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WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
IMPROVED PRIMAL SIMPLEX ALGORITHMS
FOR SHORTEST PATH, ASSIGNMENT AND
MINIMUM COST FLOW PROBLEMS
Ravindra K. Ahuja
and
James B. Orlin
Sloan W.P. No. 2090-88
November, 1988
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
IMPROVED PRIMAL SIMPLEX ALGORITHMS
FOR SHORTEST PATH, ASSIGNMENT AND
MINIMUM COST FLOW PROBLEMS
Ravindra K. Ahuja
and
James B. Orlin
Sloan W.P. No. 2090-88
November, 1988
•*5s©>iSif"'
OfTV.^"
4ZQ^^
UB^^
Improved Primal Simplex Algorithms
for
Shortest Path, Assigmnent and
Minimimi Cost Flow Problems
Ravindra K. Ahuja* and James B. Orlin
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, MA. 02139, U.S.A.
*
On
leave from Indian Institute of Technology,
Kanpur
-
208 016,
INDIA
Improved Primal Simplex Algorithms
for
Shortest Path, Assignment and
Minimum
Cost Flow Problems
Abstract
we
In this paper,
present a
new
primal simplex pivot rule and analyse the
worst-case complexity of the resulting simplex algorithm for the minivun cost flow
problem, the assignment problem and the shortest path problem.
We
consider
m arcs, integral arc capacities bovmded by an integer nimiber
networks with n nodes,
number C. Let L and U denote
lower and upper bounds respectively, and Cj: denote
U, and integral arc costs bounded by an integer
the
nonbasic arcs
the
at their
reduced cost of any arc
Then our pivot
-A/2 or any
rule
is
e
U
(i, j)
feasibiUty of the basis
rule then replace
performs
(i, j).
as follows:
with
Cj:
A be
Select as
^ A/2
maintained.
is
A by A/2.
0(nm U
Further, let
;
there
is
that the simplex algorithm solves these
to
enough
Cj:
<
0(m^
this
this rule
U
logC)
in
0(n^ logC) pivots and
These algorithms use the same data structures that are typically
flexibility for fine
an
e L with
C.
and shortest path problems we
problems
implement the primal simplex algorithms
to obtain
is
no nonbasic arc satisfying
logC) pivots and can be implemented to run in
show
used
(i, j)
value
that the simplex algorithm using
Specializing these results for the assignment
time.
initial
select the leaving arc so that the strong
time.
0(nm logC)
whose
an entering arc any
When
We show
a parameter
0(nm logC)
for
network problems and have
tuning the algorithms in practice.
We
also use these ideas
label correcting algorithm for the shortest path
problem
with arbitrary arc lengths, and an improved implementation of Dantzig's pivot
rule.
Subject Qassif ication.
Networks/Graphs
Keywords
:
Improved simplex algorithms
network flow problems
Network flow algorithms. Minimum
Assignment problem. Shortest path problem.
Matchings.
for
cost flow problem.
Simplex algorithm.
Scaling,
we
In this paper,
new
present a
primal simplex pivot rule and analyze the
worst-case behavior of the resulting simplex algorithm for the
problem.
minimum
cost flow
These results are also specialized for the assignment and shortest path
problems.
We consider a directed network G = (N, A) with node set N and arc set A. Each
€ A has a non-negative integer capacity uj: and an integer cost cj; We denote
arc
(i, j)
by
n,
m, U,
maximum
incident
eind C, the
number
of nodes,
number
of arcs,
absolute value of an arc cost, respectively.
on node
the set of incoming
i, i.e.,
denote the degree of node
i
for each
i
The primal simplex algorithm
maximum
We represent by
and outgoing
arcs at
node
and
arc capacity,
A(i) the set of arcs
Let deg(i)
i.
=
|A(i)|
e N.
for the
minimum
cost flow problem, subsequently
referred to as the network simplex algorithm, has been extensively studied in the literature.
Researchers have also been interested in developing polynomial time network simplex
algorithms for the
minimum
cost flow
problem and
its
Polynomial time
special cases.
primal simplex algorithms have been developed for the shortest path problem, the
maximum flow problem.
minimum cost flow problem is
assignment problem and the
simplex algorithm for the
to Orlin [1984]
which performs 0(n^ logn) pivots
cost flow
We now
a dual simplex algorithm
due
minimum
cost
for the uncapacitated
Developing a polynomial time primal simplex algorithm for the
flow problem.
miiumum
The only polynomial time
problem
briefly
is still
an open problem.
review the literature devoted to
this area of research.
Dial,
Kamey and Klingman [1979] and Zadeh [1979] showed that Dantzig's pivot
pivoting in the arc with minimum reduced cost) for the shortest path problem
Glover,
rule
(i.e.,
with
non-negative arc lengths performs 0(n) pivots
Roohy-Laleh [1980]
in his
when
started
from an
vmpublished Ph.D. thesis developed an alternative simplex
pivot rule for the assignment problem for which the
number
of pivots
[1983]
developed a simplex method for the assigimient problem
pivots
is
in
is
O(n^).
Hung
which the number of
0(n^ log(nC)).
Orlin [1985]
showed
that for integer data the primal simplex algorithm
maintaining strongly feasible bases performs
rule
all artificial basis.
and 0(nm
U
log
(mCU))
developed by Dantzig
0(nm CU)
for Dantzig's pivot rule.
[1983].)
When
pivots for any arbitrary pivot
(This result
was independently
specialized to the shortest path
problem with
arbitrary arc lengths
and the assignment problem, the algorithm performs 0(n^ log(nC))
pivots.
Akgul
developed primal simplex algorithms for the shortest path
[1985a, 1985b]
and assignment problems
that
Using simple data structures,
perform O(n^) pivots.
