Polynomial
Algorithms
for
Linear
Programming
over
the
Algebraic
Numbers
Ilan Adler
and
Peter A. Beling
of Industrial
Engineering and Operations Research
University
of California,
Berkeley
Berkeley, CA 94720
Department
Abstract
We
an
derive
method
that
algorithm
solves
based
linear
icients are real algebraic
encoding
polynomial
on
the
number
of its minimal
we prove the algorithm
runs in time polynomial
the dimension
the input
of the problem,
coefficients,
gebraic
extension
ficients.
This
arithmetic
and
which
bound
the input
in
symbolically,
The
linear
fact,
and
ratio-
only.
both
interior
coefficients
rational
ally
ing
put
modeled
and
the
an
of
number
that
an instance
needed
A linear
consist
algorithm
dimension
vectors
abstracted
only
is defined
to encode
and
in
the
these
be the
entries
algorithm
is usually
perform
total
in binary
is said
operands.
in
1979.
In
[7] and
[6] solve
LPs
coefficients
and the total
encoding
of the problem
do not extend
in
are real
form.
is defined
by a finite
esponding
in
notion
An algorithm
the number
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24th ANNUAL ACM STOC - 5/92/VICTORIA,
B. C., CANADA
e 1992 ACM ().89791 -51 2.7192/0004/0483...+”
1.50
bounded
483
real number
of digits,
of the theory
cannot
rithm
is known
case,
be rep-
there is no corr-
operations
in the problem
results
methods
real-number
for both
depend
LP.
time if
it performs
is
dimen-
no polynomial-time
for general
of
Prob-
is said to run in polynomia/
of elementary
by a polynomial
point
of the
of the size of a real-number
By this definition,
Complexity
operation
of the nature
as in the rational
string
sion.
terior
that
over the real numbers.)
but because a general
resented
arithmetic
regardless
that
programming
of a machine
(See [2] for a treatment
lem dimension
of
(i.e., numbers
linear
terms
any elementary
time,
general computation
and
to run
method
rational),
modeled
in constant
size of
number
in
method
way to LPs whose coefficients
can perform
sub-
is defined
the
rational-
was demonstrated
these results
are not necessarily
num-
matrices
and
of input
Unfortunately,
of
in time that is polynomial
In the case of real numbers
in-
comparison.
instance
instance,
to
to
in the
numbers.
Tur-
of addition,
a problem
the
the
of rational
division,
of entries
is usu-
Problem
operations
define
programming
from
is permitted
multiplication,
to be the
bits
to
numbers
of computation.
elementary
traction,
The
terms
model
is assumed
bers,
only
in
machine
rational
point
of bits in a binary
any obvious
with
(LPs)
ellipsoid
Karmarkar’s
data.
programming
be polynomial
by Khachiyan
Khachiyan’s
in the number
Introduction
Linear
paper
with
number
1
encoding
the course of
solvability
programs
in a landmark
coef-
using
algorithm
polynomial-time
number
al-
even if all input
during
size of the instance.
size of
the degree of any
holds
is performed
nal numbers
the encoding
contains
a polynomial-time
polynomial,
size. Typi-
that the binary
generated
op-
by a polynomial
and encoding
required
size of any number
to be the bit
of elementary
is bounded
dimension
cally, it is further
the
size of the coeflcients
it performs
in the problem
coeff-
By defining
numbers.
size of an algebraic
ellipsoid
whose
programs
if the number
time
erations
algo-
LPs.
the ellipsoid
in a fundamental
and inway
on upper
certain
and lower
numbers
related
LP. If the problem
a function
computed
method
known
[12].
construct
the problem
of
the
polynomial
existing
eral
special
of
solved in polynomial
gorithm
of variables
strongly
the rational
Adler
and Beling
of the interior
gebraic integers.
a variant
strongly
ficient
can
aland
the number
matrices
In section
to discrete
that
a variant
subring
of the al-
paper,
braic
are
the ellipsoid
algebraic
numbers,
numbers
are generally
our result
in spite
is a notion
ogous in form
not
of the fact
rational.
of problem
and function
a rational-number
Specifically,
number
size of the rational
that
tional
of a finite
numbers.
upper bound
that
and in this
during
data.
only.
problem
Finally,
(we
the
alge-
set of algebraic
we establish
a chain
that
of
leads from
to a problem
method
that
in polyno-
4, we show how we can use
to obtain
complexity
assumptions
In section
all input
ear programming
size of
the full
for LPs
integers
with
equivalences
In section
can perform
bounds
algebraic
by the ellipsoid
results
theory
In sec-
bounds
about
5, we discuss
and computations
algorithm
in section
how one
in the lin-
using rational
6, we conclude
un-
the form of
numbers
with
some
remarks.
of bit
define its minimal
algebraic
coefficients
extension
the course
sense can be viewed
time.
the input
key to
is anal-
of the
2
of the ra-
The degree of this extension
of problem
problem
der several different
we mea-
on the degree of any algebraic
can occur
measure
numbers
these
than
programming
our earlier
in terms
We also view the problem
as members
rather
In particular,
of terminol-
of the paper.
by the working
polynomial
mial
to the binary
problem.
are real
un-
manner:
and number
basic complexity
the linear
are real
The
size that
sure the size of an algebraic
polynomial.
integers
method
that
a brief review
3, we derive
fash-
of computation.
algebra
tion
can be solved
we show that
in a symbolic
in the following
we use in the remainder
coefficients
real-
the same results
model
from
initially
allows
Later we show that,
numbers
2, we provide
numbers).
in image processing.)
