
a feasible
contains B(G)
two
uniform adjacent
algorithm nodes
with
~ 1/3.
be adjacent that
an edge
Ui is
degrees Therefore
Sketch:
that
isomorphic
of the
We {u,
to G,
We can show
128
argue v, w}
now
that such
there that
is a hypergraph all
three
❑ statement
(2)
above.
nodes
a
as the
❑ Let
A5.
G a graph
equal
these
We claim
U1, . . . . u~ such the
= B(T(A))
Consider
order
cannot
D is not feasible.
B(A)
of a
under
of N.
to the
ity
3 with
has
holds.
maximum
nodes
value
lemma.
D
loss of generality
Clearly,
Lemma
through
Ni
the lemma
E(D(i,
a hypergraph
expectation.
of rank
algorithm
according
Assume
to each other, Ni
be a hypergraph
Ni ) < 1/2 for all i, then
increasing their
A
over
uniform
so is it expectation,
s n/2.
Proofi D(i,
Let
B =
nodes
Lemma Lemma
of the the
lemma
B(A)
hypergraph
earity
this
❑
following
i.e.,
the and
variables.
the
of the values
of the
G2 do not
degree.
is taken
all random
Consider sum
algo-
1/2
G1 and
same
from
>
value
A of the
i of a
homogeneous
D(j,
B,
on its view
that
algorithm
we have
n of the Define
is over
Thus,
index
view
have
uniform
A is invariant
following
uniform
unlabeled
gives
of H
A homogeneous
(the
we
algorithm
identities
i are
❑
and
only
Note
variables,
to nodes
variables
First,
in a uniform
D be a feasible
A,
expectation
every
u
xi.
be a (feasible) >
i, j,
two
variable
Let
instance
(labelings)
if A has rank
each
that Ui one
every
all of them.
maps
graph
the
to
of a node
1, i.e.,
graph.
that
unlabeled
G)
by
the
n/3, where
N.(A)
from
is smaller
is a function
1/3 =
now.
xi depends
n. The
D
follows.
B on an instance
is a feasible
every
nodes.
of the hyperedges
u removed
view
value
view
in
of A have
no isolated
is S(A)
consisting
u, with
of the
give
value The
for simplicity
hyperedges are
will
achieve
in A is the hypergraph
the
all
there
algorithm
(variable)
n is the
that
3 and
assume
by
of the
distributed
= B(N).
of the
A3.
There A be a hypergraph.
bound
of i, or the N)
algorithm
Lemma
fact.) Let
means nodes
rank.
that
stronger
lemma
that
B(i,
all permutations an
such
The
relating
identity
i.e.,
given rest
to a variable
on the
values
lower
Recall
given
its view,
safe
~ 2/3 D(A).
2/3,
i, which
we can pick
1. The
1/2.
lemma
permutations (2)
is at least Thus,
to at most at most
the
not
3 and the
i =
We
value
homogeneous
S achieves
For and
algorithms.
things:
instance
(feasible)
algorithm
of Theorem
of lVi
i nodes.
by one.
a Study
Administrative
APPENDIX We will
degree
has at least
A with have
view
Lemma
A6.
There
n = 2k + 1 nodes (1)
some
D(A)
is a hypergraph
such
A of rank
3 with
that:
homogeneous
feasible
algorithm
D
has
value
= k;
(2) for
every
feasible
permutation
Proofi
Let two
sets:
and
a “periphery” i =
i on
the
graph
hypergraph
l,...,
kis
Let
D be the
value
O, and
D(Wi,
Ni)
=
where
algorithm
1 if
its
now
of the up
the
node
(vi,
uo,
These
variables
to
value
of
B(Gj
) ~ 1/3
by
Note
that
Ni
of each
node
a A vi,
for
there
D’(r(A))
D by the
obvious (2)
have by
value value
disjoint
By
algorithms has
the
match
in
whose
the
sets A4.
of Its
u2,
the
A5.
pair
we with
occurs
ccmtains
(k+
1)/2
variable
The
permutation
lemma
Vj
hy-
= k; algorithm Thus,
in
total
has
value
variables
algorithm D’
❑
follows.
m of the
is a homogeneous relabeling.
pair
. . . . (~~+1i2,u~-l,u~).
most
other
con-
that
each
A
A3
The
property
view
u3), at
Lemma B.
hypergraph
Every
Lemma
vi and
Lemma
and
(v2,
every
to Gi,
Gi
vi
B.