Akgul's algorithms run in O(n^) time using simple data
reduced to
0(nm
structvire,
and
this
time can be
+ n^ logn) using Fibonacci heap data structure due to Fredman and
Tarjan [1984]. Goldfarb,
Hao and Kai
for the shortest path
problem whose number of pivots and running times are
comparable
to that of
Among
[1986] describe
another primal simplex algorithm
Akgul's algorithm.
the polynomial time dual simplex algorithms are the algorithms by
Roohy-Laleh [1980] for the shortest path problem, by Balinski [1985] and Goldfarb [1985]
for the assignment problem,
Recently, Goldfarb
and by Orlin
and Hao
maximum
algorithm for the
[1988]
[1984] for the
minimum
cost flow problem.
developed a polynomial time primal simplex
This algorithm performs
flow problem.
can be implemented to run in OCn^m) time. Tarjan [1988] showed
trees to further
improve the running time
This paper
is
to
0(nm
how
Let
Cj:
is
U
by incorporating a scaling techmque
Then our pivot rule is
< -A/2 or any (i,j) e U with
Cjj
is
C.
feasibility of the basis is
1.1.)
When
use dynamic
is
Our
analyzed.
in Dantzig's pivot rule.
denote the nonbasic arcs at their lower and upper bounds respectively, and
denote the reduced cost of any arc
value
and
based on the results contained in Orlin [1985] where the worst-case
essentially obtained
L and
to
pivots
logn)
behavior of the primal simplex algorithm with Dantzig's pivot rule
rule
0(nm)
there
is
Further,
(i, j).
let
A be a parameter whose
as follows: Select as an entering arc
Cj:
maintained.
> A/2;
(
We
(i, j)
e L
with
select the leaving arc so that the strong
discuss strong feasibility in detail in Section
no nonbasic arc satisfying
this pivot rule the scaling pivot rule
any
initial
this rule
then replace
We
A by A/2.
and the simplex algorithm using the scaling pivot
call
rule
the scaling network simplex algorithm.
We show
0(nm U
flow problem in
time.
that the scaling
network simplex algorithm solves the minimum
logC) pivots and can be implemented to run in
Specializing these results for the assignment
that the scaling
pivots and
and shortest path problems,
network simplex algorithm solves both of these problems
0(nm
logC) time.
0(m2
in
cost
U logC)
we show
OCn^ logC)
These results on the number of pivots are comparable
to
that of Orlin [1985] for Dantzig's pivot rule, but the
running times here are better by a
factor of n.
Intuitively, the reason v^hy the scaling
network simplex algorithm
is
good
is
that
the algorithm selects an entering arc with "sufficiently large" violation of the optimality
conditions.
helps to
This causes a "sufficiently large" improvement in the objective function and
show
that the
number
"sufficiently large" violation can
Among
other results
that solves the shortest path
also
show
we
be picked up with
obtain a scaling version of the label correcting algorithm
problem vnth arbitrary arc lengths
Dantzig's pivot rule can be implemented in
The
little effort.
that the primal simplex algorithm for the
heap operation takes 0(min
results in this
©(m^
U
paper self-contained,
Magnanti and Orlin
1.
in
miriimum
0(nm
logC) time.
cost flow
We
problem with
logC) heap operations where each
(log n, loglog(nC)) time.
paper rely on properties of the network simplex algorithm,
strongly feasible bases and the "perturbation technique".
this
Further, the arc vfith
of pivots are "sufficiently small."
we
Though we have
tried to
refer the reader to the papers of Orlin [1985]
[1988] for a
more thorough discussion on
make
and Ahuja,
these topics.
The Minunum Cost Flow Problem
In this section,
we
describe a network simplex algorithm that solves the
minimum cost flow problem in 0(nmU logC) pivots and can be implemented in
0(m2 U logC) time. The analysis of this algorithm, as specialized for the assigiunent
problem and the shortest path problem,
1.1
is
presented in Sections 2 and
3.
Background
We
consider the followdng node-arc formulation of the
minimum
cost flow
problem.
Minimize
^^i\
(i,j)€
subject to
A
^^^^
'Si'
Sxii
{j: (i,
j)
€
X
-
V)
<
In this formulation,
node
demand
a
is
i
<
Xjj
Ujj
if b(i)
We
node.
b(i),
foraU i€ N,
(lb)
e A.
(Ic)
A
e
(j, i)
{j:
=
Xji
,
for each
(i, j)
then node
>
assume
i
X
that
iG
minimum
cost flow
cost flow
problem has a
a supply node
is
=
b(i)
We
0.
and
if
<
b(i)
assume
also
then
that the
N
feasible solution.
The
feasibility of the
minimum
problem can be ascertained by solving a maximum flow problem.
The simplex algorithm maintains
solution of the
minimum
cost flow
The
partition the arc set A.
a bcisic feasible solution at each stage.
problem
B denotes
set
is
denoted by a
triple (B, L, U);
the set of basic arcs,
i.e.,
A
bcisic
B L and
,
U
arcs of a spanning tree,
U respectively denote the sets of nonbasic arcs at their lower and upper bounds.
We refer to the triple (B, L, U) as a basis structure. A basis structure (B, L, U) is allied feasible
and L and
if
by setting
(i, j)
Xj:
=
for each
(i,
j)
and by
L,
e
setting
e U, the problem has a feasible solution satisfying (lb) cind
A
dual solution to the
and a vector
bcdance constraint in (lb)
henceforth assume that
A
minimum
cost flow
c of reduced costs defined as
is
7t(l)
redundant,
=
Cj:
we
=
can
x^j
= Ujj
for each
(Ic).
problem is a vector n of node potentials
- Jt(i) + 7t(j). Since one of the mass
cj;
set
one node potential
We
arbitrarily.
0.
feasible basis structure (B, L,
U)
is
called
an optimum basis structure
if it is
possible to obtain a set of node potentials n so that the reduced costs satisfy the following
optimality conditions:
=
each
(i, j)
e B,
(2a)
q: > 0, for each
(i, j)
g L,
(2b)
(i, j)
e U,
(2c)
qj
Cjj
<
0, for
0, for
each
Given a basis
7t(l)
or
=
(2c),
structure, the
node
potentials can
be imiquely determined by setting
and then using the (n-1) equations of (2a). A nonbasic arc
whichever is applicable, violates its optimality condition.
called the xnolation of the arc
(i, j).