In thk
that
than necesssry.
algebraic
lose no generahty
and arise frequently
can be used to solve LPs whose coefficients
we work
In fact, this assump-
that
whose
of this algorithm
(LPs of this kind
convolution
and arithmetic.
ogy and concepts
for LPs whose coef-
are circulant.
input
convenience,
The paper is organized
in both
of the
defined by the input
of computation
der a rational-number
are
our algorithm’s
in the dimension
ion, we can achieve materially
scheme [19] leads to a
algorithm
a model
by handling
can solve LPs whose
of the Tardos
to show that
and the size of the data.
tion is stronger
in which at
are polynomial
The combination
and the size of the data.
time is polynomial
number
quan-
of the degree of the exten-
we work
As an analytic
be
‘rea-
of the LP. These
We use these bounds
under
us to obtain
the basic solutions
of
coefficients,
interesting
on certain
are a function
running
(see
to being
tool permits
sion in which
for
these algorithms
method
involving
and
integers
and lower bounds
to
sev-
in each inequality,
to a particular
polynomial
related
LPs
[1] have shown
point
belong
thk
upper
bounds
and the real senses.
coefficients
with
tities
from
algorithms
problems
In fact,
they
sonable’
is
time
different
for LPs in which
polynomial:
in its own right,
is not
the neces-
In [10] a polynomial
appear
is fixed,
In addition
is a tool
on linear
algebraic
LP, the degree of the extension
real-number
time.
in [11] one is given
3.1).
has shown that
is given for feasibility
most two vsriables
proposition
bad compared
is quite
LPs, Megiddo
classes
and can be
[20].
that
rational-number
involving
the running
dimension
and lower bounds
forms
time, but no poly-
an approach
upper
of our construction
polynomial
If the data
is arbitrarily
feature
are
case it is possible
in which
method
for obtaining
of the
the lower bounds
in this
An essential
of
the bounds
to compute
in polynomial
examples
Using
time.
for computing
In fact,
the ellipsoid
solutions
size of the data
possible
sary upper bounds
that
to basic
in polynomial
it is still
nomial
on the magnitude
data is rational,
of the bit
rational,
with
bounds
Algebraic
Preliminaries
is an
In thk
number
section
minology
algorithm,
theory
as a supplementary
that
of the paper.
dimension.
484
we give a brief
and concepts
from
review
algebra
we shall use throughout
Proofs
and detailed
of the terand number
the remainder
discussion
of the
results
stated
algebraic
here can be found
number
We begin
theory
by stating
shall use to describe
texts
the basic
Let
a polynomial
in indeterminate
terminology
F(t)
t with
coefficients
coefficient
if qd =
mial
1.
of F,
F
is reducible
if there exist polynomials
coefficients
and strictly
rationals
rational
positive
with
related
Definition.
numbers
that
A complex
such that
Clearly,
are intia subset
is central
many
polynomials.
one polynomial
Proposition
as being
2.1
CY is the
polynomial
The
Let
root
with
proposition
F with
root
of
whose
existence
minimal
Let
polynomial
gree of its minimal
quantity
as deg(a).
polynomial,
is asserted
in
By the fundamental
G has d (possibly
We call al,...
, CYdthe
complex)
(Note
the conjugates
of a include
are distinct,
minimal
polynomial.
and that
thk
roots,
and a.
a
operations
We define
Let a be an algebraic
this set
that
number,
subfield
cent ains both
We de-
of the complex
a and the rationals
Q(a)
Q.
to be deg(a).
property
d.
important
every
of degree
~
c
at most
CXd-l for
a unique
exten-
for our purposes.
Let CYbe an algebraic
2.3
Then
of single algebraic
Q(cx)
d.
has the representation
The
gates
theorem
CO?@@fM
form
number
of
an algebraic
is
Moreover,
every
~ = q.+
qla
set of rational
/3 E
+ ...+
coefficients
qo,ql>. ... q&l.
CYto be d, the de-
One can show that the conjugates
numbers
Proposition
number
with
and we write
numbers
of arithmetic
to be the smallest
numbers
Q(a)
polynomial.
~1,...,~d.
number
V
if 1 E
of a subfield.
fine Q(a)
degree
polynomial
number
algebraic
rational
The following
of an algebraic
of algebra,
that
numbers
numbers
numbers.
using
extension
number.
G of degree d. We define the
number
The
sions is particularly
a be an algebraic
degree of the algebraic
2.2
of the complex
up’ by a sequence
q&l
Definition.
of the complex
of the complex
‘built
irreducible
as the minimal
of its minimal
arith-
extension
of the raWe call Q(cY) a single algebraic
tional numbers by a. We define the degree of the
coefficients.
2.1 is known
in terms
rational
importance.
monic,
of a. We define two key attributes
number
is the
algebraic
be an
rational
under
w.
in terms
we distinguish
of particular
a
V
A subset
Definition.
these,
of a unique
polynomial
to our work.
a is an azgebraic
number
Among
are closed
in-
standard
Given an algebraic number a, we shall often be
interested
in the set of all numbers that can be
= O.
each algebraic
for sets that
Proposition
over
numbers,
number
F(cY)
Before
operations.
/3, @ c
coefficients
if there exists a polynomial
number
Then
rational
classes are
degrees such that
F is irreducible
to the algebraic
of the complex
coefficients
terminology
subjield
Polynomials
other
property.
6 V, we have –a, Q + ~, @ G
V. A subset TV of the complex numbers is called
if 1 E W and if,
a subfield of the complex numbers
for any a,~ E W with a # O, we have –a, l/a, a+
the rationals.
mately
its members;
the
and if, for any a,~
polyno-
the
otherwise
= F1 (t) F2(t);
over
enjoy
arithmetic
several such classes, we review
Definition.
this
F1 and F2 with
among
on the basis of this
troducing
numbers
the class is closed under
is called a subring
num-
and we write
We say F is a monic
as deg(F).
operations
metic
We define the degree of F to be d, the index
quantity
F(t)
we
+ qo be
= @d + . . . + qlt
classes of algebraic
that
defined
nals, R for the reals, and C for the complex
of the leading
Certain
property
Q where qd # O. (We use the stanZ for the integers, Q for the ratio-
~
dard notation
bers.)
on
polynomials.