Gi
core node
node
nodes
isj UI),
to each
algorithm.
contribute
has value statement
Let
assigns
uniform
graphs
that
edge;
A6,
such
= k.
feasible
core
as an
Lemma
that
peripheral
graphs
D(A)
any
a peripheral
the
For
of degree 5.2).
is feasible
to consider
peredges
. . . . v~.
known
isomorphic
k
the
on A is clearly
pair
VI, graph
exercise
view
each
view
Since
can
[L]
the
to
algorithm
the
struction
is well
which
assigns
degrees,
it suffices
it
nodes
UO, . . . . u~,
Gi.
O otherwise.
Consider
labelled
is a
~ n/3. the
1 nodes
a regular
example
C there
C(T(A))
Partition
of k +
be
L;
(see for
i=
value
Gi
of nodes
that
number. L
of k nodes
1, ..., k, let set
algorithm
such
odd
a “core”
exists
be the
nodes
k be any
into
each
distributed
r of the
D’
is obtained
Lemma
in that from
A6 implies
above.
129
PROGRAMMING
WITHOUT
THE
(Extended
Christos
ABSTRACT:
We study
the
decision-makers
who must
select
of a linear
program,
available sible
to each
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possible, able
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When
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when
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optimum ated
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in
terms
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The
matrix
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bers;
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for
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and
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parameter
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we bound
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and
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coefficients, other
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Finding
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set
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commercial
title
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25th
To copy
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and the is given
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variables
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In
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developing
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In particular,
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general
interested
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Usually the
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[MMs].
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also consider
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121
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specific
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advantage,
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in such question
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1 Department of Computer Science and Engineering, University of California at San Diego. Research supported by the National Science Foundation. 2 AT&T Bell Laboratories, Murray Hill, NJ 07974.
that
sides,
of utility
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problem
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provided
Notice
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percentage
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1. INTRODUCTION
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different
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The
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Abstract
Papadimitrioul
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H.
MATRIX
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Managing
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erating we have
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rive
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tern
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Competitive
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n observed
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ar-
decisions
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world.
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k agents
agent
Intuitively,
work
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of all fractions
efficiency
W).
regime,
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To state
example,
only
utility
by
coop-
appropriate
that
wi’s
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decision-making
of the
each
based The
fk,
decisions
several
tasks
fractions
sum
on ma-
executed
exceed
the
their
machines
between
tasks
machines,
decide
tasks
by several
no circumstances
machines.
be maximized. in Section
can
pertinent
it must
to which
it
fraction
under
scheduler
executed send
the
must
to
be executed
Suppose
the
revered
between [Ar],
Pa].
by
be
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providing
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However,
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Assignment
also
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at
decisions
distributed
generality,
information
Distributed
work
of information
that
analysis
based
Section
are
communication
agents,
of the value
of the
system
in Section
A much
is how
economic
and
or competing
estimation
of service
same
ascan
partitions
of information. Theory
not
available
value
in Economic
question.
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of the
project
problem
to high-
rate
would
process?
mic
by
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on
the
be ac-
information.
the resource
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the resource
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network
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performance
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of Information.