(i, j)
not satisfying (2b)
In such a case,
|
qJ
is
The network simplex algorithm maintains
iteration
and successively modifies the basis structure
optimum
The
basis structure.
computations
brief
a feasible basis structure at each
summary
becomes an
it
special structure of the basis enables the simplex
be performed very
to
via pivots until
we
In the following discussion,
efficiently.
give a
and the data structure required
of the network simplex algorithm
to
implement the algorithm.
The
B of the minimum
basis
this tree as "hanging"
or are
1)
node
i
1.
The
problem
stores the
first
has a unique path connecting
i
node
consist of
node
node
i
i,
and
i
itself, its
descendants of a node
For node
.
tree,
i
node
1
and
search of the
node
unique path connecting any two nodes.
amd the corresponding
violating
We refer to
its
optimality condition.
this cycle as the basis cycle.
arcs.
arc.
An
arc
which
structure.
If
(k,
*
(p, q)
new
of a
on.
the
is
node
The
i
in 0(| D(i)
this
arc (k,
i
set of
denote the nodes of the
The thread
in a "top to
/)
|
)
its
indices
way through
bottom" and
time.
[1980].
path as the
It
starts
It
to the basis
"left
for
For a detailed
In the basis there
is
a
basis path.
wdth an
initial
basic
selects a nonbcisic arc
forms a unique
cycle.
The algorithm sends a maximum possible amount
any of the lower and upper bound constraints
a blocking arc,
flow in the basis cycle
say
(p,q),
is
called a blocking
from B, This gives a new basis
then the node potentials also change. Let T^ and T2 be the two
subtrees formed by deleting arc (p, q) from the previous basis
In the
and so
in D(i).
structure (B, L, U).
blocks further increase of
The algorithm drops
say that pred(i)
The thread indices can be formed
as follows.
Adding
of flow in the bcisis cycle vdthout violating
on
1.
We refer to
bcisis
nodes
and Helgason
The network simplex algorithm works
/)
We
i.
stores the
The thread indices provide a means
node
description of the tree indices see Kermington
(k,
We
successors,
its
visiting
tree.
visiting (or finding) all descendants of a
feasible flow x
The predecessor index
1.
The descendants
an ancestor of nodes
called
finally returning to
first
node
and the depth index
)
node
at
consider
and a thread index,
a sequence of nodes that walks or threads
the nodes of the tree, starting at
and then
is
to
pred(i).
successors, successors of
Node
define a traversal of the
by performing a depth
node
the successor of
is
i
We
tree.
upward pointing (towards node
these indices are zero.
1,
induce a subtree rooted
i
subtree by the set D(i).
to right" order,
spanning
associate three indices with each
it
path (other than node
in that
of arcs in the path
predecessor of
We
1).
a
a predecessor index pred(i), a depth index depth(i),
Each node
thread(i).
is
tree arcs either are
downward pointing (away from node
in the tree:
number
from node
cost flow
basis, potentials of all
nodes
in
B where T^
contains node
Tj remain unchanged and potentials of
1.
all
.
potentials of
lower bound
all
nodes
in the
they decrease by an
in
.
T2 change by a constant amount.
previous basis structure, then they increase by an amount
amount
|
amount
We say that a basis structure
of flow from
any node
(B, L,
in the
U)
An
no upward pointing arc of the spanrung
downward
pointing arc can be at
its
is
strongly feasible
is
given in Figure
node
to
equivalent
tree
way
cam be at
1
its
basis.
1
(1,5)
(0,5)
An example
we can send
else
a positive
along arcs in the tree
of a strongly feasible basis.
is
upper bound and no
An example
(2,2)
1.
,
of stating this property
(3,4)
Figure
if
lower bound. In the subsequent discussion
be using both the definitions of the strongly feasible
feasible basis
Cj^^l
|
its
a strongly feasible basis at every
spanning tree
without violating any of the flow bounds.
that
arc at
Cj^^l
Our network simplex algorithm maintains
iteration.
D was a nonbasic
If (k,
we
shall
of a strongly
The network simplex algorithm
next basis
bound and node
to
node
(k,
k, the arc (k,
/)
,
and the
basis path
from node
node w. For example,
and
(7,5),
in Figure
and the leaving
W be the basis cycle
arcs
the entering arc
1,
is (7,5).
It
can be
the above rule guarantees that the next basis
shown
is
path from node
node w. After updating the
to
unique, then
is
it
leaves
algorithm selects the leaving
arcs, then the
arc to be the last blocking arc encountered in traversing
at
/
lower
(k, I) is at its
Let
/.
the blocking arc
If
more than one blocking
there are
If
entering arc
to the basis tree. This cycle consists of the basis
/)
flow, the algorithm identifies blocking arcs.
the basis.
that the
w is the first common ancestor of nodes k and
formed by adding arc
w
Suppose
also strongly feasible.
is
selects the leaving arc appropriately so that the
W along
is (9,10),
orientation starting
its
the blocking arcs are (2,3)
(see, e.g.,
Cunningham
We
strongly feasible.
[1976]) that
use the following
property of a strongly feasible basis.
Observation
node
w
Ln every degenerate pivot, the leaving arc lies in the basis path
1.
node
to
k.
This observation follows from the facts that the entering arc
capacity
and
basis path
that a positive flow can
from node
/
to
is
that of arc (k,
the
same except
The
/).
12
/
/)
is at its
/,
/
and the
in a
Hence no
node w.
to
upper bound then the
arc
on the
criteria to select the
that the orientation of the cycle
W
is
node k
to
node w, the
cycle consists of the basis path from
w to node
from node w to node
from node
be sent from node
has a positive
(k,
w can be blocked in a degenerate pivot.
node
In case the entering arc (k,
leaving arc
from
arc (k,
/).
defined opposite to
basis path
In this case, the leaving arc lies in the basis path
degenerate pivot.
