M.J.T LIBRARIES
-
DEWEY
c&^
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
The Parsimonious Property of Cut Covering
Problems and its Applications
Dimitris Bertsimas
and
Chungpiaw Teo
WP#
-
3649-94
MSA
January, 1994
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
The Parsimonious Property of Cut Covering
Problems and its Applications
Dimitris Bertsimas
and
Chungpiaw Teo
WP#
-
3649-94
MSA
January, 1994
M.I.T.
LIBRARIES
The parsimonious property
of cut covering problems
and
its
applications
Dimitris Bertsimas
*
Chungpiaw Teo
^
January 1994
Abstract
We
consider the analysis of linear programming relaxations of a large class of combinatorial
problems that can be formulated as problems of covering
cuts, including the Steiner tree, the
and the survivable network
traveling salesman, the vehicle routing, the matching, the T-join
design problem, to
name
a few.
We
prove that
all
of the problems in the class satisfy a deep
structural property, the parsimonious property, generalizing earlier
simas
[3].
We
identify
two
set of conditions for the
work by Goemans and Bert-
parsimonious property to hold and
two proof techniques based on combinatorial and algebraic arguments.
sequences of the parsimonious property
in
offer
We examine
several con-
LP
relaxations,
proving monotonicity properties of
giving genuinely simple proofs of integraHty of polyhedra in this class, offering a unifying under-
standing of results in disjoint path problems and
We
also propose a
new proof method that
in
the approximabihty of problems in the
utilizes
worst case bounds between the gap of the IP and
the parsimonious property
LP
values.
Our
class.
for establishing
analysis unifies
and extends
a large set of results in combinatorial optimization.
'Dimitris Bertsimas, Sloan School of
Management and Operations Research Center, MIT, Cambridge,
MA
02139.
Research partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from
Draper Laboratory.
^Chungpiaw Teo, Operations Research Center, MIT, Cambridge,
MA
02139.
Introduction
1
We
consider the following class of problems defined on a graph
following integer
G =
{V,E) and described by the
programming formulation
IZf{D) =
minimize
SeCE^ea^e
subject to
Ee€*(t) ^e
=
f{i),
D CV
e
i
T,e€S(S)^e>f{S},SCV
where /
:
2^ -+ Z+
is
a given set function, 6{S)
selecting different set functions f{S)
problems, including the Steiner
join
and
different sets
tree, the traveling
we replace constraints Xe € Z+ with Xe >
Goemans and Bertsimas
[3]
0.
We
{e
=
{i,j)
e E\
D we can model
i
e
S,
V
€
j
\ S}.
By
a large class of combinatorial
salesman, the vehicle routing, the matching. T-
and survivable network design problem, to name a
Let IPf{D) be the underlying feasible space.
is
=
We
few.
denote the
LP
relaxation as Pf{D), in wliich
denote the value of the
LP
relaxation as Zf{D).
studied the survivable network design problem, in which the objective
to design a network at minimal cost that satisfies connectivity requirements (for each pair {i.j)
of nodes in V, the solution should contain at least
Tjj
edge disjoint paths) and considered an integer
programming formulation of the type IPj{D) with f{S) =
the following property, which they
call the
max(,_j)gi(5)
rjj,
parsimonious property, which
in
D=
0.
They showed
our notation can
l)c
stated:
Theorem
Cifc
+
1
[Goemans and Bertsimas
Ckj for all i,j,
[3]]
// the costs Ce satisfy the triangle inequality {c,j
k € VJ, then for the survivable network design problem {f{S)
=
<
max^^^^^s) ^e)
Zj{D) = Zj{%).
In other words, the degree constraints are imnecessary for the
LP
relaxation in the stirvivablc net-
work design problem. They further examine several sometimes surprising structural and
properties of the
LP
relaxation,
and examine the worst case behavior of -^^4^
algorithniic
for the survivable
network design problem. Goemans and Williamson
[5]
show
interesting worst case
Our goal
in this
paper
is
bounds on the
[4],
Williamson
et.al.
[16]
and Goemans
ratio -^^my-
to understand the class of problems for which the parsimonious propertj'
holds and examine several implications of the parsimonious property. In this
way we shed new
to a large collection of results in discrete optimization and graph theory, imderstand their
origin
1.
and generalize them
We
et. al.
in interesting ways. In particiilar
continue the program started in
[3]
set of conditions
it
on the
way we prove that a
f{S), for which the parsimonious property holds. In this
of classical combinatorial problems satisfy
common
our contributions in this paper
by identifying a
light
are:
set function
large collection
including the matching problem, the T-joiii
problem, a relaxation of the vehicle routing problem, some disjoint path problems, some
matching problems,
[4]
satisfy
it.
We
etc.
In particular
also find that
the property does not hold.
on
We
if
all
problems considered
in
b-
Goemans and Williamson
the set function f{S) does not satisfy this set of conditions,
offer
splitting techniques originated in
two proofs of the property: a combinatorial proof based
Lovasz
[8]
an algebraic proof based on hnear programming
and used
[3]
generalization of the parsimonious property to integer
Goemans and Bertsimas
Goemans
duality.
developed this generalization using the techniques in
in
.
The
[6]
[3]
and
has also independently
duality proof reveals a further
programming programs
as well (the
dual parsimonious property).
2.
We
use the parsimonious property to prove interesting monotonicity properties of the
LP
relaxation for problems in this class.
3.
We
use the parsimonious property to give genuinely simple proofs of the integrality of some
polyhedra Pj{D): the T-join problem, special cases of the Steiner tree problem including the
shortest path problem
4.
We
and the shortest path
tree problem.
further extend the parsimonious property under
implications in the disjoint path problem.
results in this area
We find
more general conditions and examine
that this extension
is
its
the source for several
and provides a unifying framework to understand these
results.
5.
Wc
new proof technique
a
ofTcr
the ratio
^ Agv
Our proof technique
.
of problems that
We
leads to a
it
D
new approximation bounds
is
structured as follows. In Section 2
we introduce the
/(S) that imply the parsimonious property and examine
in this way.
In Section 3
we derive
Problem IPf(D)
classical combinatorial
problems that can
as well as the dual
interesting monotonicity properties of this
it
to the analysis of the disjoint path problem. In Section G
examine applications of the peirsimonious property to the
Section 7
We
for
problems as consequences of the parsimonious property. In Section 5 we further extend the
parsimonious property and apply
rise to
is
properties of the set function
we prove the parsimonious property
integral parsimonious property. In Section 4
class of
[4]
^H).