Definition.
go,... , qd
in most
(see, e.g., [5], [16], or [17]).
say
of a.
a itself.)
next
tension
in terms
that
Proposition
degree d with
they share the same
member
number
F
of an algebraic
proposition
of every
the
of the conjugates
2.4
conjuex-
of the algebraic
Let a. be an algebraic
conjugates
~j,
with
of/3
bers of the collection
485
the
algebraic
defines the extension.
be a polynomial
conjugates
characterizes
of a single
where j = 1,...,
rational
= F(a)
{F(cYj);
number
coefficients.
are the
distinct
j = 1,...,d}.
of
d. Let
Then
mem-
As a natural
single
generalization
algebraic
extension,
of the notion
of a
we have the following
algebraic
definition:
Let al,...,
Then
we define
Q(cq,
. . . . an)
Rather
the
cr~ be algebraic
numbers.
algebraic
extension
muJtip/e
to be the smallest
al,. . . ,CXn ~d
Then
field that contains
teger
surprisingly,
Proposition
2.5
bers
with
there
exists
I’1~=1 dj
every multiple
extension
is
for
... dn,
an algebraic
in proposition
the definition
Q(a)
respectively.
number
number
algebraic
Then
O whose
2.5 is not
number
of a single
on the other
Proposition
such
2.6
that
“””
is an
light
a,
existence
unique.
it
is asIndeed,
is evident
algebraic
Let
hand,
is uniquely
CY and
Q(a)
=
~
be algebraic
Q(/3).
qlt
+
if
and
inator
following
number.
only
if the
has in-
proposition,
number
one can
as the quotient
of an al-
and m integer.
+ qo be a polynomial
with
Let
H(t)
qo, . . .> !/d-l.
the product
za is a root
Then
d–1
q&~t
denom-
if a is a root
of F(t)
+
coefficients
Q, and let z be a common
E
for
of H,
= td + .Zq&ltd–l
+
the coefficients
of
+ .zdqo. Moreover,
are all integers.
Although
tient
=
as can
the rationais
td +
that
deg(cr)
Then
integer
algebraic
=
F
num-
be an
of a over
2.8
from
defined.
a
algebraic
integer
. . . + zd–lqlt
extension
Let
of the
qo9. . . . qd-1
always
an algebraic
of two algebraic
algebraic
integer.
however,
under
integers
the quo-
is not, in general,
The algebraic
addition,
number,
integers
an
are closed,
multiplication,
and nega-
tion.
deg(/3).
In light
defining
of proposition
2.6, we are justified
the degree oj a multiple
algebraic
to be the degree of any equivalent
Combining
propositions
every member
Proposition
in
subring
extension
extension
function
gebraic
Corollary
2. I
cY~) be a mtdtiple
Let Q(crl,...,
of degree d.
CKn) is an algebraic
most d. Moreover,
the representation
number
At this point
every
coefficients
we introduce
numbers
programming,
that
prove
than the full set of algebraic
Definition.
A complex
integer
if there
integer
coefficients
exists
number
a monic
such that
A
to be the set of all al-
and we define
to be A. n 1?,
&
integers
that
are
also real
As a final
preliminary,
we define
some notation
matrices
concerning
and the set of all column
components
for
belong to
r-vectors
K. Given a matrix
M, we use 111~ to denote
the transpose
whose
or vector
of M.
subset of
to
polynomial
3
Linear
Programming
Algebraic
easier
over
Integers
numbers.
We consider
a is an algebraic
F(a)
the following
integers.
cY~) has
we turn
to be somewhat
to work with
a
matrices and vectors.
Given a set K,
we use Krxs and .Kr to denote the set of all r xs
qo, . . ., qd–1.
when
form
number
“ + q&ldd-l
a particular
will,
aL
to have
for the algebraic
We define
integers,
integers
numbers.
/3 E
of degree at
c Q(cq,...,
P = qo + qlo +””
set of rational
the algebraic
Then
there exists an algebraic
0 of degree d such that every/3
a unique
algebraic
numbers.
the set of all algebraic
extension
Q(Crl,...,
The
convenient
notation
Definition.
alge-
braic number.
gebraic
prove
shorthand
can be ex-
of a single
2.9
of the complex
It will
single extension.
2.3 and 2.5, we see that
of a multiple
pressed as a polynomial
linear
2.7
Proposition
6 of degree at most
= Q(a + 1). The degree of a single algebraic
extension,
The converse is not true,
polynomial
gebraic
num-
is also an
coefficients.
In
Q(w, . . . ,~~) = Q(o).
SUCh that
any
bers
all,.
an be algebraic
integer
the next result.
view an algebraic
Let al,...,
degrees
algebraic
a
minimal
Q.
every algebraic
number.
Proposition
also a single extension.
serted
that
be seen from
Definition.
The
It follows
the following
(P)
F with
max
St.
= O.
486
linear
program:
CTX
Ax
<
b,
the
where
A c ~~
c c ~~,
with
‘n
the
complexity
It is important
ity
of the
full
ber
integer.