Scaling Pivot Rule
We now
The
describe the network simplex algorithm with the seeding pivot rule.
algorithm performs a number of scaling phases. The algorithm maintains a parameter
which remains unchanged throughout a scaling phase. Hence a phase
is
called a A-scaling phase.
integrality of
A
Initially,
it is
in the subsequent phases
we
let
A = Z ^°SC
I
.
arc
is
selected so that the next basis
nonbasic arc has violation
less
than A/2 then A
is
is
it
strongly feasible.
replaced by
to maintain
in the A-scaling phase,
algorithm selects any nonbasic arc vdth violation at least A/2 and enters
The leaving
of the eilgorithm
any A ^ C, but
sufficient to select
A/2 and
a
A
the
into the basis.
When
every
new scaling phase
;
When A
phase begins.
<
1,
the integrality of costs
am optimum flow. Hence, the algorithm performs
A
A/2
1
that the algorithm has obtained
+ flogC! scaling phases.
formal description of the scaling network simplex algorithm
we
In the algorithm description,
least
shows
is
algorithm
some node
incident to
is
given below.
define a node set S such that any arc with violation at
in S.
SCALING NETWORK SIMPLEX;
begin
U) be a
let (B, L,
Ji(j)
Cjj
A:
feasible basis structure,
be the corresponding node potentials and
be the corresponding reduced
costs;
= Z^logCl
whUe A > 1 do
begin
S
:
= N;
while S ^
do
begin
select a
node
i
in
S and delete
for each nonbasic arc
(k,
/)
it
from
in A(i)
S;
with violation at least A/2 do
begin
add
arc (k,
/)
to the basis creating a cycle
augment mjiximum
possible flow in
drop the appropriate blocking
W;
W and
arc;
update the node potentials and add
to
S each
node whose potential changes;
end;
end;
end;
end;
The following observations can be made about
Observation
for
2.
I
the algorithm:
The algorithm maintains the invariant
which some nonbasic
arc in A(i)
may have
that the set
S contains
violation at least A/2.
all
nodes
i
.
This property follows inductively from the facts that at the beginning of each
S = N,
scaling phase
and whenever the
Nodes are deleted from S only when each
Hence when S
Observation
incident arc has a violation less than
A/ 2.
3.
it is
empty, the A-scaling phase ends. This also shows that at the end of the
is
The A
is
an upper bound on the
of a scaling phase (except the
a phase.
to S.
phase, the algorithm terminates with a
last scaling
node changes,
added
potential of a
first
At the beginning of the
phase).
first
minimum
maximum
cost flow.
arc violation at the beginning
This property need not be maintained during
scaling phase, the
nuiximum
arc violation
may be
as high as (n-l)A.
Observation
4.
The algorithm scans
number
Let P^(i) denote the
is
added
S
to
at the
deg(i) =
of updates of
|
7t(i)
A(i)
|
arcs
during the
beginning of a phase and whenever
that the total scanning time during the A-sceding phase
is
whenever node
added to S.
A-scaling phase. Since a node
its
i
is
potential changes,
m + X deg(i)
•
it
follows
P^d)
N
ie
Also observe that there are two bottleneck operations in the scaling network
simplex algorithm:
to
augmenting flows
update node potentials
We
is
in cycles;
and
(ii) aire
dominated by the arc scanning
scanning time. The time
time.
next analyze the complexity of the scaling network simplex algorithm.
analysis depends
13
(i)
on the
perturbation technique that
The
we discuss next.
Peturbation Technique
The
perturbation technique
is
a
well-known method
simplex algorithm for linear programming.
for avoiding cycling in the
The technique
right-hand-side vector so that every feasible basis
is
non-degenerate and an optimum
solution of the perturbed problem can be easily converted to an
original problem.
Cuimingham
[1976]
showed
techiuque for the network simplex method
technique.
is
slightly perturbs the
optimum
solution to the
that a specific variant of the perturbation
equivalent to the strongly feasible basis
Orlin [1985] extended Cunningham's result to allow for
more general
perturbations.
The minimum cost flow problem can be perturbed by changing the
supply/demand vector b to b + e We say that e = (e^ , £2 /•••/£„) is a feasible perturbation
.
if it satisfies
the following conditions:
10
e^
(i)
>
for all
n
= 2
i
;
n
^t^<\;and
(ii)
i=2
(iii)
=- 2-
ei
i
One
£i
= 2
possible choice for a feasible perturbation
is
ej
= 1/n
for
i
=
2,
.
.
.
,
n (and
thus
e^
= -(n - l)/n
positive
Another choice
).
is
Ej
= a' for
i
=
2,
.
.
.
,
n with a chosen as a very small
,
number. The perturbation changes the flow on basic
that perturbation of
arcs.
It
can be easily shown
b by a feasible perturbation e changes the flow on basic arcs in the
following manner:
Observation
If
5.
B and
D(j)
is
descendants of node
then the pertiirbation decreases the flow in arc
(i, j)
by
(i, j)
j,
is
a
dov^ward
pointing arc of tree
the set of
Ev
I
S^^k
^
•
k € D(j)
L
<
Since
*-
Ck
k G D(j)
Observation
If
6.
descendants of node
(i,
<
Since
ke
Z
Ev
i,
"^
*^^ resulting flow
^'
j)
is
is
non-integral
and thus non-zero,
an upward pointing arc of tree B and D(i)
then the perturbation increases the flow in arc
^'
*^^ resulting flow
is
is
by
(i, j)
the set of
e.
J^ Er.
I
k € D(i)
i)
non-integral and thus non-zero,
D(i)
In the perturbed
problem flow on every arc
non-degenerate pivot and the leaving arc
establishes a connection
is
is
non-zero, hence every pivot
uniquely identified.
The following
between a primal simplex algorithm in which strongly
is
a
result
feasible
bases are maintained for the original problem, and the ordinary primal simplex
algorithm for the perturbed problem.