The paper
be modelled
for this class
does not use reverse deletions, but only shortest path computations.
use the parsimonious property to prove
with
new approximation algorithm
compared with the algorithm proposed by Goemans and Williamson
simpler to implement as
6.
that utilizes the parsimonious property to find bounds on
a
we introduce a new proof technique
new approximation algorithm
to
integrality of certain polyhedra Pf{D). In
bound the
ratio "z^t^- This proof technique gives
problems considered
for the
we
in
Goemans and Williamson
[4].
further examine applications of the pzirsimonious property to the approximability of Problem
IPj{D).
Parsimonious
2
set functions
In their study of the approximability of problems in the class I Pf {</}),
(for the case that
f{S) takes values
in {0, 1})
and Williamson
et.
al.
Goemans and Williamson
in [16] (for
the case thai f{S)
takes values in Z+) introduce the following set of conditions for the set function f{S).
Conditions
=
A
(proper set functions):
1.
/(0)
2.
Symmetry:
3.
Propereness:
O.
f{S)
= f{V
If /I
n
/?
\ S) for all
=
0,
SCV.
then f{A
U B) <
max{/(/l), /(/?)).
[4]
We
next introduce the following set of conditions:
Conditions
B
=
0.
(parsimonious set functions):
1.
/(0)
2.
Symmetry:
3.
Node
4.
Quasi-supermodularity (QS): For aU 5,T C
f{S)
= f{V\S)
Subadditivity (NS):
for all
An
If
S QV.
{x}
=
0,
then f{A
K,
STl
U
T
{x})
< f{A) +
fi{x}).
^^
Either
/(5)
+ /(T)</(5UT) + /(5nT)
or
f{S)+f{T)<f{S\T) + f{S\T).
We
also introduce the general subadditivity condition:
Subadditivity:
If /I
nB =
The QS property was
0,
then f{A
U B) < f{A) +
f[B).
paper of Goemans
also introduced in the recent
et. al.
[5].
who used
f{A).
We
will
llic
term weakly supermodular.
Finally
we introduce a
Conditions
=
C
set function /:
O.
/(0)
2.
Symmetry:
3.
Weak
4.
on the
(weakly parsimonious set functions):
1.
that
third set of conditions
/(5)
= f{V
\ S) for all
Subadditivity (WS):
a; is
If
S QV.
^n
{x]
^
then f{A
0,
U
<
{x})
then say
a weakly Steiner vertex.
2-Quasi-supermodularity (2-QS): For every three mutually
are crossing
li
A\B, B\A, Ar\B
Compared with Conditions
B
axe
nonempty)
/
two of them
satisfy the
(parsimonious set functions), weak subadditivity
node subadditivity, while the 2-QS property
there are set functions
at least
crossing sets (two sets A,
is
a relaxation of the
satisfying one of conditions
B
or
C
QS
property.
but not the other.
is
QS
B
property.
stronger than
In other words,
We
next show that Conditions
Proposition
if
f
proper,
is
Proof
If
:
/
[Goemans
1
it
is
et.
B
al.
are
[5]]
more general than Conditions A.
lyct
be a
f
is
f{S U T), f{S\T), f{T\S), say f{S n T)
max(/(5 n
/(S)
+
fiT)
In Figure
we
/
T),
f{S
\ T))
= f{S
< f{S \T) + f{T \
\ T),
S).
considered.
As we show
in
node subadditive. Among the terms f{S
attains the
By
minimum.
and f{T) < max(/(5 n T), f{T
The other
(we consider symmetric
1
0.
Tlien,
quasisupermodular and node subadditive.
a proper function, then clearly /
is
=
symmetric, set function with /(0)
set functions)
we draw the
properness, f{S)
= f{T \
\ S))
cases follow similarly from
T),
Pi
symmetry
S).,
of /.
<
and so
D
relations of the various conditions
the next sections, the parsimonious property holds for set functions
satisfying either conditions A, or B, or C.
Proper
weakly
parsimonious
parsimonious
(NS+QS)
(WS+2-QS)
Figure
1:
Relations of parsimonious, weakly parsimonious aind proper set functions.
Remarks:
1.
The
the
T
following simple observation
QS
property.
containing
v.
We
select
is
usually useful in checking whether a given function has
any node v and check the
By symmetry
of /,
QS
properly only
we can extend the QS property
to
for
all
those sets
5 and
T.
S and
2.
Conditions
B
are strictly
more
general,
i.e.,
there are set functions which are parsimonious
but not proper.
3.
The symmetry
we can
conditions are without loss of generality.
redefine the following symmetric set function: f{S)
Notice that /Z,-(0)
feasible in
2.1
=
IPA^) and
/Z/(0) and Z,-(0)
=
the function /
= f{V\S) =
is
not symmetric,
max[/(5), /(V''\5)].
Z;(0), since the optimal solution of /P/(0)
is
vice versa.
Examples of problems
In Table
1
below we review several
classical combinatorial
formulation /P/(0) for / satisfying both Conditions
C
If
A and B.
naturally arise in the study of the disjoint path problem.
Problem
problems formulated using the cutset
In Section 5
we show how Conditions
D=V and
with
1
E.€sfc(0= 2k+l
l-?|>2,
,
I
Notice that the function /
definition
A, B,
it
is
is
if
b{i)
parsimonious property,
The capacitated
Given a graph
like to
not proper, because for
violated. However, the function f{S)
node subadditive
satisfy the
is
5={i}, V\{i}
b{i)
\
tree
G =
{V
it
\}.
QS. While /
While
does satisfy
L)
{0}, E),
demands
d,,
€ V, a depot
i
is
also
QS, since
{0})
for
uT) =
S,T
cV
2
2
^'^^\(^^^)
containing
'
is
0€5
it is
not proper but
S C
V.
modeled
Interestingly, the vehicle routing
cV
it
satisfies
such that
Conditions
S f\T =
(b
{0})
< /(5n
=
f{S)
+
f{T).
f{T).
problem
as
IPj{D) with
problem can be modeled
{V U {0},E), demands
+
{o})
0,
traveling salesman problem can be
we want
and we would
of the type IPj{%) witii
< /(5n
The
Q
E
e
^'^^^^^' =
/(5) + J{T),
traveling salesman and vehicle routing
G =
e
Ce,
valid
The
of capacity
costs
A
/(5 U T) + f{S nT)
Given a graph
0,
a popular relaxation of the vehicle routing problem.
j^S U T) =
It is
I}.
symmetric and subadditive as we show below: For S,T
/((Sn
problem does not
most
obvious that J{S) does not satisfy Conditions A, since
clearly
\ {i} the
at
^
It is
+
K
demand
cutset formulation of the capacitated tree problem
B:
€ {a,a
is
not subadditive for general sets
cost such that each subtree from the depot has
Q. The capacitated tree problem
It is
is
in general the 6-Tnatching
b{i)
it if
whose union
disjoint
problem
minimum
design a tree of
e {a,a +
is
A,B
dt,
i
e V, & depot
D = V
in
=
2 for
all
our framework as follows.