). Because
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of each
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results
Value
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as part
possibly
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problems
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nature
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arrive
Linear
Our
ratio
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priced
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voice
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on a specific
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to
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managers,
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computers,
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ing
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1 The relevance of competitive analysis to Economic Theory should come as no surprise. Essentially what we now call the competitive ratio was first proposed by the famous economist Leonard Savage in the 1950’s [Sa] under the term “regret ratio. ”
linear
in advance What
are
122
difference
of this
mation the
ratio
regimes
“extra
between
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is a defensible
information”
comparable
estimate
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value
ter
of
colored
or communication.
the Outline. plete
We
study
information.
which
each agent
all constraints the
“safe”
achieves
where
d is the
in any row can
achieve
ilar
results
the
icients
of the
the
are
hold
bound
to
argument,
optimal
matrices,
have
On of
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of nonzero umn the
straints),
and
show
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instance),
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algor~hms, for
all
In Section pattern
that feasible
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use
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matrix of the
in which
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5).
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matrix
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algorlthm
D on ‘instance measures
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of A D
%j.
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Define
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competitive
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opt(A) = mAax — D(A)
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us denote
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in which
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of Theorem
that our
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Suppose
matrix
on every
With
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CONSTRAINTS
that
(again,
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value
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d variables
property:
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assume to
coefficient
let D(A)
to our
safe
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distributed
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- a rather
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for
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OWN
no use of this),
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following
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giving
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known
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to
is optimal.
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a complete
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itive
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trix
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give
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le-
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true
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number
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distributed
to homogeneous
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coincide,
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and
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associated
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Section
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compare
constrained
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Finally, lower
based
hyperedge
with
there
ratio,
graph.
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adversary
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involves
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examples
easy-to-compute
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d =
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intricate
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and
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tive
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in Subsection
the
Besides
to
It
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per
when an upper
clique
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show
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of colors
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by
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algorithm
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tralised other
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number
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in which
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parameters
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agent:
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lem.
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rithm
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rank
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are involved
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or
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coefficients
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node.
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set
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above;
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identical
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problem:
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discussed
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as its
establishing
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involving
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technique
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of nodes,
of
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set
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agents,
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(Theorem
any
maximum,
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problem
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by
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number
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case in which
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smaller
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optimum
hypergraph
matrix,
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We show
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variable.
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for a variable,
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algorithm
constraint
straint
basic
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involving
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In Section
The
of the
D’
algorithms
several
123
ranges which
over
all distributed
specialize
to
the
algorithms instance
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but
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course
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stances
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deal
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the
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column,
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and
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For
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the
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the
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for
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i; each
to
this
case).
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are
m
involving
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to maximise
Similar
of Theorem
of the
There
to assign
in order
executed.
proof
the
each
wish
ith
provided
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fraction
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where the
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The
to the
task
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this
n sets,
modeling
agent
each
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d tasks).
sum
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techniques
1 yield
the
as
follow-
result:
Theorem
2:
assignment
problem
that
any
are
a single mts
distributed
Xicv would
~(i, be
constraint
the
value
of d – 1 variables
n}
1 to the
that the
algorithm
involving to
identities
in
algorithm
some
lY
d), there
and are
corresponding set
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V).
+ ~i~V~A performance
xv
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of instances
1.
the
distributed
ratio
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V
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must
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V) look
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have
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Algorithm: :=
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is a new
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applicable
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distributed
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feasible
The
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If a safe
on
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Theorem
7. K(DIC).
maxc
Sketch:
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Note
agents
and
constraint
Examples
Two
In this
in
where
the
C).
any
constraint
In
Figure
a triangle, triangle
forces
6 and
colorings
two
maximum
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(although
the
at
fills
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straint, erality
and (all
constraints
alzl
a cycle,
better
simple
by
suppose three
that
agents
as follows:
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and
Obviously,
if the
solution
of this
is this:
first
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~.
optimum
was achieved.
value
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s
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can be done
Z1 to the
prevails,
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then
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of zl.
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as a hyA,
order).