Lemma
1.
minimum
For any basis structure (B, L, U) of the
[1985]).
follamng statements are equivalent:
(Cunningham [1976], Orlin
cost flow problem, the
(i)
(B,L,U)
(ii)
(B, L,
U)
is
strongly feasible.
is
feasible if b is replaced
by b + e for any feasible perturbation e
.
,
11
Proof,
less
=>
(i)
than
Since
(ii).
(B, L,
U)
is
upward pointing
strongly feasible, each
arc has flow
upper bound and each downward pointing arc has flow greater than
its
decreases the flow on
upward pointing
the flow on
The perturbation increases
downward
arcs
by
at
most one unit and
Hence
pointing arcs by at most one vmit.
zero.
U)
(B, L,
is
a
feasible basis structure for the perturbed problem.
(ii)
=>
(i).
Consider a feasible basis structure
non -integral
in the basis has a positive
U)
(B, L,
flow.
If
we
for the perturbed problem.
consider the
same
Each arc
basis tree for the
on upward pointing arcs decrease, flows on downward
pointing arcs increase and the resulting flows are integral. Consequently, x^: < Uj; for an
original problem, then flows
upward
pointing arc
Xj;
(i, j),
>
for a
downward
pointing arc
(i, j),
and B
strongly
is
feasible for the original problem.
We
algorithm.
shall
now
analyze the worst-case complexity of the scaling network simplex
Our arguments
the perturbed problem.
on the scaling network simplex algorithm as applied
rely
However, because
Lemma
of
actually be done; maintaining strongly feasible bases
same as
since the pivot sequences are the
14
the perturbation need not
sufficient to realize these
bovmds
for the perturbation technique.
Complexity Analysis
Let
z denote the objective function value of the
the begiiming of
(B, L,
is
1,
to
U) denote
function value.
I
(i,i)€
Further,
the basis structure.
Cjj
Xij
A^'
^^
=
I
z - z*
qj
xj:
(i,i)e A^'
^'
(i, j)
e
I
(i,j)6
A
'
respect to the objective function
'
i
Xii
'
+
€
z* denote the
bovmded by
mUA
I
-
7i(i)
x^j
X
e
A
'
N
I
ie
set of
(i, j)
let
(i,i)GA
^
Consequently for a given
is
+
Cji
A
cost flow
problem
Let x denote the corresponding flow and
the A-scaling phase.
We first show that
minimum
{j: (i, j)
.
let
objective
Observe that
I
{i,j)e
e A)
optimum
at
Jt(j)
Xjj
A
'
{j: (i, j)
€
AT
7c(i)b(i).
N
node
potentials,
C::
x.:
'
'
is
the total
improvement with
equal to the total improvement vdth
12
£
respect to the objective function
(i,
S
the objective function
C::
A
(i,j)e
Cj:
x^j
Further,
.
the total
improvement
in
A
j)e
Xj:
bounded by
is
improvement
the total
in the
'
'
foUow^ing relaxed problem:
X
minimize
<Hi
'Si
'
'
A
(i,j)€
^"^^
subject to
<
<
x^:
Ujj
for all
,
(i, j)
For any basis structure (B,L,U) of
optimum
solution of problem
=
by
setting
in
B unchanged. At
phase),
all
Xj:
for each arc
(3)
(i, j)
e A.
(1)
by setting
e
U
with
(3b)
and a corresponding flow, we can obtain an
Cj: < 0,
Xj: = Uj: for each arc (i, j) e L with
Cj:
>
0,
and by leaving the flow on other arcs
the beginning of the A-scaling phase (except
arc violations are
no more than A; hence
the objective function by at most
mUA. We have
this
when
it
the
is
first
readjustment of flows decreases
thus shov«\ that at the begiiming of a
scaling phase
z-z» <
Since in
non-increasing,
phase A ^
the
(4)
primal simplex algorithm, the objective function value
remains true during the A-scaling phase as well. In the
(4)
C and
mUA.
mUC
(
or -
objective function value of the
mU
C)
minimum
is
an upper bound
(
scaling
or lower bound) on the
We
cost flow problem.
first
is
have thus established
the foUowdng result.
Lemma
2. If z is the objective function value of the
beginning of the A-scaling phase, then z - z* < 2 m U A.
minimum
M
We now
these pivots.
consider the non-degenerate pivots and the time needed to perform
EXiring the
A-scaling phase each non-degenerate pivot
one imit of flow, and since the entering arc has violation
decreases the objective function value by at least
there are at
cost flow problem at the
most
4mU
takes 0(n) time to
augmentation time
units.
It
A/2
at least
units, the pivot
follows from
Lemma
2 that
non -degenerate pivots during a scaling phase. Each such pivot
augment
in
A/2
at least
augments
the flow
O(nmU)
time.
and update node
potentials.
Hence the
node
is
Further, since each
potential
total
updated
13
0(mU)
times,
it
follows from Observation 4 that the arc scanning time
is
0(m^ U)
per
scaling phase.
We
number
next consider the
We
perform these pivots.
is
a positive integral multiple of
1
.
.
/n,
.
and hence each pivot
carries at least
Lemma
follows from
It
O(nmU) degenerate
2 that there are
1
/n
for the original
problem decreases the objective function value of the perturbed problem by
units.
on
1/n). For this perturbation, the flow^
,
Thus during the A-scaling phase, each degenerate pivot
units of flow.
to
use arguments based on analyzing the perturbed problem
first
with perturbation e = (-(n - l)/n, 1/n, 1/n,
each arc
and the time needed
of degenerate pivots
at least
A/2n
pivots during a
scaling phase.
We now
compute
we
For this purpose,
the time spent in the degenerate pivots.
n
use the perturbation defined by e =
X
deg(i)/2m
= 2
(i
deg(n)/ 2n), where deg(i) =
feasible perturbation.
We
first
|
A(i)
|
It
.
can be easily verified that
be handled
It
(k,
/) is
at its
lower boimd. The case when arc
w
(
the
first
degenerate pivot, arc
upward pointing
(p, q) is a
.