It
(P)
numbers
—
— then,
volves
not
into
only
to show
the input
We
this
that
will
the
tools
the complexity
mediate
functions
specialize
to real
of inequalities
are not
bers).
of the
step
define
(there
toward
the
that
the
(iii)
to
roots
of the
norm
S(a)
write
where
= max{lcql,.
we use the standard
G(t)
of the
where
~ is the complex
The
braic
complex
of degree
following
proposition
and metric
Proposition
conjugate
properties
3.1
term,
(iii)
(iv)
the
for
that
by an ar-
from
the definition
we may
integer,
assume
we can also
It is clear that
by t to obtain
the
smaller
degree, contradicting
Hence,
coefficients
1.
we have IZo\ ~
between
we see that
JJ$=l laj I ~
implied
a (monic)
the two
.zO= JJf=l
Using
aj.
the proof
It
the inequality
by the definition
of conjugate
we then have ICYI= lcq I ~ (S(a) )l–~,
of the proposition.
which
•1
fi,
useful
alge-
use proposition
results
inequalities
norm.
concerning
whose
of the notation
any inte-
Definition.
jkth
S(ap)< S(a)s(p),
the matrix
entry
3.1 to derive
matrices
coefficients
gers. These results
gers a and b,
Let M c A’”
M
several
and systems
are algebraic
are most easily stated
introduced
of M.
ICYl
< s(a),
If a # O, then [al > (S(a) )l–d,
G(t)
of G(t).
of G(t),
< S(a)
norm,
A. Then,
+ ]blS(/3),
we have
used in (i).
zo, . . . . z~–l.
by matching
We shall
main
distinct
Z., must not be zero, since if it were
of strictly
expansions
of /3).
of the conjugate
Let a, /3 c
(i) S(WI + I@ < lalS(a)
(ii)
gives
Thus,
of a, where
coefficients
1. But,
We
]/31 for the mag1~1 =
=
= (i! - a~)(i! - a2) 0.0 (t - ad) be the
the irreducibility
Iajl
~ (i.e.,
the
follows
is obvious
zero we could divide
. . . la~l},
number
S(P)
= td+ zd_ltd–l+”. o+ Zlt+ Z. for some
completes
nitude
statement
to that
statement
constant
of a to be:
notation
of
+ b~p(oj)l}
Since a is an algebraic
integer
a
min-
of a.
and
of aa + b~ are
polynomial
a = al.
follows
S(a)
that
conjugates’
maxj{[F’a(Oj)l}
of this
similar
This
polynomial
conjugates
rational
+ lblS(~).
minimal
num-
of the
=
are the
of {aFa (Oj) + bFP (Oj) }.
s(a).
(iv) Let G(t)
which
we define
al ,. ... ctd be the
with
we know
of
bounds
goal,
deg(0),
S(a)
The proof
(ii)
systems
integer
Fp
2.4,
S’(aa + b~) = maxj{la~~(oj)
im-
this
1,.,.,
conjugates
members
of a.
conjugate
=
< lalS(a)
Our
entities
and
Since aa + b~ = aFa(6) + bFp(6)
is also a rational
polynomial
in 6, we also know
gument
of complex
F.
proposition
maxj{[Fp(Oj)l}.
in
is no need
progriwns,
Let a be an algebraic
let
Hence,
use in
we discuss
in terms
magnitude
polynomial
Definition.
linear
of
6.
later.
and lower
until
defined
As a first
measure
of a G ~
and
generally
Oj, j
in-
is polynomial
upper
numbers
that
(P).
the
where
the
integers
we shall
~ have
bers of {Fa (Oj) } and {F” (Oj) }, respectively,
an
It is straightforward
that
polynomials
By
a and
Q = $’a (0) and ~ = F’(8)
from
a LP
be defined
some
that
an al-
num-
and the ob-
of problem
goal is to establish
on certain
d, and
integers.
transformation
O such
exists
mem-
are algebraic
problem
number
representations
2.1, there
of a and ~ are the distinct
and
given
corollary
the conjugates
Recall
integer
By
coefficients.
we can transform
equivalent
develop
for
instead
algebraic
integer,
an
gebraic
is to
an algebraic
the constraints
measures
now
analyzing
numbers.
necessarily
algebraic
that
integers
if we are
(i)
unique
the coefficients
by an appropriate
problem
section
we lose no general-
algebraic
that,
in which
by multiplying
jective
that
Proof.
tJ c A.&,
(P).
of an algebraic
follows
but
in this
we can view
quotient
the form
imal
the
2.8 that
as the
the
with
set of algebraic
proposition
rank,
of problem
to note
by working
column
Rn. Our goal
and z 6
bound
full
of
inte-
in terms
below.
and let Mjk
We define
the conjugate
denote the
norm
of
to be
Z’(M)
=
rn,y{s(Mjk)}.
where d =
Let
deg(a).
487
I denote
the rank
of M.
Then
we define
the
conjugate
Proposition
size of the matm”z M to be
L =
L(M)
= tlog(lz’(M)),
L(M,
vertex
nent
where
by log
ditionally,
we mean
we define
the
the
base-2
be the degree of the multiple
Q({~j~}),
logarithm.
Ad-
of the matriz
degree
algebraic
programs.
notation
when
Let (Q) denote
MU s g}, where
Let W denote
g c AL,
the set of all entries
We define the conjugate
norm
Let 1 denote
the rank
L(M,
gram
(Q)
braic
as deg(M,
of M.
{max
fTv
respect
degree
Q(V),
(Q) to be
Proof.
The statement
follows
3.2 and Cramer’s
rule.
Propositions
analytical
of linear
3.1 through
tools.
almost
Using
of a matrix
to our later
Proposition
[7] or the interior
of the multiple
this
alge-
quantity
notation
with
in time
inequfllties.
point
this section,
3.2
determinant
that
are funda-
Let
B
~
~rxr,
and
let
If B is nonsingular
L
=
then
for problems
with
and Steiglitz
[15].
of real numbers
operation.
