28
t414
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ON THE CO>G'LEXITY OF FOUR POLYHEDRAL
SET CONTAINMENT PROBLEMS
by
Robert M. Freund
and
James B. Orlin
Sloan W.P. No. 1527-84
January 1984
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
ON THE COMPLEXITY OF FOUR POLYHEDRAL
SET CONTAINMENT PROBLEMS
by
Robert M. Freund
and
James B. Orlin
Sloan W.P. No. 1527-84
January 1984
L
Abstract
A nonempty closed convex polyhedron
X = {x:Ax<b}, where
(A,b)
,
is called an H-cell, or in
where (U,V) are given, in which case
The problems of determining if Xcy, where
an H-cell and Y is a closed solid ball,
X
X
is a closed solid ball and
Y
(ii)
X
is an H-cell and
Y
(i)
X is
is a W-cell,
is a W-cell, are all shown to be
NP-complete, essentially verifying a conjecture of Eaves and Freund.
Furthermore, the problem of determining whether an integer point lies in a
W-cell is shown to be NP-complete, demonstrating that regardless of the repre-
sentation of
X
X
This note discusses the computational complexity of certain
set containment problems.
or (iii)
can be represented either as
are given, in which case
the form X = {x:x=UA+Vp ,EX .=l,X>0,yr-0}
is called a W-cell.
X
as an H-cell or W-cell, this integer containment problem is
NP-complete.
0749548
,.
1.
Introduction and Preliminaries
A nonempty closed convex polyhedron
can be represented either in
X
the form X = {x:Ax<b}, where (A,b) are given, in which case
an H-cell
(H
for halfspaces)
,
are given, in which case
(U,V)
X
is called
or in the form X ={x :x=UX+VtJ ,EA .=1,X >0,y >0} where
X
is called a W-cell
(W
for weighting of points)
The computational complexity of many problems related to polyhedra depend on
the polyhedral representation as an H-cell or a W-cell.
a linear program, which can be stated as maximize
is a polyhedron.
If
X
For example, consider
subject
c x
xeX, where
X
is an H-cell, this is the usual linear program, whose
solution time, while polynomial, is by no means negligible.
However, if
X
is
represented as a W-cell, the linear programming problem becomes trivial.
As
another example, consider the problem of testing if
where
X
is a polyhedron.
If
X
xeX
for a given
x,
is an H-cell, the problem is trivial, whereas if
X
is a W-cell, the problem reduces to solving a linear program.
This note discusses the complexity of two types of problems.
problem is the set containment problem (SCP)
X
(resp. Y)
is a cell
,
that of determining if
,
t
.
2
(x-c)<r }, in which case
There are nine forms of SCP corresponding to
a particular form of SCP will be denoted
Y
is a B-cell.
,
In Eaves and Frexind
linear program for the six forms (HH)
e .g.
[
(WH)
,
by (W,B)
,
,
where
]
,
(BH)
,
(WW)
,
(WB)
showing that these problems are solvable in polynomial time.
also conjectured that the forms (HW)
,
(BW)
1
,
X
and
Y
X
is a W-cell
SCP is shown to be solvable as a
1
,
(resp. Y)
X
For notational convenience,
each being given as an H-cell, W-cell, or B-cell.
and
XcY, where
defined to be either a polyhedron (an H-cell or a W-cell)
or a closed solid ball of the form {x: (x-c)
is called a B-cell
The first
,
and (BB)
,
thus
Eaves and Freund
and (HB) are "intractable."
In
,
Section
2
of this article, we show that these three forms of SCP are NP-complete,
thus essentially confirming the conjecture.
Section
problem (ICP)
that of finding an integer point in a given polyhedron
,
is a W-cell.
X
case that
addresses the computational complexity of the integer containm ent
3
Karp
3
[
corresponding ICP is NP-complete.
when
showed that when
]
is an H-cell, the
X
Herein, it is shown that ICP is NP-complete
is a W-cell.
X
The notation used is standard.
The Euclidean norm of
xeR
be n-dimensional Euclidean space.
Let R
is represented by
the dimension is clear from the context.
mxn
in the
X
Let Q
matrices and n-vectors, respectively.
||
x
j|
,
.
Q
Let e = (l,l,l,...l) where
be the set of rational
Define {a,b} ={xeR
:
x =a or b,j=l,...n}
J
2.
Three NP-complete Cases of the Set Containment Problem
.
Consider the following version of the integer containment problem:
Given
(ICPl)
A
:
e
q"^"
that satisfies Air^e?