0, costs Ce,
to find tours of the vehicles from the depot of
8
and f{S)
e
€
minimum
E
and vehicles
cost, such that
.
demand
the
in
We
a relaxation of the vehicle routing problem.
example f{S)
Notice that the capacitated tree problem
each tour does not exceed capacity.
=
'
2[
'^^
While
].
can strengthen the formulation
this set function
subadditive,
is
it is
3
The parsimonious property
The
cut-set formulation introduced in the previous section captures
mization problems studied in the literature.
parsimonious property holds for the
section,
LP
1
The primal proof
in
[3]
2.
The dual proof
two ways:
A
3.1
Lovasz
[8]
and used
uses linear
programming duality and extends an observation
class of
re
We
split V at {u,
be a feasible solution
in P/(0)
w} by some
A>
S,u,w ^ S and x{S{S)) = f{S).
way
in the following
for the
—A
x{v,w) <— x{v,w)
—A
We call
x
such a set
—
also use the notation x{6{A, B))
2 If S,T are
S\T,T\S
either
(ij
or
S n T,S
LI
tight sets,
are tight,
T
x
and f
{
is
6 {S
are tight, x{6{S
where u,v and
a tight
set.
We
QS, then
T))
\T,T\S)) =
0.
=
sets.
5 C V
denote by
^e={i,j},ieA,j£B^e and x{6{S))
n T,S U
are vertices in G.
A.
unless there exists a set
5
w
:
x{v,u) *— x{v,u)
+
0,
need a preliminary lemma regarding properties of tight
Lemma
(ii)
>
with x{v,u),x{v,w)
splitting operation will preserve feasibihty of
We
[2]
problems P/(0).
x{u, w) <— x{u, w)
We
of Frank
primal proof of the pairsimonious property
Let
of S.
in
to prove the parsimonious property for the survivable network design problem;
matching problem to the general
The
of the classical opti-
In the remainder of this
an extension of edge spUtting techniques introduced
is
write for
thus interesting and indeed surprising that the
It is
in
we
not QS.
many
relaxations of these problems.
we prove the parsimonious property
if
is
=
S
such that v e
the complement
x(6{S, S)).
Proof
Since /
:
is
QS, wc
first
consider the case
J{T
+ /(5 \T) >
\ 5)
+
f{S)
In this
f{T).
instance,
f{T\S) + f{S\T)
<
x{6{T\S))+xi6{S\T))
=
x{6{S)) +x{6{T))
=
f{S)
+
f{T)
+2x{6{SnT,SuT))
+ 2x{6{SnT,SuT).
Uencex{6iS\T)) = f{S\T),x{S{T\S)) = f{T\S) and x{6{SnT,SuT)) =
holds.
On
the other hand, the case f{S
U T) + /(5 n T) > /(5) + /(T)
0, i.e.,
condilion
gives rise to condition
(i)
(ii),
using an identical argument.
We
can now prove the central result of this section.
Theorem
and f
3 (Parsimonious Property)
// the cost
a parsimonious set function, then
is
Zf{D) = Z;(0),
Primal proof:
a u in
D
that has x{6{v))
A, while maintaining
(If
>
is
feasibility.
and so
+ f{S \
>
some
positive A,
5nT
is
nT,S UT)) >
a tight set.
5
By
x{v,u)
>
5 be
x.)
If
Since /
0.
the minimallity of 5,
with x{v,w)
{v)), violating the
a minimal tight sot
S
is
T
is
another
tight set
QS, condition
is
(ii)
of
contained in T. So there
u.
>
0, else
f{S)
node subadditivity of
^mm{x{6{S)) - fiS)
Because of the triangle inequality,
<
Let
/.
=
We
x{6{S))
can then
=
x{6{v))
+x{6{S\
split v at {u.
w} by
where
A<
x'{6{v))
0.
This contradicts the minimallity of
In addition, there exists a u; in
i^}))
>
is
such set does not exists, then we can decrease x{v, u) by some positive
a vmique minimal tight set containing v but not
f{v)
D.
f{v); Let u be such that x{v,u)
containing v but not u, then x{6{S
2 holds
for all
Let x be an optimal solution in P/(0), with 5Zee£2;e minimal. Suppose there
containing v but not u.
Lemma
function c satisfies the triangle inequality,
x{6{v)) and x'{6{t))
-
:
V
€ S,u,w ^ S,xi6{S))
-
f{S)
>
0}.
this operation yields another feasible optimal solution x' witli
x{6{t))
\U ^v,t eV. This
Remarks:
10
contradicts the minimallity of x.
1.
In the splitting process,
if
x,f are both even and
integral,
we can choose
A
to be
corresponds to the classical edge-sphtting notion and has played an important role
We
cormectivity problems.
G
Corollary 1 Let
be
>
This
in
many
this discussion in the following corollary.
an Eulerian multigraph and xg
he
an even, parsimonious
x{d{v))
summarize
1.
set function.
If
xg
be the incidence vector of
G. Lei f
and
a feasible integral solution to IPf{%)
is
f{{v}) for some v e V, then there exists {u,w} and an edge splitting operation
new graph G' and
of V at {u,w}, yielding a
a corresponding incidence vector xg' that
a
is
feasible integral solution to IPj{%).
This corollary generahzes the following result of Lovasz
and Bertsimas
[3].
G
Corollary 2 Let
from
disjoint paths
to j
i
and v
be a vertex of
• rG'{i,j)
=
rciij) ifij
•
=
mm{rG{i,j),degG{v) -
Proof
Set f{S)
:
The
function.
=
x{6{v))
Thus
=
>
there exists
new graph G' ) such
{u,w} and an edge
that
max{rG{i,j)
xg
f{v). Since
:
2) for j ^^ v.
e
=
{i,j)
G
of graph
G
is
is
G S{S)}.
with xg'{S{S))
>
Note that /
is
a feasible integral solution
a parsimonious
in
IPf{9), with
Eulerian, f{S) and x{6{S)) are even for each
there exists an edge splitting operation of v.
feasible solution
Then
G.
of edge
i- v.
incidence vector
degciv)
maximum number
an Eulerian multigraph, rG{i,j) be the
be
splitting operation of v at {u,u}} (obtaining a
rG'{vJ)
and a refinement due to Goemans
[8]
f{S).