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order
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number,
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of
than
size of a linear
Thus,
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+ a2x2
of two
side
ratio.
examples
(see Figures
chromatic
competitive
on the
is represented
not
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answer
the
solution
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gap,
aszs
in two.
to agents number
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Figure
has The
as defined
lustrates
by Theorems
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The
+ a3z3 al
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< az, without
know First,
optimum ~
small
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three
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these
algorithm the
top
per
con-
constraints
in
Figure
complexity
hypergraph.
each
colorings
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bound
algorithm:
with
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answer
X3 takes constraint
intricate
problem
even
total
order
maximum
D IC
of Theorem true
2 is more
of the
The
moreover,
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126
example the
The
10SS of genbreak
2.
the
two; The
bound
between
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side right-hand
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s
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compute
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7, to be three.
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can
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straint
their
+ azxz
agents
maximum
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is
maximum
way
1 the
the
of the
other
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it forms
in Theorem
K
split
constraint
alzl
three
and
of colors
ratio
structure
the
large
lows:
that
let
number
the
an intricate
with
and
and
a of variables
thus
exist
constraint
pergraph
x~l,
variables
the
1.
❑
we present
quite
r
right-hand
Suppose
of D,
is an upper
may
subsection
behave
as well:
variant
assignment
3.3
ratio
two
The
competitive
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of the
conknow
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variable
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this
coloring,
per
then
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system,
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achieves
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ail
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largest
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rapidly:
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proceed.
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rithm
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distributed
to be close
coloring
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the
and
decrease
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wasted
is a legal
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induction There
(1,2,
variables
them
was
maximum,
other
achieved.
side
be
a,..
to %1 less than
is not the
say
considers
the
coefficients
>>
straints
argument
achieves
is two-colorable.
globally
compatible,
and for
il-
such
is again However,
and the up-
7 is three. here is ~, achieved
~ of the constraint (1,4,
5); zs takes
by the following (1, 2, 3); *5 takes
~ of the
constraint
(6, 2); Z7 takes variables
~ of the
whose
constraints
constraints:
It
for
turns
this
that
is a little
within
part
Special
plus
two
known (not
extra
algorithm
$ of the optimum;
of the proof.
The
lower
We
izations
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them,
show
bounds and
now
of Theorems
from
that
the
all
three
cases yielding
(the
proofs
are
6 and
correct
there
the
coincide
if the uncertainty
this
but
specializations
1:
competitive
When
ratio
in
agents
there
is two
which
only
appear,
in which pears.
6 and
the
intuitive
to
some
are
two
if and there
Corollary
2:
variables,
competitive
then
constraints
the ratio
the
competitive
component
to the
is two.
is a con-
with
one
of
[BOW]
that
and
utility
ratio
with
same
agent.
all
When
each
then
number
more
ap-
❑ than
two
is one if and (Multigraph
variables
the
variable
ratio
ng,
pp.
X.
Deng
Proc. 1992.
side
[HLP]
competitive
variables
T.
We studied
and
J. M.
problem
of a linear
this
understood.
OPEN
to a dif[L]
maxi-
appearing
L.
IFIP
We
analysed
(Section
2), we do not
know
pattern)
of the
of the nonzero help
(Section
the
On
basic
other:
In
in
M.
A.
pp.
the structure
which
Moscow,
first
we might
(and
ratio).
informais [Ps]
one
of the matrix
but
there
223-234,
side,
problems it would
left
open
[PY]
H.
Proc.
“Joint
for
ATS-
ACM
SIG-
and
Exer-
1992. Problems
A.
ACM
McGeoch,
D.
D.
On-line
Symp.
322-333,
Sleator,
Problems,”
on Theory
of Comput-
1988. “The
Value
at the World
August
1992.
Papadimitriou,
Optimization:
C.
H.
Value
one
Making,”
not [Sa]
by this
be interesting
Pacifici
1979. L.
of Information,”
Congress Paper
to
of Economics, appear
in
the
L.