Question: Is there a tt€{-1,1}
This classical integer linear inequalities problem is NP-complete, even if
restricted to be
(HB)
,
:
(A,b,c ,r^)e
Question
(HW)
Is
:
Xj^Y,
(q'"'"^
(HW)
,q"',q" ,Q^)
where X={xeR'^ Ax<b} and Y ={xeR
:
Given: (A,b,U,V)e(Q"^^",Q"',Q^^
Question
(BW)
,
can be stated formally as:
Given
Given
:
(c
Question
:
,
r^ ,U, V)
e (Q^"
Is X^Y,
whf^rc X={x'^r'^:
:
,
(x-c) ^(x-c) <r ]?
and Y={xeR":x=UX+Vy
,Q^ ,0""* ,0"""^)
Y={ xgR" x=UA +Vy
:
.Q-^^P)
Is X^Y, where X={xeR" :Ax<b}
:
is
as there is an elementary transformation from the number parti-
The three set containment problems of interest, foinns (HB)
tion problem.
and (BW)
2,
m
o^'X
=l
,
(
A
x-c)
,
y>0}
(x-c)<r
?
}
and
,
e*^X=l ,X
,y>0}?
—
.
Note that problems
(HB)
this for the problem (HB)
(HW)
,
,
and (BW) are all elements of NP.
X^Y, then there exists either an
note that if
,
v of X, or else there exists an extreme point
extreme ray
an element of B.
then there
If the fr>rmer is true,
consisting of rows of A, and an index
j
To see
x of X that is not
a submatrix of M of A
i;;
such that v is the unique solution to
The size required to record v then is polynomially bounded in
Mv=0, v.7^0.
siibvector d of e,
and the size of x is polynomially bounded in the size of A.
-
The test that
d for some sxibmatrix M of A and
If the latter is true, x=M
the size of A.
(x-c)
2
t (x-c) >r
is
obviously polynomially bounded in the size
of A as well.
is an element of NP,
To see that problem (HW)
The test that v^{y :y=UX ,X>0} or the test that
there exists x or v as above.
x^
is equivalent to solving a linear program, which is polynomially bounded
in the data
(v,U)
or
(x,U,v)
Finally, if for a giv^jn instance of (BW)
Y has no interior in
IR
than r
2
.
hyperplane containing this facet to
th*;
If the former is true, then the system irU=ae
This test is polynomial in the data (U,V)
tTt^O.
Then, either
suppose that X^Y.
,
or there is an (n-1) -facet of x such that the shortest
,
Euclidean distance from
polynomial in this data.
U'
first note that if X^Y
has a solution
and the size of
ir
will be
the latter is true, there exists a submatrix
rf
consisting of columns of
,
irV=0
,
larger
c is
V
and a submatrix
U,
consisting of columns of V,
such that the hyperplane containing the facet in question is determined by
a solution
Z={x|
IT
(u,a)
.x=a}
.
to irU'=ae, TrV'=0,
ttt^O,
The size of the unique solution
polynomially bounded in the data (U,V)
from c to
/ la—IT- c
is given by*/-^^
zr-
»
rr
•
class NP.
th<'
to the above system is
and the shortest Euclidean distance
,
-
data (a,iT,c,r
2
)
—
a—iT-c
-'—z
IT
IT
-
polynomially bounded in
,
(tt,cx)
The test that
I
Z
where the hyperplane in question is
_
•
>
r
2
.
,
is also
IT
and so problem (BW)
is in the
,
Also note that if X^Y
,
then it follows from the convexity of
separating hyperplane theorem that there exists x^X, and
(i)
y<a for all yeY, and (ii)
IT
tt
'x
it,
and a
Y
such that
a
>o.
Our main result in this section is the following:
THEOREM
1.
The set containment ]iroblems
(HB)
(HW)
,
,
and (BW) are NP-complete.
Before proceeding to the proofs, we define a few more terms and we state an
elementary property concerning linear programs defined over the rationals.
For each matrix A, let P(A) ={x:Ax<e, -e<x<e}
.
Thus the integer contairiment
problem ICPl can be stated as follows: Does P(A)n{-l,l}
For a given rational matrix A, we will let
r.ax(A)
=(t)?
denote the maximum
absolute value of a niimerator or denominator of a component of
(The numerator and divisor
max(2/ll,-14/2) )=14.
For two sets
S,T,
e.g.
have a common divisor..)
be the infinium of the distance between
d(S,T)
let
can
A;
In the proofs, we will use the
the two sets, where the supremum norm is used.
following elementary lemma.
LEMMA
PROOF.
1.
-
If P(A)n{-l,l}" =
<^,
Let z*= d(P(A) ,{-1,1}'^)
z*(y)= d(P(A) ,{y})
z*=min (z*
(y) :ye
z*(y)= minimum z
subject to
;
{
then d(P
- ^"^^
(2max(A) ^^
(
{-1
,
l}'^) >
(x.-y.) >^0
j
= l,,..,n
t^j-yj)
j
=i
,
(A)
,
and
then
-1, 1}
),
and
,
z
+
xeP(A)
->-0
,n
-1
)
nl
)
.
.