By
The graph G' obtained
in this
5 C
way
V'.
is
a
the max-flow-min-cut theorem, the graph G' has
the required cormectivity.
2.
QS and node
Notice that for the parsimonious property to hold / needs to be
The
full
or
+
fc
1
subadditivity
is
both
is
not needed.
QS and node
The 5-matching problem,
subadditive, while
subadditive.
11
if
b{v)
e {k,k
for
+
subadditi\'e.
example with
l,k
+
2}
it
is
b{v)
not
=
k
node
Although the condition that the cost function
we next show
c satisfies the triangle inequality
that for problems of the form /P/(0), P/(0),
ensure that this condition
is
i.e.,
/Z^(0) and Zy-(0) denote the respective solution value with
Theorem
P
<
c(e).
c!{g)
—
c{g) for each
Let
3.2
=
Z;(0);
/Z}(0)
=
be a shortest path (with respect to
edge g on P.
another optimal solution (since
when
denote the shortest
the triangle inequality. Let
as the objective function.
IZjifD).
Let x be an optimal solution to Pf(0) (or IP'A^)). Consider an edge e
:
(/(e)
in
(/
satisfies
(/(u, f)
4
Z}(0)
Proof
d
c'(e)
A
The dual
restrictive,
with no degree constraints, we can
met by the following transformation. Let
path between u and v with c as the length function. Clearly
seems
=
c(e),
If
x{e)
c'{c)
>
=
0,
we can
c) linking
u and v.
=
{u,v) such that
Then P ^
and
[e],
reroute the flow through the path P, resulting
d{P)). Repeating this procedure, we have x{c)
>
only
D
thus proving the theorem.
dual proof of the parsimonious property
of
P/(D)
is
as follows
:
DZ/(D) = maximize
subject to
T.scvyiS)fiS)
T,S:eeS(S) y{S)
<
c(e), e
E
e
y{S)>0,ScV,S ^D.
Let
DPf{D) be
the dual polyhedron and
DZf{D) denote
the optimal objective value.
parsimonious property using a dual argument, we only need to show that
solutions to
feasible to
if
for all
DPf{D), we can always choose one with
DP}{^). Let
/t, /i
Theorem
€
T be a collection
y{v)
>
of sets (subsets of V).
among
all
To prove
the
dual optimal
for all v
€ D. This solution
We call
this family of sets
is
then
laminar
T either Af^B = ^,orAcB,OTB(zA.
5 // the cost function c satisfies the triangle inequality, and f
function, then
Zj{D) = Z;(0),
12
for all
D.
is
a parsimonious set
Dual proof:
that the set
Let y be a dual optimal solution in DPf{D).
T
:= {S
and
Thy S\T,T\S
A
T containing v,
e
:
0}
replace
A
y{A)
Let p(e)
V
c{v,
Z!s:ee<5(s) y('S')-
^ 0,
=
By
we
i.e.,
y{V
w
w) >
dual
\
A)
QS
property,
we may assume
5
replace two intersecting sets
&v e D
<
such that y{v)
0.
For
all
set
- y{V \ A) + y{A).
solution with
no member of J^ containing
A=
we can
We
v.
Note that
w)
—
Thus the modified
c{u, v)
solution
optimal solution with y{v)
Notice that
if
=
c{u,
<-
y{v)
y{Au{v})
*-
y{Au{v}) +
y{A)
^
yiA)-A.
is
>
dual feasible.
for all v in
>
v)
we only need
By
p{u, w)
Let
A
a
is
he a maximal
:
+A
y{v)
w) — p{u,
assume that there
modify the dual solution as follows
A
to consider edges of the form {v, w)
not in A. Note that by the construction of ^, p{u,w)
c(u,
We may
c(e).
increase y{v) otherwise.
imn{—y{v),y{A)).
for feasibihty of this modified solution,
is
<
feasibility, p(e)
c{u,v), since
of J^ containing u. Let
To check
where
=
such that p{u,v)
member
we can always
T. Suppose there exists
V \A,
the
laminar.
is still
u £
by
laminar, since
way we obtain another dual optimal
In this
J^
is
S n T,S U
or
we
>
y{S)
By
-
p{u, v)
—
p{v,w) +p{u,v) -2y{v). Hence
—
p{v,
w) - 2y{v) >
p{v.
w)
+
2A.
repeating this procedure, we can construct a dual
D.
A
y in the above proof takes only integral values, then
can be chosen to be
integral.
This yields an integral analogue of the parsimonious property in a dual sense.
Let
DIZf{D) denote
Theorem
the optimal objective value over
6 (Dual Integral Parsimonious Property) ///
triangle inequality, then DIZf{il>)
3.3
On
DPf{D) with
is
integrality constraints on y{S).
parsimonious,
andc
satisfies the
— DIZf{D).
the minimallity of conditions for the parsimonious property
We
remark
QS
or the node subadditivity property.
in this subsection that the parsimonious property does not hold
13
if
we
relax either the
=
Consider the set function / on 3 nodes as follows: f{vi)
Then
clearly
/
is
QS, but
it
f{S)
follows:
=
if
1
and symmetric. In
this instance,
then x{6{{vi,Vj}))
<
otherwise.
is
empty.
=
|5|
Pf{V)
or
1
is
f{S)
3,
=
2 otherwise.
again empty, since
ii
Then /
>
x{Vi,Vj)
is
clearly subadditive
and x{vi)
=
= L
x{vj)
2.
Monotonicity properties
4
Let / be a parsimonious set function defined on V, and
fw{S)
that
= /(5n
fw
is
V)
for
S C V
UW. We
call
fw
W
let
be a
fw{S) =
if
from
set disjoint
Vu
an extension of / to
again a parsimonious set function. Note that
5 C
W
W
.
.
It
is
Lot
I'.
easy to check
For this reason, we
\V the set of Sleincr vertices.
For instance, when f{S)
problem, with
function.
=
1
for all
S C
V, the extension
W the set of Steiner vertices.
Goemans and Bertsimas
Shmoys and Williamson
We
=
/(5)
1,
the other hand, subadditivity alone does not guarantee the parsimonious property. Define /
on 4 nodes as
call
=
not node subadditive. In this case the parsimonious property does
is
not hold, as the polyhedron Pf{V)
On
f{{v2,V3})
[15] for
[3]
When
f{S)
for the survivable
the Held and
=
1
fw
corresponds to
for all
odd 5
in
tlie
V fw
,
Steincr tree
is
the
I
'-join
network design problem and independently
Karp bound proved the
following monotonicity result.
extend this result to the class of parsimonious functions.