We refer
derived
under
S(det(B))
(iv)
2il-~JL
< I det(B)l
The
properties
apriori
analysis,
that
performs
ad-
division,
model
and
it as real-number
of the theory
we
and
time
as the
per
real-
to algorithms
algorithms.
of general
(See
compu-
over the real numbers.)
program
(P)
the input
consists
of an instance
of the
of the following
items:
< 2L.
statement
of conjugate
As a consequence
tain
this
by
< 2L,
(ii)
Proof.
to
the
data
in constant
of computation
We assume that
< d,
that is centered
multiplication,
comparison
model
part of
rational
we have a machine
number
dimension,
We shall loosely follow
of the complexity
subtraction,
linear
(iii)
a procedure
analysis
assume that
method
[6] to solve problem
method.
tation
deg(det(B))
of the ellipsoid
size. In the remaining
we outline
given
our basic
we can show how to
on the ellipsoid
dition,
work.
3.3 constitute
in the problem
degree, and conjugate
[2] for a treatment
(ii)
from propo•1
method
polynomial
Papadimitriou
we now derive some char-
and d = deg(B).
L(B)
trivially
them,
any variant
For the purposes
mental
< d,
sition
modify
g, f )).
and we write
fixed notation,
acteristics
compo-
Gj = ~j[~,
g, and f.
in M,
We use analogous
to systems
Having
every
in the form
:
We define the conju-
program
be the
g, f ).
@ is a
and f E d;.
we define the degree of the linear pro-
extension
Then
and let
Suppose
of the linear program
g, f ) = tlog(fT(M,
to
< g}.
@j of G can be written
(ii) deg(cxi) < d, deg(~)
(P)
Additionally,
g).
g, f ) = rnn~{S(a)}.
size of the linear
gate
R’ ]Mv
c
g c A;,
deg(M,
extension
(Q) to be
T(M,
Let M c A~s,
d =
where
w deg(M).
discussing linear
the problem
M G d;’,
of {v
3.3
and
M to
~d We write this qu~tity
We use similar
g)
bounds
norm
easily
from
given in 3.1.
of the last proposition,
on the magnitude
gate norm of the vertices
ing coefficients
follows
belong
of polyhedra
the
•1
we ob-
(iii)
deg(A,
b, c),
L(A,
C).
Our
time
b,
strategy
solvability
polynomial
and conju-
problem
(P) to a problem
whose defin-
method
to A.R.
488
for
establishing
of (P)
the
is to establish
transformations
polynomiala chain
leading
of
from
that cm be solved by the ellipsoid
in polynomial
time.
We begin
rem
by noting
of linear
gram
(P)
closed
that,
given
programming,
is no harder
the duality
solving
than
the
solving
a system
it gives us a report
that
the instance
a feasible
to the instance,
Proposition
rank,
g
deg(M,
3.4
c
Let M c d~$
let 6, A c
g) < 6 and L(M,
exists a real-number
if
is infeasible
or
Then
there exists
that solves the linear
mial in m, deg(A,
Proof.
program
By writing
sibility,
algorithm
of primal
follows
bounded
from
the corresponding
by noting
that
system
of linear
Proposition
umn
above by a polynomial
associated
g c
d = deg(M,
ones.
by solving
Let
M
A;,
c
and
gorithm
a closely
d~s
with
let L
=
full
a solution
to the other
and
4
by Papadimitriou
173–174]
for
tems with
and Steiglitz
rational
of that
[15, lemma
result
coefficients.
the coefficients
The
main
tions
8.7, pp.
concerning
We
and polynomial
require
we can state
one
more
the main
preliminary
result
the resulting
that
system
is
it can be solved by
result
Other
about
differ-
In
problem
before
every
489
section,
degree
different
and
define
Input
input
a solution
bounds
that,
bounds
problem
input
In many
of
situa-
apriori
from
ways
conjugate
form,
to
and
information
coefficients.
we discuss
Because
for prob-
on the degree
may not be known
if possible,
pro-
in addition
the instance,
includes
assumptions
coefficients.
possible
complexity
size of the instance.
be deduced,
this
several
of the section.
that
the individual
problem
we derived
the assumption
these bounds
so must
sys-
•1
forms.
section,
of (P)
and conjugate
given
ence lies in the use of the properties
of conjugate
3.1) to derive lower bounds
on
norm (proposition
linear
from
we can always struc-
to
cedure and associated
lem (P) under
in time polynomial
is an adaptation
an analogous
It follows
a
(cf. [15] or [4]) in polynomial
Extensions
an instance
The proof
is, in
of solving
•1
In the previous
inr.
Proof.
to that
Models
to one system, finds
system
method
closed in-
problem
col-
L(iW, g)
al-
a solution
of linear
open inequalities.
so that
3.4, it suf-
of a polynomial
3.5, thk
well-bounded
in time
b, c).
of proposition
3.3 and 3.6, that
the reduction
(P)
and L(A,
systems
algo-
time.
~ g
given
program
reducible
of linear
the ellipsoid
related
there exists a real-number
that,
and polyno-
a real-number
b, c),
By proposition
sufficiently
of lin-
if and only if the closed system Mv
Moreover,
(proposition
of the section.
the existence
for solving
propositions
Let e, denote the r-vector
of all
2dLMv < 22dLg + er
Then the open system 2
is feasible
result
In light
polynomially
system
(P).
g).
is feasible.
coefficients
lies in the use
norm
exists
deg(A,
(Outline)
turn,
in
with
There
in m,
equalities.
open inequalities.
3.5
rank,
rational
•1
solves the linear
that
algorithm
size of
we can solve a system
ear closed inequalities
3.6 is an adap-
on linear
rithm
ture
we note that
lower bounds
3.1
Proof.