We now claim that z* (y)> (2inax(A)
be a point in
X*
y
To see this, let
.
and without loss of generality
,
is an extreme point of the feasible region of the above
x*
Therefore
linear program.
program and
of minimum distance to
P(A)
we may assume that
for any Ye{-l,l}
(nl))
x* = B
where
f
is a row basis of the linear
B
is a vector of O's and I's of the right-hand side components
f
corresponding to
denominator of
B
and
B
can be written as
B
.
C
is an integral matrix.
is
det(C). Because
-1
a denominator for
B
B = d
where
C
Because B
- n^
d^max(A)
is a common
d
-1
= dC
-1
-,
= °
/adj
(C)
j^^
(r^\
I
and
2
,
,
,
,— , n +1,
,
max(C) ^ (max (A)
)
,
we obtain
,
dct(C)<max(C)"n: < {max(A)'^
Thus this bound on det
provides a bound on
(C)
PROOF THAT
let
e
=
d(P(A) ,{-l,l}
AeQ
.
Let
Suppose first that
-e<x<e
d(x,{-l,l}'^)<e
conclude that
nl
.
,
=
(2max(A)
(n"^+
n.')
Note that
.
A.
be an instance of ICPl, and
Y = {yeP^ry^y
and let
Then
XcY.
x*
<
n-e}
Consider the
.
XcY.
||
x
||
<
for any
n-e<n
and thus
xeP(A)
.
and
.
2max(A)
<
which is polynomial in the size of
,
A
Let
.
X = P (A)
X^
Conversely, suppose that
Since
M(A)
let
of determining if
(HB)
(j)
,
IS NP-COMPLETE
[M(A)]"
n'.
H
.
3
(HB)
P(A) n{-l,l}'^ =
)
is 0(n log(l+max(A))
M(A)
instance of
max(A)
which is a denominator for any component of
,
Henceforth, for each
the size of
=
"^"^)'^n:
x x>n-c
,
t^
xeP(A)
it follows that
It follows that
P (A) n{-l ,1}^^
Let
.
())
.
d(P
S
(A)
be selected so that
|x.|>l-e
,{-1 1}") <e
,
,
for each
j
x^
and thus
and thus by Lemma
1,
we
.
PROOF THAT (H,W)
Let
NP-COMPLETE.
IS
"
Y={y:.>: |y.|<n-e}.
-
-1
e= [M(A)]
.
I
and let
X=P(A),
Let
polynomially represented as the W-cell
U= (n-e)
[
I
,
(n-c) (-1)
]
J
I
XcY
P(A) n{-l ,1} =
<n-e and thus
and let
Note that
may be
Y
of determining if
(H,W)
XcY.
must satisfy
xe.P(A)
Then any
.
,
{y :y=UA ,A>0,e X=l} by letting
Now consider the instance
.
Suppose first that
eIx
be an instance of ICPl
A
(})
^
Suppose next that
XjzTJ
Let
.
be chosen so that
xeY
E|x,|>n-E
-e<x<e
,
l-e<[x
it follows that
and thus
j=l,...,n
for each
<1
|
Since
.
3
J
j_
d(x,{-l,l}
)
Se
P(A)n {-1,1}" /
that
d(P
Therefore
.
PROOF THAT (B,W) IS NP-COMPLETE
Let
X={xeR'^: ||x
II
•)
that
X^
Thus if XcY
,
,
such that
uniquely given by
t
it
,X
12
,X
and so
>_0,e X
yeY'
veP
and let
(p ,
v/y
P(A) n{-l,l}"=
-
x=(n-e)
loss of generality that
-1/2
1
1
1|tt
x=(n-e)
can be
Y
.
X
X
by TTeP(A).
.
To show that
1
1
||
XjcY.
n{-l,l}
veX.
and so
Also note
We conclude in this case that
•
4).
X^
In this case there exists
.
Moreover the value of
.
(A)
v v=(n-e)
-
x>l
(A)}
= M(A)
e
tlt2t3
=1}
+e
+e
3
,X
of determining if
Note first that
.
Next consider the case that
7r6P(A)
we conclude
1
and use linear programming duality to show that
y'fTi'
||v||>l
X
ueP
:7r^y<l for all
—t 3
2
P (A) n{ -1, l}"?^
11^11" v
V v=(n-e)
.
Y={yeR"
Consider the instance of (BW)
v=(n-e)~
and thus by Lemma
,
by premultiplying any
Y'cY
Suppose first that
Let
<e
be an instance of ICPl and let
A
1
one can ass\ame that
y'<Y
)
Y'={y:y=X -X +A
represented as the W-cell
YcY'
Let
.
and let
^<l/(n-e)}
(It is easy to see that
,{-1,1}
S
.
4)
(A)
-1
tt
IMI
whenever
t
•
xeX
tt^O
.