Theorem
as above.
7 Lei f,g
be
parsimonious functions defined on
Suppose fw{S) < g{S) for
satisfies the triangle inequality,
all
S C V
L)
W
.
V and VuW
If the cost
respectively
function c (defined on
then
Zf{V)<Zg{VUW).
Proof
Zf{V)
= Zf^iVuW)
=
Zj^{^)
<
Zg(fh)
=
Zg{V U W) (parsimonious
(parsimonious property)
{g
dominates fw)
14
and fw defined
property).
VU
\VJ
n
Notice that the monotonicity result does not hold for IZf{D) in general. This
the fact that the parsimonious property does not hold for integral solutions.
since the parsimonious property holds for dual integral solutions,
is
On
due
in part to
the other hand.
we show next that the
following
monotonicity result.
Theorem
fw <
8 Let f,fw,9 be defined as above and
9-
Jf c (defined
on
V U W)
satisfies the
triangle inequality, then
=
DlZji^)
Proof
:
Clearly
Let {y{S)
:
DIZf^{9) <
5 C F} be
DIZg{ID), since
DIZf{(D).
On
y'{S)
=
From
iorv
A
DIZg{</)).
We
9-
show next that DIZ}{%)
DIPj^{W)
= J2T:Tnv=s
if
2/(^))
{y{S)
:
y'{S)
-
2/(5)
if
y'{v)
=
-M
\{veW,
y'{S)
=
We
.
is
is
DIZj^{W) >
DIZf{(/)).
optimal for DIZf^{(l>), then by constructing
feasible solution to
conclude that
Therefore,
= DIZf^{W) >
DIPj{^), with the same objective solution.
DIZ0) =
DlZf^id).
Weakly parsimonious functions and the
natural question
\\'g
5 CK,
otherwise
S C V U W}
we obtain a
DIZj,„{^).
as follows:
the dual parsimonious property, DIZf^{<J})
the other hand,
=
M be a large positive number.
W, Escvuwy'{S)fw{S) = Escv y(5)/(5).
in
Hence £»/Z/(0) > DIZj^{<Ji).
5
fw <
<
an optimal solution to DIP}{^). Let
construct a feasible solution to
Since fw{v)
DlZf^iUi)
disjoint
path problem
whether node subadditivity and quasi supermodularity (QS) are the most
general conditions on the set function
show that the parsimonious property
/
still
for the
parsimonious property to hold. In this section we
holds for weakly parsimonious set functions (Conditions
C
introduced in Section 2) and observe that these relaxed conditions provide a unifying understanding
of several results on the the disjoint path problem.
We
next prove that the parsimonious property holds for weakly parsimonious functions.
15
Theorem
f
is
9
D
Ia:1
be the set of weakly Sieiner vertices.
and
weakly parsimonious, then
= zjm.
zj{D)
Proof
:
The proof
>
f{v),x{v,u)
similar to the proof of
is
where x
0,
By
contains v but not u.
52-
If c satisfies the triangle inequality,
Then
We
all
an optimal solution
is
the
=
/(S.)
is
a
x{6{S,))
in 5i
it;
=
Pi
3.
Let v e
D,u e
V, and suppose x(6{v))
most 2 such minimal
exist at
u must contain one of these two
Sa with x{v,w)
x{S{S, \ {v}))
>
Consider the minimal tight sets that
in Pf{9).
2-QS property, there
tight sets containing v but not
show next that there
Theorem
>
0.
sets,
say S\ and
sets.
Assimiing the contrary, then
+ x{6{{v},S:)) -
>
xi6{{v}, 5.))
f{SA{v})+x{6{{v},S;))-x{6{{v},S,)).
From weak
subadditive, f{Si \ {v})
>
/(S,); hence,
x{6{{v},Sl))<x{6{{v},S,)).
Since
we have assumed
that x{6{{v}, S\ DS2))
=
0,
we
rewrite the inequality for
\ 5i))
+ x{6{{v},Si U 52)) <
x(<5({v}, 5, \ 52)).
xi6{{v}, 5, \ 52))
+ xi6{{v}, 5, U 52)) <
x{6{{v},S2 \S,)).
x{6{{v},S2
z
=
2
and obtain
U 52.
Therefore,
1.
and
Hence x{6{{v},S\ U
there exists a
u; in
52))
S\
<
n52
0,
which
is
with x{v, w)
a contradiction since x{v, u)
>
0.
By
in
Similar to Corollary
Corollary 3 Let
even,
2-QS
G
and u € 5i
splitting at v using u, w,
optimal (because of the triangle inequality) solution.
optimal solution
>
By
we obtain another
feasible
repeating this procedure, we obtain an
Pf{D), thus proving the theorem.
1
the above proof actually yields the following:
he
an Eulerian multigraph and Xc
set function.
If
xg
is
be the incidence vector of
a feasible integral solution to IPj{9)
some weakly Steincr vertex v e V, then
there exists
{u,w} and an edge
G. Let f
and x{6{v)) > /({;})
16
an
/"'"
splitting operation of v at
{u,w}, yielding a new Eulerian graph G' and a corresponding incidence vector xq' thai
integral solution to IPf{(/i).
be
is
a feasible
As we show
next, this corollary provides a unifying
2-QS functions and the
Given an undirected graph
to
understand several seemingly unrelated
(EDP) problem.
results for the edge-disjoint-path
5.1
way
G =
path problem
disjoint
{V, E), a collection of source-sink pairs {si, ii},
EDP problem asks whether there exists a collection of edge disjoint paths in G,
to
H
corresponding sink. Let
its
Let xc{e)
=
e
if
1
G
G
and
=
1
if
e
G
XG{S{S))>XHi6{S))
There has been an extensive literature
G
cycle
problem.
Let
,
{sk, tk}, the
each joining a source
...
,
{sk, ik)}-
is
the cut-criterion:
5 CK.
[2]
and Schrijver
both necessary and
Kn,Cn denote
also denote the disjoint union of
[13])
that finds
sufficient for the existence
respectively the complete graph and the
m
copies of
Kn by mKn- The
following
known:
Theorem 10
cut-criterion
The
We
on n nodes.
results are
EDP
for all
is
example Frank
(see for
and H, so that the cut-criterion
of a solution to the
.
//.
Clearly, a necessary condition for the existence of these paths
conditions on
.
denote the demand graph, with edge set {{si,t\),
x//(e)
let
.