.
this perturbation is
,
a
We
(k, /) is at its
assume
that the
upper boxmd can
similarly.
follows from Observation
from node
.
determine the flow augmented in the perturbed problem
corresponding to a degenerate pivot in the original problem.
entering arc
deg(3)/2m,
deg(2)/2m,
,
common
1
ancestor of nodes k and
(p, q) is either
arc at
downward
its
downward
a
lower bound.
It
/)
to
node
pointing arc at
k.
its
Since this pivot
is
a
upper bovmd or an
follows from Observations 5
and
6 that
if
pointing arc, then in the perturbed problem the flow in (p, q)
X
decreases by H(q)=
that the leaving arc, say (p, q), lies in the basis path
deg(k)/2m
,
and
q) is
if (p,
an upward pointing arc then the
k € D(q)
flow in
(p, q) in the
X
perturbed problem increases by H(p)=
deg(k)/2m
.
In either
k 6 D(p)
case, the flow
is
either
around the basis cycle
node p or node
q.
in the perturbed
problem
is at least
Since the entering arc has violation at least
decreases the objective function value
of
the perturbed
H(v),
A/2
,
problem by
where v
this pivot
at
least
H(v) A/2 units.
In this pivot operation, potentials of nodes in D(v) are changed
are
added
to the set S.
From Observation
4, this
and these nodes
pivot operation increases the scanning
14
time by at most 2mH(v)
time by
.
We
have thus shown that the pivot increases the scanning
most 2mH(v) while decreasing the objective function by
at
Consequently, the scanning time
As
objective function.
we
is
0(m^ U)
and
/,
or
we
/
method described
backward
we
until either
(k,
Glover and Klingman
locate the first
common
ancestor of nodes k
In the latter case, the time spent in this
identify a blocking arc.
In a degenerate pivot,
in Barr,
no flow
is
We
We trace the predecessor indices
/).
be charged to updating node potentials in D(v) which
time.
in the objective function is at
for degenerate pivots per scaling phase.
an entering arc
[1979] to identify the basis cycle for
and
improvement
in the
consider the time needed to identify a degenerate pivot.
incorporate an additional test in the
of nodes k
H(v) A/Z
most (4m/A) times the improvement
at
the total possible
most 2mUA, the scanning time
Finally,
is
at least
is
method can
dominated by the arc scaiming
augmented so we need not consider
the
augmentation time.
We
Theorem
0(nmU
have thus showTi the following
1.
result:
The scaling network simplex algorithm solves the minimum
logC) pivots and
Remark. Although
running time applies
0(m^
the analysis
to
cost flow
problem in
U logC) time. M
carried out prinnarily
is
the original problem.
on the perturbed problem, the
The reason
for this is that
we
require the
time for a degenerate pivot to be proportional to the size of the subtree on which
potentials change,
and
irrespective of
problem, the running time per pivot
we have
2.
large
is at least
not bounded this value over
all
is
the basic cycle.
In the perturbed
as large as the size of the basic cycle,
and
pivots.
The Assignment Problem
In this section,
assignment problem.
in
how
0(nm
we
specialize the scaling
We show
that the algorithm
network simplex algorithm
to the
performs 0(n^ logC) pivots and runs
logC) time.
The assignment problem
G=(Nj u N2, A) such
minimum
that
is
a
minimum
A c Njx
cost flow problem:
problem on a bipartite network
N2. This problem can be stated as the following
cost flow
15
^
Minimize
Xj:
A
'
(5a)
Cj:
(i,j)6
subject to
X
=1,
Xj:
Z
{i:aj)G
A
non-zero flow equal to
1
problem
X
'
^i\^\
e a'
(i, j)
Glover and Klingman
Lemma
3.
the
If
number
x
in (5), a set of indices
X
=
'
solution of the assignment problem.
bound on
defined by B, the set of basic arcs.
is
For a discussion of the graph theoretic properties of strongly
.
Observe that
=1}.
(5d)
problem has 2n-l arcs out of which exactly n arcs have
We associate with any feasible flow x
Xjj
(5c)
forall(i,j)G A.
feasible bases of the assignment problem, see Barr,
A:
(5b)
AJ
basis structure of the assignment
basis of the assignment
ie Nj,
=-1, foralljeN2,
Xjj
Xjj>0,
A
for all
G A)
Ij: (i, j)
(i, j)
^ij ''ii
G x'
The following
of pivots performed
X
[1977].
defined by
We denote by
x* an
X={(i,j)
optimum
'
result enables
us
to obtain a tighter
by the scaling network simplex algorithm.
the assignment solution at the beginning of the A-scaling phase, then ex
is
e
-
ex*
< 2nA
Proof.
Let n be the node potentials corresponding to the assignment
for each
(i,j)
Z
=
(i, j)
6
g
X we note
,
Cj: Xji
X
'
=
^
-
A
ex
^
for each arc
Since
Cj:
=
that
i
XNj
^<i)
I
+
e
j
G
(i,j)
g A.
Therefore
<6)
^^P-
N2
At the beginning of the A-scaling phase (except when
Cjj
x.
it
is
the
first
phase),
16
S
(i, j)
''
'
Combining
ex*-
=
CjXj*
e X»
i
(6)
and
(7)
we
XNj
j
first
6
Ji(j)^-nA.
(7)
N2
get
ex - ex* <
For the
I
+
JtW
€
nA
(8)
.
A ^ C and
scaling phase,
lemma
the
follows from the fact that
an upper bound on the cost of any assignment solution and -nC
cost of
any assignment
We now
phase.
is
a lower
nC
is
bound on the
solution.
count the number of non-degenerate and degenerate pivots in a scaling
Each non-degenerate pivot
in the
A-scaling phase carries one unit of flow
A/2
decreases the objective function value by at least
Hence there are
units.
at
and
most 4n
non-degenerate pivots in a seeding phase. To bound the number of degenerate pivots,
we
consider the perturbation e =
Hn
- l)/n,
1/n,
.
.
.