.
which maximizes
.
xeX
it
tx
and
.
is
Thus we may assume without
It follows that
IttII
=
^=-n^-n
(n-e)
^
1-6 <|tt.|<1
-e^TtSe,
for
d(P(A), {-1, !}")<£
3
.
j
(n-e)^
>
||tt|| tt^x
1
1
it
1
,
1
and thus
= l,...,n and thus
|1it|J
d(Tr,{-l,l}
^
>
n-e.
S ince
We conclude that
)<r..
and thus by Lemma 1, P(A)n[ -1, l}"7^0. D
The Complexity of Finding an Integer Element of a Polyhedron .
(see for example Garey and Johnson
It is well known
[2
]
)
that the
problem of determining whether there is an integer point in an H-cell is
In this section we show an analogous result for integral
NP-complete.
Consider
containment in a w-cell.
Given
ICP2:
^nxk,
Q
(U e
)
Is there an integral n-vector
Question:
X={xeiR'^
The problem
THEOREM
2.
PROOF.
Note first that
demonstrate that
Input
e^X=l, A>0}?
is NP-complete.
ICP2
ICP2eNP
since if
ireX
is integral, then we can
ICP2 is NP-complete, we carry out a transformation from
0-1 knapsack problem.
Integers
.
where
by solving a linear program in polynomial time-
ireX
To show that
the following
x=XJX,
:
TreX,
a,
,
.
.
.
a
,
,
b
n
Question
Is there a vector
.
such that
ye (0,11
a.y. = b?
^
i=l ^
T,
The above problem is known to be NP-complete.
Suppose that
a,
,
.
.
.
,
a
,
b
is an instance of the above knapsack
problem.
We transform this instance into a problem in modular arithmetic as
follows:
Are there vectors
\,s satisfying:
n
(
Z
a.X. - bX
(la)
is integral,
,t)
J=l
is integral
(n+l)X.
(2n)~
(X.+s.-X
,,)
3
n+1
ST + ... + S +X, +
1
n 1
...+X
2
s,X>0
_LT
n+1
for j=l,
is integral
'^
= 1
.
.
.
,
n+1,
for i=l,...,n,
(lb)
(Ic)
(Id)
(le)
.
.
First note that
-
(la)
2n+l colunms each of which
is a special case of ICP2 in which
(le)
in
i'i
Suppose first that
n+1
= l/(n+l)
ye'O,
and let s .=l/(n+l) -X
(n+l/2n)X
Since
0<X
X
^,
n+i
,,
n+1
.
Finally, let
(1)
If we subtract each of the n
of constraint (Id)
from (2n
.
X,s satisfy
Suppose next that X,s satisfy (1).
constraints of (Ic)
,
we obtain the constraint
-l/2n is integral
,^1, we conclude from
n+1
(If)
(If).
that
= l/(n+l)
We conclude from (Ig)
if and only
(1)
is feasible for the knapsack problem.
l}
It is easy to verify that
.
has
knapsack problem.
for j=l,...,n
Let A.=y./(n+l)
X
thf'
U
|R
We claim that there is a feasible solution to system
if there is a solution to
,
(Ig)
.md (Id) that
(Ic)
,
X.4S. = l/(n+l)
for
j=l,..., n
(Ih)
and by (Ih) and (lb) we conclude that
X
.
=
From
or l/(n+l)
(Ig)
,
(li)
for
j
= l,...,
(li).
n
and (la) we conclude that
y=(y ,..., y
)
is feasible for
the knapsack problem, where y.=(n+l)X., j=l,..., n, completing the proof. D
Summary.
In certain cases, the representation of a polyhedron as an H-cell
or a W-cell drastically affects the computational complexity of the underlying
problem.
For the set containment problem, involving polyhedra and/or closed
solid balls, the problem is solvable as a linear program and hence is
,
(W,B)
,
and (B,B)
This note shows that the remaining three cases (H,B),
(H,W)
,
and (B,W) are
polynomial, for the cases (H,H),
(B,H),
(W,H)
,
(W,W)
,
see
[1]
NP-complete.
As regarding the determination of an integer point in a given polyhedron,
this note has shown that the problem is NP-complete, irrespective of the
representation of the polyhedron as an H-cell or a W-cell.
,
References
[1]
Eaves, B.C., and R.M. Freund, "Optimal Scaling of Balls and Polyhedra"
M athematical Prograimning 23 (1982), 138-147.
[2]
Garey, M.R. and D.S. Johnson, Compu ters and Intractibilit y
and Co., San Francisco, 1979.
[3]
Karp, R.M., "Reducibility among Combinatorial Problems", in R.E. Miller
and J.W. Thatcher (eds.) Complexity of Computer C omputations Plenum Press,
New York, 1972.
,
W.H. Freeman
,
^Ok
022
Date Due
NOV.
2
FEB17U94
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