If
is
G+H
is
Eulerian, and
is
either a double star or a
necessary and sufficient for the solvability of the
case in which //
is
The C5
EDP
case
is
The K4
case was proved by
due to Lomonosov
[9].
See
[2]
and
K4 or
a C5, then
tlic
problem.
2K2 was proved by Rothschild and Whinston
a
case follows easily from their result.
independently.
H
Seymour
[13] for
[12].
[14]
The double
star
and Lomonoso\-
nice proofs
[9]
and exposition
of these results.
At
first
sight these results
appear to be unrelated without a unifying characteristic.
use the theory developed in this section to identify the unifying characteristic of
contained in Theorem
10.
The
central reason
the 2-QS property. In particiilar
it is
is
all
We
could
the above results
that the set function xh{S{S)) in these cases has
easy to prove the following proposition:
17
Proposition 2 The
function
set
K3 and K^-
a 3A'2 or disjoint copies of
K4
xh (6(5))
has the 2- QS property
This in turn holds
if
if
and only
and only
if II
does not contain
if II
is
a double star or a
or a Cs-
In order to sec
how Proposition
be used to prove Theorem
2 can
10, let us rewrite
the cut
condition as follows:
XG+Hi6iS))>f{S) = 2xH{6iS)).
Under the assumptions
Theorem
of
graph (by assumption), while /
nodes
in
G
that
is
do not belong to
10 and using Proposition
an even, 2-QS
2,
xg+h
corresponds to a Eulerian
D = V\{s\,ti,
our context D is the set of
set function. Let
a source-sink pair. In
.
.
,Sk,tk} he the
.
weakly Steincr
vertices.
Applying Corollary 3 we can then perform edge-splitting operations on the edges of
obtain a
new graph G'
or sinks.
The
G
that satisfies the cut criterion, but with edges incident only to the sources
proof involves showing that G' has the set of edge-disjoiiil-paths joining
rest of the
each source-sink pair. This follows from a tedious case by case analysis which we omit here, as
is
to
unrelated to the theme of the paper.
of edge-disjoint-paths in
G
By
reversing the edge-splitting operations,
that meets the cut criterion, thus proving
Theorem
we obtain
it
a set
10.
Applications in proofs of integrality of polyhedra
6
An important
show
programming
integrality of the associated polyhedra for integer
common
torial
direction of research in integer
proof technique
is
is
the development of techniques to
programming problems. Perhaps the most
algorithmic. Researchers develop an optimal algorithm for a combina-
optimization problem, which at the same time shows integrality of a proposed formulation
for the
problem. In this section we show that the parsimonious property loads to non-algorithmic,
genuinely simple proofs of integrality of some polyhedra Pf{D), yielding new simple proofs of some
classical results as well as
A
some new
results.
milestone in combinatorial optimization
matching polyhedron. This
is
the proof of integrality
result follows directly
as the perfect matching polyhedron
is
(Edmonds
of the perfect
from the integrality of the T-join polyhedron,
a face on the T-join polyhedron.
18
[1])
Surprisingly,
wo can
derive the integrality of T-join polyhedron from that of the perfect matching polyhedron, using the
parsimonious property.
Theorem
11 Let f{S)
=
1
iJ\Sr^T\
otherwise.
odd,
is
=
Conv{IPj{^))
Proof
:
Let fr be the restriction of / to T, defined on
matching polyhedron on T.
By Theorem
4,
We
we may assume
Then
P/(0).
5C
=
show next that IZj{^)
T. Note that IPf^{T)
2/(0) for
c satisfies the triangle inequality.
all
The
is
just the perfect
integral cost functions
c.
following inequalities are
immediate:
/Z;(0)
From
<
/Z/^(0)
/Z/^(r).
the integrality of the perfect matching problem IZfj.{T)
property Zf^{T)
=
Zf^{^)
= Zf
{(!)),
=
Zfj,{T).
Prom
the parsimonious
yielding that
/Z/(0)
The
<
reverse inequality holds trivially
<
Zf{<D).
=
and so IZj{^)
Zj{^), which shows integrality of the T-join
polyhedron.
In the next section
we
briefly review
another (and
in
our opinion quite powerful) proof technique
that proves the integrality of the perfect matching polyhedron directly.
The
shortest path polyhedron can be treated as a Steiner-1-cormectivity polyhedron on two
terminal nodes. Integrality of the polyhedron also follows easily from the parsimonious property.
We
generalize this result, using the parsimonious property, and
for the Steiner-2-Connected
Theorem
12 For
polyhedron with at most 5 terminal vertices
the Steiner-2-Connected problem
on
Conv{IP0)) =
Proof
:
It is well
known
show that the cut
([11])
that the
parsimonious property the result follows
TSP
at
most
is
set formulation
integral.
5 terminal nodes,
PfiH)).
polyhedron on at most
5
nodes
is
integral.
From
the
D
easily.
19
We next consider the multi-commodity flow problem with a single source s, multiple sinks
D=
and with no capacity constraints. Let
{s,t\,t2,
.
,tk}- Algorithmically, the
{t\.t2-
problem reduces
to the computation of the corresponding shortest paths between the source and the sinks.
that
if
for the
D=
V^
the problem
problem with f{S)
from Johnson
Theorem
is
the shortest path tree problem.
= Y.u&s
if
1
5
=
^ 5 and /(5)
We
f{S)
,
Note
show that the cut-set formulation
is
This result also follows
integral.
[7].
13 For the uncapaciiated multi- commodity flow problem with a single source and mul-
tiple sinks,
C(mv{IPf{<ll))
Proof
:
We only
PfiY) has only a
need to show that
I
=
P}{<Ji).
Zj{D) = Zf{D) when
single integral solution with x{s,
t,)
=
c satisfies the triangle inequality.
for
1
each
z
=
1, 2,
.
.
.
A:,
,
Since
the result follows
D
immediately.
7
Applications in worst case analysis
In recent years there has been a lot of interest in the approximability of combinatorial optimization
problems. Typically researchers propose a heuristic algorithm for an integer programming problem
(a minimization
problem) and compare the value of the heuristic to the value of the
(or to the value of a dual feasible solution of the
example of
this
approach
is
the 2(1
-
proposed in Goemans and Williamson
.4)
and taking values
in
{0,1}.
distinct characteristic of their
in the solution are deleted.
polyhedron Pf{V)
In this section
functions.
is
A
^)
[4]
LP
relgixation).
for the
problem
I
Moreover,
is
relaxation
very nice and \ory general
approximation algorithm (T
=
[v 6
V^
:
f{v)
—
1})
Pf{9) with / being proper (Conditions
corollary of their result
method
A
LP
is
the bound
-^^v ^
2(1
-
-rfi).