❑
Next
= {v c
difference
of conjugate
fices to demonstrate
(P) in this way are in-
quantities
systems with
Theorem
polynomial
we can re-
and conjugate
dividually
fea-
closed inequali-
ties. Proof of the proposition
from
Let K
g).
given for the analogous
We now state the main
of objectives
the row dimension,
the system constructed
and let
mial forms.
b, c).
to linear
degree,
g c AL,
in time polyno-
and equality
programming
in r, 6,
proof
The main
3.1) to derive
values as a system of closed inequalities,
duce linear
concerning
(cf. [4] or [15]).
of the properties
(P)
E A~S,
of proposition
result
g, 6,
the conditions
dual feasibility,
proof
of a standard
that, given M,
and L(A,
b, c),
The
tation
there
a real-number
Let M
g) and d = deg(M,
Suppose
and A, solves MU < g in time polynomial
and A.
L(M,
Proof.
with full column
R be such that
g) < ~.
algon”thm
we
of inequalities
3.6
R8 IMv < g}. Then if K is bounded and nonempty
the s-dimensional
volume of Ksatisjies
vols (K)
>
2-4sdL .
as appropriate.
and
A:,
L =
pro-
of language,
solves
solution
linear
a set of linear
As a matter
inequalities.
say an algorithm
Proposition
theo-
of obtaining
size bounds
about
the form
we cannot
the
discussion
under
of the
anticipate
is more
illustrative
than
through
exhaustive.
an example
and conjugate
the given
of this
from
size,
data,
number
theory
that
input
that
We conclude
ing how our results
integers
that
tools
for
by show-
First
note
multiple
size of
from
is
1+c),
L = L(A,
b, c).
For every
4.1.
in addition
know
to the actual
a G V,
suppose
numerical
a set of four integers
by a function
z2,
Z3 such
the problem
of the
to us. As it will
turn
for thk
is the total
purpose
representation
members
that
where
Recall
S(a),
a useful
input
number
of the integers
hat
measure
of bits
in a
CYG V!. Hence, we may first bound
gate norms
of the individual
calculate
a bound
input
thk
on L from
these
of the conjugate
norm
(proposition
(deg(fi))(deg(fi)
=
8,
used
the
that,
W,
example,
<
Izol + 1211s(/2’)
z2&+
+
bounds
quanti-
of finding
square-root
of bounding
(or bounding)
an effective
F(t)
of another
integer.
= t2 – p is the
minimal
gate norm
+ [231s(/6).
W,
the
are, in
numbers
algebraic
It
490
in
observations
be quite substantial
(cf.
for many
bound
S(a)
norm
that
(P)
in m and E.
given in example
4.1 illus-
input
forms.
for each a E U. If
for a in terms of other alof the conju-
to that
of those algebraic
smaller
Q(W).
of boundintegers.
a set of algebraic
num-
than W itself — such that
extension
the product
integers.
based on our
3.1 implies
we use the properties
algebraic
includes
d is at most
It is easy to show that
insignificant
polynomial
d, we identify
bers — hopefully
norms of the
of@
=
@
to reduce the problem
To bound
to that
such that p is not the
polynomial
=
on d somew-
and Q(ti,
Although
strategy
L, we first
the multiple
integer
deg(fi)
(W)(W)
for d and L, theorem
gebraic integers,
terms.
Let p be a positive
square
the conjugate
that
we note that,
The ad hoc analysis
trates
3.1) we
L is thus reduced
)(deg(fi))
the savings from
can sometimes
ing the conjugate
‘I’he problem
from
the degree of
the bound
@
can be solved in time
z3/ii)
lz21s(ti)
evThis
4.2).
To bound
S(ZIJ+ Zlfi+
that
W).
It follows
than
fact
since
we know a representation
=
n,
= 2.
As a final observation
and
have
s(a)
It is obvious
2.5, we then have
we have d < 4.
simple
example
ties. Proceeding
along these lines, we note that if
a = Z. + Z1W + .22@+ 23 m, then by the properties
of the
~ denote this last quantity
<
Q(ti,
degree
coefficients
@.
we can tighten
of this kind
of
the conju-
coefficients
W,
Letting
by noting
Hence
We use
function
to
fact, the same, and so must have the same degree.
that deilne the
L is a nondecreasing
in
degree
Q (iii).
d is no larger
= deg(fi)
extensions
this number.
that
W).
Actually,
available
bound
of the
the
to Q(fi,
Q (W,
C
we have
deg(w)
conjugate
quantities
of T! in terms of the square roots.
E to denote
then
out,
W,
value of a, we
.zo, ZI,
Our first goal is to bound
binary
definition
d equals
extension
2.1 that
d < ~
that,
o! =z0+ZI&+Z2&+Z3&.
size, L,
the
and using proposition
in A, b, and c.
this
‘
d, the degree of (P).
of the problem
Q(W)
corollary
we shall use I@to denote the set of all
Example
by
program,
algebraic
Q(W,
Additionally,
lz31fi,
Using
for us to bound
the form
implies
entries
lz21ti+
ery a ~ IP also belongs
d = deg(A,
= @.
then gives
defines L, it is straightforward
that,
of a linear
conjugate
I–@l}
L s ml?.
It remains
the algebraic
we shall used to denote
the
that
show that
are @ and –W.
inequahty
for any a E Ill.
the formula
numbers.
this section,
in our earlier
holds
which
We then
the section
from
of@
= max{l~l,
s(a) < Izo[ + Izllfi+
the use of
the degree and L to denote
(P);
nature
some results
forms.
the conjugates
Substitution
from
The
illustrate
generalize
to the algebraic
Throughout
apparent
that
Hence, we have S(@)
degree
serve as useful
for other
several examples
these results.
not
leads us to consider
bounds
consider
although
follows
by working
the problem
are easy to bound.
analysis
obtaining
We begin
in which
generated
We then
by these
can claim
of the degrees
that
of these
The
input
form
considered
special
case of the form
integer
and
that
to use the
to find
we must
jugate
know
norm
caveat;
any
attempt
where
conjugate
norm
it
bounds
We begin
we must
from
auxiliary
the
with
along
In addition
ing:
lines
(i)
~(t)
be a polynomial
We
define
maxj
integer
with
the
{ lzj [}.