J.
Princeton
for
127
K.
Steiglitz
Algorithms
Combinato-
and
Complexity,
1982.
Papadimitriou,
M.
Yannakakis
“On
of
in
Distributed
Decision
Information Proc.
of Distributed
are many
technical
G.
Control
Nodes,”
Prentice-Hall,
coefficients. the
C. rial
know
second
Lazar,
September
proceedings.
as we saw, this
In the
“Distributed Information”,
Madrid,
Algorithmsfor
talk
of Computi-
Incomplete
A.
C. H. Papadimitriou invited
(zero-nonzero
though
coefficients
[Pa]
the vari-
on Theory
Combinatorial
20th
“Target Variables,”
Papadimitriou
North-Holland,
pp.
Winkler
Random
Admission
S. Manasse,
ing,
frameworks, the
the structure
matrix
competitive
3) we know
actual Clearly,
work.
constraint
problem
two
each
for
of partial
is an important
to
the value
values
on the basis
orthogonal
the
trade-offs
to exchange
of Information,
P.
Congress,
and
Lovasz
Ott,
with
Switching
cises,
PROBLEMS
of selecting
program
We believe
not
a
leads
1992.
G. H.
Hayman,
COMM,
[MMS]
AND
the
essentially
did
is
so that
1984.
Symp.
691-698,
12th
Proc.
little
agents
Economics
J.
ACM
Scheduling
•l
DISCUSSION
tion.
instance
are the
the
Programmed
Decision-making
only
is the
The
with 24th
of the
at each
the
is assigned
competitive
of nonequivalent
constraint.
ables
in Theo-
for
achieved?
Press,
Brightwell,
Based
3: agent,
G.
Proc.
agent
is one.
Otherwise
Arrow Univ.
to a constraint
has
of the
Kenneth Harvard
“Competitive 4.
What
among
the
bounds
for each resource
ratio.
or
“organiza-
decisions
competes
What
complete,
of error,
what
consequence
partitioning
competitive
other
El
Corollary
mum
no constraint
is bipartite,
assigned
any
When
connected
ferent
margin
by our
obvious
extensions
is not
are suggested
communication
Shooting
is a path with
[Ar]
the
[DP]
ratio
some
is it
6
then
if there
associated
associated the
agents,
only
variables
but
a variable Otherwise
every
coefficients
within
of agents
a better
to the
and results:
Also,
fact
information
of Theorems
the
why
Furthermore,
related model
distribution?
One
number
between
character-
known
example,
NP-hard?).
basic
about
7?
3 (for
com-
of a hyper-
REFERENCES
Corollary
the
principles”
rem
ratio
problems
a known
tion
computational
Is it
of our
are
have
small
7):
straint
they
even
precise competitive
in Section
open
that
7 can
for
following
are many
and interpretation
competitive
the the
problem?
B).
bound
down
as explained
a decidable
to C), and to
pin
of computing
graph
~ of
known
to
plexity
Cases
seen that
important
and
is not
instance
program
remaining
is a distributed
program
between
ratio.
the
~ of (4,7)
remaining
easier.
We have
fairly
The
a linear
to all three,
constraint
involved
Three
differ
(4, 7).
to solve
X2 from
the
there
linear
is the most
3.4
on
Z4 from
out
solves
one
(6, 2) (this
one
now
are known
The
constraint the
constraint
z ~, X2, X4 have
10th
ACM
Computing,
Savage Univ.
The
Symp.
on Principles
pp.
61-64,
Foundations
of
Press,
1957.
the
1991. Statistics,
[Se] A.
Schrijver
Theory
gramming, H.
[Si]
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of Linear
Wiley,
Simon
and
Ni
Administrative
of Decision-Making Organization,
maximum
Pro-
Integer
1986. Behavior;
Processes
3rd
ed.,
in
Free,
sketch
We need (1)
For
for
every
the proof
to show every
two
1976.