A
a reverse deletion step, in which edges that were added
for the
matching problem the bound
is
exact (the matching
integral).
we propose
The proof method
a
new proof method
gives rise to a
does not use reverse deletions and therefore
that shows that
new (and
it
is
in
^ ^L
2(1
- 4^)
for
proper
our opinion more natural) algorithiii thai
easier to implement.
20
<
Moreover, wo remark that
tk}
method
the proof
quite powerful as
is
can prove integrality of the matching polyhedron.
it
also be used to prove the integrality of the multicut formulation for the
It
can
minimum spanning
tree
problem and the branching polyhedron. Finally we use the parsimonious property to bound ^ Yu/
if
c satisfies the triangle inequality.
A
7.1
We
proof technique to bound the ratio -^4^
—
consider problem /P/(0) with / being a
Our proof technique
14
T=
{v e
V
f{v)
:
=
1}.
uses the crucial observation that a minimal solution to the problem must be
a forest, and thus has at most
Theorem
proper function. Let
1
IT"!
—
1
edges.
(Goemans and Williamson
If
[4])
f
is
a
—
1
proper function
iZf{<ii)<2{i--^^)Zjm.
Proof
:
For the purpose of contradiction we assume the contrary. Therefore, there exists a coimter-
example on the
5Ze€£;'^(^)
's
number
least
of nodes, with
be checked that
is
/'
a u with f{v)
is still
proper.
=
0.
By
Since the optimal solution in /P/'(0)
We may
Let /' denote the restriction of f on
further assume that
the parsimonious property Z^(0)
V\
{v}
.
It
can easily
the minimallity of the counter-example,
is
also feasible in /P/(0), /^/(0)
by using the shortest path distances Z/(0)
By
c integral.
minimal.
Suppose there
4,
/ proper and
=
=
Zj{{v})
Z^(0) and Z//(0)
=
=
< IZf
{</}).
From Theorem
Z^,(0). But, Z^,(0)
=
Z'^{{v}).
Z^/(0). Therefore,
/Z;(0)<2(l-|ii)Z;(0),
which
If
is
a contradiction. So
there
is
an edge
as a supernode,
we
e
=
we may assume
{u,v)
restrict the
e
E
with
Ce
=
f{v)
=
0,
for all v.
then by contracting this edge, and treating {u. v}
problem to one of
coimter-example, there exists a solution that
1
strictly smaller size.
satisfies
21
the theorem.
By
By
the minimallity of the
introducing the edge
(u.v).,
with no extra cost since
and the theorcnn
Now
y{v)
let
=
0, if
necessary,
holds. Therefore,
=
vious paragraph).
4
Ce
^ for all v
By
wc obtain
we may assume
Ce
a solution feasible to the original problem
>
for all e.
and consider the cost function
the mininiallity of
where
c'
c^
=
-1
Ce
(Cg
>
from the pre-
there exist x, y' such that x £ JPf{9), 5Zs:e6«(S) v'i'^)
c,
-
and
J2cf,xe<2{i-^^) Y,
Since /
-
is
1,
le corresponds to a forest and therefore,
X;
The
<
Xe
iri
=
+
y'
y.
-
1
=
2(1
we have shown
E
y*
is
is
=
y{v)
2(1
-
^) X:
f{v)y{v).
wc can assume f{v) =
that
1.
E
y'{S)=
y{S)
+
l
+ \<c', + \=Ce
S:ee6{S)
dual feasible. Therefore,
^
This
^
Note that
S:e€S{S)
and so
i^)
last equality holds, since
Let y'
y'is)f{s).
CeXe
= 5:
c'^Xe
+
E
^^
<
2(1
-
-)-)
\T\
J]
y'{S)f{S).
again a contradiction and the theorem follows.
Remarks:
1.
The
dual variables y constructed
We
if
f{S) >
is
bounded above by
0.
in
the proof arc half-integral.
can refine the previous theorem as follows.
2(1
- tL)
We
We
call a cut
6(S) an /-cut
have shown that IZf{9)
times the maximtmi half-integral c-packing of /-cuts.
observation has an interesting implication for the TSP.
It
is
well-known that,
if
This
the cost
function c satisfies the triangle inequality, the Christofides heuristic constructs a solution
with objective value (denoted Zc) not more than 3/2 times of the optimum.
has been strengthen further by Wolsey
that
Zc <
(^
-
[17]
and Shmoys and Williamson
^)Zy(0), where / corresponds to the
inequality by replacing Z/(0) with the vahie of the
by DZ;(l/2)). Note that
all
TSP
function.
maximum
We
[15]
ro.'^ult
who showed
can strengthen the
half-integral c-packing (denoted
cuts are /-cuts in this instance and 2 DZ/(l/2)
22
This
< DZ;(0) =
Zf{0).
From the above
above by 2(1
minimum spanning
discussion, the solution to the
— ^)DZf{l/2). A
minimiun matching on the
set of
known
well
odd nodes
is
on matching
result
tree
bounded
is
(see [10]) says that the
bounded above by DZj{\/2). Hence
^c<(3-||^)DZ;(l/2).
2.
The proof
only works for proper and not the more general parsimonious functions, because
we want the parsimonious property
subadditivity
QS
3.
is
not
to hold even
we need /
sufficient. Therefore,
—
property, which for the case of
functions
1
The above proof technique can be used
if
is
we
Therefore, node
create supemodes.
to satisfy the
full
subadditivity and the
exactly the class of proper functions.
to prove the integrality of the matching polyhedron,
the multicut formulation for the minimimi spanning tree problem and the branching polyhedron. For the matching polyhedron the difficult step
handled using techniques from
As an example of a
formiilation of the
IZmcut
=
[10].
The
is
final step is easy, since
different application of the proof
MST.
Let
n=
{S\,
.
.
.
5|n|}
minimize
subject to
the case with
method
Ce
=
Yle^e
=
0,
which can be
V ~ ^i'
y(^')-
us consider the muUicTit
let
be a partition of V.
HeeE CeXe
T.ees(Si,Sj); i<t<j<|n|a;e
>
|n|
-
1,
V
n=
{5i,
.
.
.
5|n|}
Xe e {0,1},
Let Zmcut be the
LP
relaxation and consider the dual problem.
Zmcut
=
End^l -
maximize
subject to
En:
eesiSi,Sj) vi'^)
<
>
0,
2/(n)
We
need to show that IZmcut
=
We
Zmcut-
variables as follows. Let 11 == {{1}, {2},
E(|n| -
.
.
.