,
1/n). Corresponding to each
degenerate pivot in the original problem in the A-scaling phase, the flow aroimd the
cycle in the perturbed
problem
is at least
1/n units and thus the objective function
value decreases by at least A/2n units. Hence there are at most 4n^ degenerate pivots
per scaling phase.
We
next consider the time needed per scaling phase.
0(n) time
takes
augment flow jmd
to
Thus the non-degenerate pivots take a
time
the
needed
for
Each non -degenerate pivot
increases the arc scanning time
total of
degenerate
by
at
O(irm) time in a scaling phase.
we
pivots,
consider
the
most 0(m).
To bound
perturbation
n
e
=
Z deg(i)/2m,
(i
deg(2)/2m,
.
.
.
,
deg(n)/2m
)
.
The same
analysis, as
done
for the
=2
minimum
cost flow
pivots also take
problem together with inequality
0(nm) time
in a scaling phase.
We
(8),
shows
that all degenerate
have thus established the following
result.
Theorem
2.
logC) time.
The scaling network simplex algorithm solves the assignment problem in
M
0(nm
17
The Shortest Path Problem
3.
we
In this section
analyze the complexity of the scaling network simplex
We show
algorithm for the shortest path problem.
0{vr logC) pivots and
We
0(nm
takes
that the algorithm performs
logC) time.
consider the problem of determining a shortest path from node
1
to all other
nodes in the network. This problem can be stated as the following minimum cost flow
problem:
X
Minimize
(i, j)
Cij^ij
(9a)
A
e
subject to
^
2-Xii
{j:(i,j)€
k]
A basis structure
is
basis
B
of
node
i
for all
1)
has
we
earlier,
The directed
thread indices.
arc (pred(i),i)
^ 0,
foralliGN,
demand
g A.
(i, j)
and the flow on
in the basis tree.)
1, i.e.,
(9c)
is
defined by B, the set of basic arcs.
a spaiming tree in which every tree arc
store the basis tree using predecessor,
out-tree property of the basis
equal to
1
this arc is
(9b)
ifi*l
-1,
a directed out-tree rooted at node
is
node
I
of the shortest path problem
dowrnward pointing. As
(except
=i
{j:(j,i)6'A)
xjj
The
r(n-l), ifi=l
^
Lxji
-
and the
imply that every node
I
D(i)
Hence every
I
.
i
fact that
depth and
each node
has exactly one incoming
(Recall that D(i)
is
the set of descendants
basis of the shortest path
problem
is
non-
degenerate.
In the basis
B
there
P(k) denote this path from
basis
B can be obtained by
for each
(i, j)
g B.
is
a unique directed path from node
node
1
setting
1 to
every other node. Let
node k The node potentials corresponding to the
jt(l)=0 and then using the equation Cj: - 7t(i) + jt(j) =
to
The directed
.
out-tree property of the basis implies that
jc(k)
=
-
^
(i, j)
Cjj
Hence
.
the negative of the length of the unique path
rt(k) is
node k
1
to
in B.
With
this
background we are ready
network simplex algorithm
every pivot
time,
we need
To bound
4.
Tfik)-
7c(k)
the
If 7t(k) represents the potential of
number
Here we assume
Z
7t*(k)
node k at the beginning of the A-scaling phase, then
Zqj - ^(1) + «(k)
=
Cii
node potentials
to the
€ P»(V)
(i, j)
=
1
node k and
and the lemma
follows.
fact that the largest
is at
least
-
(10)
Ji»(k).
Ji*(k)
is
the
first
seeding phase),
|
^ -nA,
In the
first
node potential
(11)
scaling phase,
A S C and
in the jilgorithm is at
the
lemma
follows from the
most nC and the smallest node
-nC.
In the A-scaling phase, the entering arc (k,
potentials of all nodes in D(/) increase
potential of
represent the
G P''(k)
|
nik) -
Cj;
Then
jc.
Jt(k)
to
At the beginning of the A-scaling ph<ise (except when it
> - A for each arc (i, j) e P*(k). Since P*(k) < n, (10) yields
number
that
<2nA.
reduced costs wath respect
potential
of non-degenerate pivots
k.
Proof. Let P*(k) denote the shortest path from node
(i, j)
complexity of the scaling
For the shortest path problem
the following result.
denotes the optimum node potential of node
Lemma
to analyze the
for the shortest path problem.
a non-degenerate pivot.
is
and the computational
Cj:
from node
G P(k)
any node can change
at
by
at least
/)
A/2
has violation at least
units.
Lemma
4 implies that the
most 4n times during a scaling phase.
of pivots are O(n^) per scaling phase.
A/2 and
Hence the
Observation 4 implies that the arc
n
scanning time
0(m
X deg(i)
+
i
(4n)
)
= 0(nm) per scaling phase.
We
summarize our
= 2
discussion below.
Theorem
3.
The scaling network simplex algorithms solves the shortest path problem
logC) pivots and
0(nm
logC) time.
•
in
0(n^
19
Scaling Label Correcting Algorithm
4.
label correcting algorithms for the shortest path
The
extensively studied.
and show
label correcting algorithm
this section,
we assume
problem can be stated as follows
length of
d(i),
label correcting algorithms
d(i)
+
the condition
below.
logC) time.
node
qj, for
The
C2 may be
Ahuja, Magnanti and Orlin [1988]
some path from source
The
<
0(mn
aU
(i,j)
In
i.
the optimality conditions for the shortest path
(see, e.g.,
d(j)
to satisfy condition C2.
to
node
for all
i
i
):
and
e N;
€ A.
always maintain condition CI and
basic approach
is
some
violated for
to maintain a set
arc
(i,j)
S of
and repeat the
eill
iteratively
nodes
i
try
which
for
iterative step given
Iiutially, S={1}.
Label Correcting Step.
from
S.
add
to
Scan each arc
S
( if j
is
If
S =
(i,j)
then
in A(i),
not already in
STOP
and
if
performing induction on the
d(j)
Otherwise
>
d(i)
select a
node
i
e S and delete
+ qj then update d(j)=d(i)+
Cjj,
it
and
S).
set S.