3 for the cased=
3.
each
Ui sum
given
value
(hypergraph)
A
value
of rank
algorithm
S(A)
D,
1, ..., k, the
will
following
show
For
every
constant
C (and
therefore
instance
A and
C’(A)
the
uniform
in particular
and
(We
algorithm
also for S) there
a homogeneous
cD(A).
feasible
will
algorithm
show
exists
D
something
in
the
following
discussion
cardinality Thus,
exactly
the
node
We will
safe
and
number
of nodes.
that
contain rank
3 then
the
algorithm variable value
view D
and D(i,
Lemma rithm.
is an ordinary
for
Al.
Let
If D(i,
Gl)
distinct
nodes
contain
any
Statement
D
(1)
G
1/2
then
nodes
above
Proof:
and
of i)
the
G2)
graphs
with
the
follows
to
a
the
for
two
n
A2.
nodes.
D(A)
Any
the
For
nodes
with
node
view.
we can for
value
order
are nodes
the
homogeneous
each
i, let
choose all
views
larger
i. N1
than
k.
otherwise distinct By
to
without
1 through
be its 1/2
the
view
and
they
nodes Al, N~
are
maximum
distinct.
holds
distributed
for
for
any
algorithm.
B such
that,
= E(D(T(A))),
all
random
for
where
permutations
of the
sum
of the node
B gives
expected this
of the
=
N,
where
A, and
For
algorithm
the i and
B
expectation the labelling
e of A.
of e is equal
values
given to
for
than
1 for
and
a hyperedge to the nodes
is equal
values
is no more
B(A)
B as follows.
(view)
J(N))),
that
sum
sum
algorithm
choices
of expectation,
D.
by the
since
every
D.
follows
D
By
lin-
is feasible,
labelling,
also
The
expectation
and
B is also feasible.
13( D(m(A)))
The
from
thus
equal-
linearity
of
❑ A4.
nodes
value
that
be their
Let
belong
views.
y such
If
view
of y in NV.
them
in
Lemma
degree
in
and
in
A.
A be a hypergraph,
let
to
and
a common
Then
node
that
that
the
NU
edge,
has
view
a node
u, v be two let
x and
of x in NU is the
N., N.
Nv has
same
degrees.
B
that Then
be
a feasible
contains B(G)
two
uniform adjacent
algorithm nodes
with
~ 1/3.
be adjacent that
an edge
Ui is
degrees Therefore
Sketch:
that
isomorphic
of the
We {u,
to G,
We can show
128
argue v, w}
now
that such
there that
is a hypergraph all
three
❑ statement
(2)
above.
nodes
a
as the
❑ Let
A5.
G a graph
equal
these
We claim
U1, . . . . u~ such the
= B(T(A))
Consider
order
cannot
D is not feasible.
B(A)
of a
under
of N.
to the
ity
3 with
has
holds.
maximum
nodes
value
lemma.
D
loss of generality
Clearly,
Lemma
through
Ni
the lemma
E(D(i,
a hypergraph
expectation.
of rank
algorithm
according
Assume
to each other, Ni
be a hypergraph
Ni ) < 1/2 for all i, then
increasing their
A
over
uniform
so is it expectation,
s n/2.
Proofi D(i,
Let
B =
nodes
Lemma Lemma
of the the
lemma
B(A)
hypergraph
earity
this
❑
following
i.e.,
the and
variables.
the
of the values
of the
G2 do not
degree.
is taken
all random
Consider sum
algo-
1/2
G1 and
same
from
>
value
A of the
i of a
homogeneous
D(j,
B,
on its view
that
algorithm
we have
n of the Define
is over
Thus,
index
view
have
uniform
A is invariant
following
uniform
unlabeled
gives
of H
A homogeneous
(the
we
algorithm
identities
i are
❑
and
only
Note
variables,
to nodes
variables
First,
in a uniform
D be a feasible
A,
expectation
every
u
xi.
be a (feasible) >
i, j,
two
variable
Let
instance
(labelings)
if A has rank
each
that Ui one
every
all of them.
maps
graph
the
to
of a node
1, i.e.,
graph.
that
unlabeled
G)
by
the
n/3, where
N.(A)
from
is smaller
is a function
1/3 =
now.
xi depends
n. The
D
follows.