=
(ii)
com-
for
some
polynomials
coefficients
height
of
the
define the
We
to be the height
Zo,...,
Let
and
height
ZIi
F
E denote
+ ZII
norm
to
the
ation
Next
we state
a well-known
jugate
norm
of an algebraic
that
total
by finding
of each member
height
4.1
h.
Then
Let
S(a)
the thethe con-
polynomial
For
the
height
the
purpose
of
of an algebraic
degree
nomial
and
Proposition
with
4.2
integer
Then
the
evey
gree at most
algebraic
suffices
Let
F
of F
1,
is an algebraic
4 and height
at most
e.g.,
proposition
As
for
we next
arises
matrices
proof
of
in the
of the
theory
use
this
To
h.
of LPs
of the
whose
last
bound
d < deg(u).
d < p.
bounds
of
F is not
give
In particular,
since
1, proposition
at most
F
4.2 implies
4P. Proposition
in turn
the
us enough
4.1 then
gives
< 4P+’ ~
Izjl.
bound
it is easy to show
that
L <
d, note
But
Q(V)
w is a root
(In
fact,
theory
Q(u),
and
so
= W – 1 and so
at most
it is possible
d somewhat
number
c
of F(t)
4.2 has degree
on L and
from
that
p. Hence
to sharpen
by using
concerning
we
our
additional
the roots
unity.)
preceding
an input
Although
S(u~).
< 4P+1, which
by proposition
4eh.
for
for all j G
3mE.
of de-
a generalization
bounds
1.
as
results
obtain
S(u~),
j=(l
4.2.
an illustration
results,
that
[13]
then
lZjlS((4)j),
height
S(CI)
have
See,
p and
poly-
height
integer
conjugate
zjw~,
p–1
polynomial
and
on the
of w~, it does
to bound
S(W~)
Using
degree
above.
and
is a root,
be a monic
in a binary
If a = ~~~~
to know
integral
integer
= tp -
wj has height
gives
degree
shows.
coefficients,
root
it
of any monic,
proposition
F(t)
information
of
4.1.
the
integer,
height
of which
the following
bounding
of bits
Zj in (ii)
on bounding
polynomial
has degree
of proposition
ZP-l
{o ,. ... p – 1}. First note that since@’=
1 for all
“=
-1,
it
is
clear
that
wj
is
a
root
of the
07...,P
J
to its height.
integer
zo,...,
j=O
< 2h.
[18] for proof
integers
p–1
that
See, e.g.,
for
ZjkJj
a bound
of Il.
S(CY) < ~
from
a be an algebraic
CY G I&
we have
minimal
Proposition
number
integers
be
relates
integer
every
zjwj.
polynomial.
result
numbers
the follow-
a set of integers
of the
We now concentrate
ory of transcendental
that
as a = ~~~~
a = ‘@
of an algebraic
of its minimal
we know
p such
a G*,
that
represent
z~ 6 Z.
polynomial
integer
every
such
by introducing
0.0-1-
that
obtain
integer.
zdtd +
and p is integer.
Zo, ... ,zp-l.
We begin
Let
~= W
to V!, suppose
can be written
and
integers.
Definition.
where
a positive
integer.
to
information
concerning
pth root
and
the degree
an algebraic
these
and con-
degree
how
primitive
form
a general
algebraic
considering
Let w be the first
is,
e2uilf’,
w =
we ultimately
bound
of an individual
terminology
algebraic
the
program,
is worth
associated
additional
reflects
4.2.
that
is clear
this
the degree
This
zj is an
outlined
with
to bound
size of a linear
Therefore,
monly
about
Example
of unity;
It
approach
problems
something
a point
these
for
of Oj, for each j.
in
conjugate
reach
bounds
integer.
general
4.1 is a
where
ZjOj)
Oj is an algebraic
in order
above
in example
a = ~
Combining
form
our bounds
3.1, we see that
coefficient
mial
are circulant.
491
in m, p, and
(P)
E.
on d and L with
can be solved
in time
theorem
polyno-
of
LPs
of the
analyzed
sults
considered
in a different
in example
manner
but
with
Recall
4.2 are
similar
re-
is a real
next
consider
knowledge
braic
integer
in the
Example
both
~ =
grees
the
of the
number
coefficients
Using
of the
gebraic
in
a binary
4.1,
that
is at most
Using
the
Using
of the
Recall
degree
the
an isolating
that
that
define
can be solved
in
there
the
in time
To
algorithm,
extension.
It
given
can find
L ,
ence,
polynomial
example
it
4.3
the polynomials
associated
cients
and integral
are monic
is easy
The results
to
the
with
to
generalize
case
in
are similar
not
with
coeffi-
necessarily
the
to the irreducible
done
case.
bit
5
Rational-number
Model
of
complexity
are derived
An
lows
bounds
under
real-number
this
assumption
can
achieve
of algebraic
sults
(somewhat
the
for
numbers
the
messy)
but
than
necessary.
same
results
symbolic
conjunction
3. We provide
below,
defer
details
that
In
of computation
in
3 and
arithmetic.
is stronger
scheme
in sections
of computation
and
model
of section
procedure
input
materially
rational-number
well-known
given
a model
exposition
to a subsequent
of such
time
given
sum,
differ-
polynomial
triplets.
of
in
We can
time.
representing
and
we can establish
programming
in much
with
similar
results,
overall
bound
is polynomial
problem
that
degree,
represent
we must
the
as is
and the
input
co-
soid
in the
and
standard
procedures
of each
occur
stay
the
during
algorithm.
rational-number
of particular
method
of working
of computation
to control
that
programming
of ordinary
a
a
be careful
linesr
subalgorithms
complication
model
numbers
a
re-
extra
that
analysis
One
the
unavoidable
a rational-number
of the
manipulation
with
we
their
numbers,
and
with
the rational
fact,
by using
an outline
a full
4
al-
under
[9],
efficients.