,
difference
{n}} and y{U)
i)2/(n).
23
Ce,
ye £
E
use an identical proof method, assuming the
The only
existence of a minimal counter-example.
l)y(n)
=
1.
is
that
we update the dual
Note that J2e^e
=
n -
\
=
AllhouKh the proof method
method
that
in
Theorem
7.1
non-algorithmic,
is
approximate solution.
to construct an
algorithm
in
also leads to an algorithmic
from the Goemans and Williamson's
differs
It
it
noods a pre-processing step to compute pairwise shortest paths. With this
it
hand, we can discard
approach avoids the
=
vertices with f{v)
all
0.
We
the
critical reverse deletion step in
the expense of computing pairwise shortest paths.
Approximation Algorithm
for
0—1
call these vertices
the Steiner vertices. This
Goemans and Williamson's
Our algorithm
in
algorithm,
at
as follows:
is
proper functions
1.
Compute
2.
Discard the set of Steiner nodes. Select an edge with the least cost. Merge the two end nodes
into a
the pairwise shortest path distances for
supcmode, delete
all
all
pairs of non-Steiner nodes.
edges joining these two nodes and update the costs of joining the
supemodes.
3.
Repeat step
last
two supemodes merge to form a
2 until
edge selected.
remaining, reduce the cost to
4.
with f{S)
more non-Steiner nodes, go
there are no
If
S
set
and return to step
c(e')
Replace the edges selected in Step 2 and 3 by
its
=
to step 4.
0.
Let
Else for
e'
all
be the
edges
1.
corresponding shortest paili
the original
in
graph G.
For the Steiner tree problem, our algorithm emulates the
MST
heuristic on non-Steiner nodes
with the pairwise shortest distaince metric. In this respect Theorem
fact that the
MST
heuristic gives a 2(1
-
tjtt)
7.1 generalizes the well
known
approximate solution to the minimum Steiner tree
problem.
For arbitrary proper functions /, as
Theorem
a feasible solution by utilizing
/,
and
for
appending
each
p,
-
i,
=
fp,{S)
p,_i {po
=
1
if
,Pn)
=
J17=\
>
7.1.
p,
0) copies of the
obtain a feasible solution which
'W(Pi.P2)
f{S)
Goemans and Williamson
'''"J''"'
is
<
Let p\
and
p2
<
otherwise.
.
.
.
.
.
.
Similarly for arbitrary
24
<
we can construct
observe,
Pn be the distinct values of
Note that
approximate solution to
within 2H(pi,p2,
[4]
/p,
/p, for
is
each
proper
i
=
1,
-
1.
2, .... n.
By
we
,Pn) times of the optimal solution, whore
QS
functions
we can use
llic
results of
[5]
to find
/Z;(0)<2W(p,,P2,...,Pn)Z/(0).
On
7.2
We
the approximability of IPf{V)
The
next study the approximability of IPf{V).
recent research activity on approximation
algorithms has so far concentrated on problem /P/(0), partly because
feasible integer solution to
IP/ {V). In
fact,
checking feasibility
is
it is
difficult to
construct a
usually NP-hard, as indicated for
the case of the Hamiltonian-Cycle problem. Using our understanding of edge-splitting techniques
and the parsimonious property, we can extend many of the approximation
/
an even parsimonious function, and
is
Theorem
15 ///
is
results to IZf{\').
when
c satisfies the triangle inequality.
an even parsimonious function, and c
satisfies triangle inequality, then
IZf{V)<2n{puP2,...,Pn)Zf{V).
Proof
Let /' be f/2.
:
Then
/' is
again a parsimonious function.
From
[5]
We
approximate solution to JPf'{9), with x\y denoting the primal and dual solution
X
=
Then the graph corresponding
2x'.
to
x
is
first
construct an
respectivel}'. Let
Eulerian, since each vertex has even degree. Note
that
n{^,...,^)=n{p,,...,pn), and
^
e
Since /
CeXe
= 2Y^ c,x',<An{^,...,^-)Y.
^
^
is
t/(5)/'(S)
-
27f (p,
,
.
.
.
,
p„)
^
S
e
even and the graph corresponding to x
is
Eulerian, applying Corollary
edge-splitting operations to construct a feasible solution to IZj{V).
cost of the constructed solution has not increased
y{S)f{S).
S
and
By
1,
we can use
the triangle inequality, the
therefore,
/Z/(K)<27i(pi,P2,...,p„)Z;(0).
From
the parsimonious property Z^(0)
Remark:
=
Zf{V) and the theorem
follows.
For the case of the TSP, the previous theorem corresponds to the well-known
doubling the edges of the
MST
solution yields a 2-approximate solution to the
fact that
TSP.
Acknowledgement:
We
would
like to
thank Michel Goemans
for pointing out his
25
unpublished work in
[6]
to us.
References
[1]
,1.
Edmonds,
(1965).
Maximum matching and
a polyhedron with
(0, l)-vertices
,
J.
Res. Nal.
Bur. Standards, (69B), 125-130.
[2]
A. Frank, (1990). Packing paths, circuits and cuts-a survey, in Paths, Flows and VLSI-Layout.
ed. B. Korte et. al, 47-100.
[3]
M.X. Gocmans and D. Bertsimas,
(1993). Survivable Networks, linear
and the parsimonious property, Mathematical Programming,
[4]
M.X. Goemans and D.P. Williamson,
strained forest problems, Proc. 3rd
[5]
M.X. Goemans, A.V. Goldberg,
(1994).
(1992).
60, 145-166.
general approximation technique for con-
ACM-SIAM Symposium
on Discrete Algorithms, 307-316.
S.Plotkin, D.B. Shmoys, E. Tardos and D.P. Williamson,
Improved approximation algorithms
SIAM Symposium
A
programming relaxations
for
network design problems, Proc. 5th
ACM-
of Discrete Algorithms, to appear.
[6]
M.X. Goemans, A note on the parsimonious property, unpublished manuscript.
[7]
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[8]
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On some
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[9]
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[10]
L. Lovasz
[11]
R.Z. Norman, (1955).
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[12]
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A. Schrijver, (1991). Short proofs on multicommodity flows and cuts,
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Comb. Theory,
Ser.
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[14]
P.D. Seymour, (1980). Four-terminal flows, Networks, 10, 79-86.
[15]
D. Shmoys and D.P. Williamson, (1990). Analyzing the Held-Karp
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[16]
D.P. WilHamson, M.X. Goemans, M. Mihail and V. V. Vazirani, (1993).
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[17]
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A
primal-dual ap-
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ACM
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Prog. Study, 13, 121-134.
27
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