B on an instance
is a feasible
every
nodes.
of the hyperedges
u removed
view
value
view
in
of A have
no isolated
is S(A)
consisting
u, with
of the
give
value The
for simplicity
hyperedges are
will
achieve
in A is the hypergraph
the
all
there
algorithm
(variable)
n is the
that
3 and
assume
by
of the
distributed
= B(N).
of the
A3.
There A be a hypergraph.
bound
of i, or the N)
algorithm
Lemma
fact.) Let
means nodes
rank.
that
stronger
lemma
that
B(i,
all permutations an
such
The
relating
identity
i.e.,
given rest
to a variable
on the
values
lower
Recall
given
its view,
safe
~ 2/3 D(A).
2/3,
i, which
we can pick
1. The
1/2.
lemma
permutations (2)
is at least Thus,
to at most at most
the
not
3 and the
i =
We
value
homogeneous
S achieves
For and
algorithms.
things:
instance
(feasible)
algorithm
of Theorem
of lVi
i nodes.
by one.
a Study
Administrative
APPENDIX We will
degree
has at least
A with have
view
Lemma
A6.
There
n = 2k + 1 nodes (1)
some
D(A)
is a hypergraph
such
A of rank
3 with
that:
homogeneous
feasible
algorithm
D
has
value
= k;
(2) for
every
feasible
permutation
Proofi
Let two
sets:
and
a “periphery” i =
i on
the
graph
hypergraph
l,...,
kis
Let
D be the
value
O, and
D(Wi,
Ni)
=
where
algorithm
1 if
its
now
of the up
the
node
(vi,
uo,
These
variables
to
value
of
B(Gj
) ~ 1/3
by
Note
that
Ni
of each
node
a A vi,
for
there
D’(r(A))
D by the
obvious (2)
have by
value value
disjoint
By
algorithms has
the
match
in
whose
the
sets A4.
of Its
u2,
the
A5.
pair
we with
occurs
ccmtains
(k+
1)/2
variable
The
permutation
lemma
Vj
hy-
= k; algorithm Thus,
in
total
has
value
variables
algorithm D’
❑
follows.
m of the
is a homogeneous relabeling.
pair
. . . . (~~+1i2,u~-l,u~).
most
other
con-
that
each
A
A3
The
property
view
u3), at
Lemma B.
hypergraph
Every
Lemma
vi and
Lemma
and
(v2,
every
to Gi,
Gi
vi
B.
Gi
core node
node
nodes
isj UI),
to each
algorithm.
contribute
has value statement
Let
assigns
uniform
graphs
that
edge;
A6,
such
= k.
feasible
core
as an
Lemma
that
peripheral
graphs
D(A)
any
a peripheral
the
For
of degree 5.2).
is feasible
to consider
peredges
. . . . v~.
known
isomorphic
k
the
on A is clearly
pair
VI, graph
exercise
view
each
view
Since
can
[L]
the
to
algorithm
the
struction
is well
which
assigns
degrees,
it suffices
it
nodes
UO, . . . . u~,
Gi.
O otherwise.
Consider
labelled
is a
~ n/3. the
1 nodes
a regular
example
C there
C(T(A))
Partition
of k +
be
L;
(see for
i=
value
Gi
of nodes
that
number. L
of k nodes
1, ..., k, let set
algorithm
such
odd
a “core”
exists
be the
nodes
k be any
into
each
distributed
r of the
D’
is obtained
Lemma
in that from
A6 implies
above.
129