Computation
The
us to ma-
as represent
numbers,
for linear
triplets
allow
in polynomial
dimension,
size of the
programming
of Lovasz
a means
3. The
in the problem
(see [3]
results
in
of the
manner,
in section
poly-
algebraic
represents
triplets
bounds
same
two
algebraic
complexity
ir-
an
algorithms
coefficients
must
or quotient
Equipped
co-
such
in time
as well
some
for
lengths
manipulating
size of the
of a linear
scheme
that
two
exists
size is
construct
size of its
from
product,
gl,
algorithms).
triplets
a triplet
there
encoding
well-known
numbers
follows
triplet
of G are suffithat
we can
as part
also compare
which
the problem
but
binary
encoding
roots
of a polynomial
the triplet
that,
the
does
by storing
binary
several
algebraic
them.
Since
encoding
fact
of such
be useful
nipulate
of
the
4.2,
in the
the
binary
are
ratio-
Q but
a, we can rePre-
machine
whose
In
with
of G.
the roots
for a survey
al-
on d and
isolate
nomial
use
degrees
of G.
if a
polyno-
contains
well-separated
in the
interval;
minimal
defines
that
and
conjugates
Hence,
an interval
root
interval
of G.
that
algebraic
of the
small
efficients
proposition
of a multiple
product
to
make
other
coefficients
polynomial
prob-
multiple
exists
be shown
ciently
E.
proposition
reducible.
d, we again
the
with
of the
by the
3.1 and our bounds
(P)
results
above
any
can
total
d < ~.
theorem
in m, ~, and
the
there
with
say [ql, qz], that
contain
It
de-
is straightforward
degree
numbers
we see that
the
it
Q(W).
we have
denote
representation
To bound
extension
algebraic
associated
E
from
d is the
extension
Hence,
let
number
sent a in a rational-number
the
of the
then
the
are distinct.
(G; 91, ~2) unambi~ously
polynomials.
size.
that
of a and
product
polynomials
Also,
2.5 states
the
the
L is bounded
fact
value
that
2.4 that
number
algebraic
G,
q2, and
of W.
proposition
lem encoding
of each alge-
!I?, suppose
proposition
nal endpoints,
of a.
of these
that
6
numerical
minimal
of bits
mial
direct
not
a
HaeW deg(a),
members
we have
polynomial
every
polynomial
Let
in which
problem.
For
4.3.
minimal
show
LPs
of the minimal
we know
from
of an algebraic
in [1].
We
the
form
concern
Gaussian
analysis,
under
intermediate
bit
size of
the
course
As
in
the
the
two
LPs,
are the
elimination.
ellip-
Also
we can ensure
control
is
that
by relating
as
these
the
size
number
to that
of the
input
ellipsoid
method,
this
can be
data.
of the
In the
paper.
492
case of the
done
with
variant
e.g.
the
aid
of the
developed
for
[15] or [4]).
rational
numbers
lipsoid.
Exact
the linear
(cf.
Briefly,
(triplet)
proposition
range
using
the
computation
[4]).
so that
But
subdeterminant
encoding
triplet
by using
that
size of the
we can
each
the
the
input
results
tend
the
eral
algebraic
linear
input
matrix
algebraic
and
extension
the
degree
by
used
matrix
in the
defined
with
(see the
previous
re-
a polynomial-time
coefficients
al-
belong
classes
to
the
can
are other
simultaneously
diago-
of matrices
of small
algebraic
subring
in [1]).
the
families
progress
using
whose
do exist
in identifying
results
from
sentations.
We
subsequent
paper.
the
plan
to
to
we can
families
from
similar
Equivalently,
is composed
such
be shown
by arguments
LPs.
ex-
investigating
of problems
for circulant
that
which
in [1] to gen-
is worth
polynomial
there
paper,
results
it
pears
by
in this
programming
necessarily
erable
its
results
numbers,
other
ndlzable
size of any
by a polynomial
scheme
whose
of the
ask whether
matrix
entries.
6
light
those
in section
encoding
LPs
to be strongly
ar-
intermediate
of the
is bounded
multiple
for
whether
3, it is easy to show
of the
In
for
elimination
elimination,
Tardos
conjunction
subring.
3.5).
is a subdeterminant
(see, e.g.,
in
gorithm
only
A feasible
Gaussian
of the
mark)
(see,
each el-
is used
subroutine.
case of Gaussian
number
variant
precision
LPs
and represent
arithmetic
be recovered
In the
finite
we use limited-precision
to calculate
separation
can then
standard
rational-number
diagonalizing
numbers
Indeed,
and
them
theory
report
(not
it ap-
that
can
considbe made
of group
on this
repre-
topic
in
a
Remarks
Acknowledgments
1) Using
tion
LPs
with
LPs
in
the
given
the
[1], where
tionals
We
also
In
[1]
it
is
strongly
problem
discussed
ity
cient
by
in
valuable
work
with
case.
ence
Rothblum,
integers
research
about
integers
funded
under
grant
this
financial
appreciate
the
by
to
to Michael
to algebraic
was
and
on extending
cyclotomic
comments
Foundation
greatly
Uriel
discussions
We are also indebted
algebraic
This
Khachiyan,
for
perceptive
from
the ra-
to
make
the
Todd
generalization
numbers.
the
National
DMS88-10192.
SciWe
support.
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may
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the
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in which
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scheme
extension
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in
quite
in sec-
algorithm
right-hand
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is independent
Tardos’
a similar
with
case,
and
to the
the
coefficients
rational
extension
tools
Tardos
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objective
this
by
whose
resulting
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