The
exponential time as showrn by
is
.
of the label correcting algorithm
The correctness
form
that this algorithm runs in
CI.
C2.
d(i) is the
incorporate a scaling technique in the generic
that the set A(i) represents the set of outgoing arcs at
For a set of distance labels
j
we
In this section,
problem have been
is
easy to establish
label correcting algorithm in its
Edmonds
[1970]
by
most generic
and Kershenbaum
[1981].
However, algorithms with improved running times can be obtained by specifying some
rules for selecting
queue runs
and adding nodes
0(nm)
in
time.
are described in Glover et
We now show
nms
in
0(mn
basic idea
number
Several other polynomial time label correcting
We call
this
algorithm the scaling
S as a
algorithms
label correcting
algorithm
label correcting algorithm.
The
cause "sufficiently large" decrease during each distance update so that the
of updates are "sufficiently small."
aj^jj
set
[1985].
repeats the iterative step given below.
2 llogCl
The algorithm which maintains the
that a scaling version of the generic
logC) time.
is to
al.
to S.
when A <
1,
The scaling
label correcting algorithm
The algorithm uses a parameter
the algorithm terminates.
A.
Initially,
A=
20
Scaling Label Correcting Step.
node
i
e S and delete
then update d(j)= d(i)+
c,;
The accuracy of
S stores
all
property
is
nodes
,
S.
S
is
Scan each arc
and add
j
by adding
j
(i,j)
is
and
not already in
i
i,
to
>
violates d(j)
violates
we need
d(i)
+ q:
A =
1
set
S=N.
>
d(i)+Cjj
Select a
d(j)
>
to
this
and the
+ A/2
S).
The
shown using inductive arguments.
we keep
S
d(j)
if
d(i)
+
Cj;
+ A/2.
Whenever
property satisfied.
set
This
a distance
Observe that
scan only outgoing arcs. This follows
whenever distance label of node i decreases, eill
Whenever a node i is deleted from S, then d(j) <
In the last scaling phase,
G A(i).
(i,j)
if
(
in A(i)
fact that
condition C2.
(i, j)
S
(i,j)
certainly true at the begirming of each scaling phase.
while scanning arcs incident to node
arc
to
which some arc
label d(i) decreases, then
from the
empty, then replace A by A/2 and
the algorithm can be
for
i
from
it
If
set
S stores
all
keep satisfying
circs (k,i)
+
d(i)
Cj:
nodes
i
+ A/2
for
for all
which some
Thus the algorithm terminates with the shortest path
.
distances.
We
treat n(k)
next
come
to the complexity of the scaling label correcting algorithm.
= - d(k) for each k e N,
decrease by at most 2nA
distance label by at least
A/2
Theorem
4.
arc lengths in
The scaling
0(nm
Thus arc scarming time
this result as a
any node decreases
is
can
label
As each update decreases a
0(nm) per
at
most 4n
scaling phase
and
theorem.
label correcting algorithm solves a shortest
logC) time.
any distance
we
K
path problem with arbitrary
Implementing Dantzig's Pivot Rule.
5.
Dantzig's rule
is
4 implies that
units, the distance label of
We state
logC) overall.
Lemma
units during a scaling phase.
times during a scaling phase.
0{nm
then
If
is to select
the entering arc
whose
a special implementation of the scaling pivot rule,
and 3 concerning
with the
the
maximum
number
violation
as
much
is
as
maximum. Thus
and the botmds
However, the time
of pivots apply.
may be
violation
0(m) per
pivot,
to
of
assignment and shortest path problems.
Orlin [1985]. Here
we show how
to
1,
2
determine an arc
which would give a
ruiming time of OCnm'^U logC) for the miiumvmi cost flow problem and
for the
Theorems
it
0(n^ log
C)
Indeed, these are the times given by
reduce the running time by nearly a factor of
n.
21
we
To obtain these improvements,
The
violations of the optimality conditions.
cis
the key of each arc.
Changing the
heap operation takes O(logn) time
Hopcroft and Ullman
[1974], or
it
if
set
Q
violation of
maximum
the heap, or finding an arc with
Q
Q
maintain a set
maintained
is
an
arc,
of arcs with positive
a heap with violation
cis
adding or deleting an arc from
one heap operation. Each
violation counts as
maintained as a binary heap
is
takes 0(log log (nC)) time
if
Q
Aho,
(see, e.g.,
maintained using the
is
data structure of Johnson [1982].
The network simplex algorithm with Dantzig's pivot
follows.
Using a heap operation, the algorithm
selects
rule
an arc wi\h maximvun
During the pivot operation, whenever the potential of a node
incident
on
The
node are examined.
that
The dominant term
performing heap operations corresponding
to this arc scanning.
same order
the
pivot rule.
Hence
minimum
cost flow
shortest path problems in
0(m^U
in
0(nm
(log n, log log (nC)}) time.
6.
Conclusions
some
better.
comparable
The network simplex
and the scaling
logC) heap operations and the assignment and
anticipate that our rule
to the best other
is
with Dantzig's pivot rule solves the
of the best implementations in practice but
We
computations
in the
have presented a variant of the network simplex algorithm
in spirit to
much
updated and the
logC) heap operations, where each heap operation takes
0(nun
We
violation.
updated then arcs
of arc scanrung for Dantzig's pivot rule
the network simplex algorithm
problem
is
violations of arcs are
necessary heap operations are performed.
algorithm has
implemented as
is
that is very close
whose worst-case bound
wUl give miming times
is
in practice that are
primal simplex pivot rule; however, our conjecture in this
matter awaits thorough computational testing.
The algorithms presented here
the similarity assumption,
i.e.,
C
illustrate the
= 0(n'') and
power of
U = ©(n*^)
for
scaling approaches.
some
k,
the scaling based
approaches are currently the best available in the worst -case complexity for
following problems
among
others
the assignment problem, the
problem, the
results in this
minimum
:
all
of the
the shortest path problem vdth arbitrary arc lengths,
minimum
cost flow
Under
problem
cycle
mean problem,
(see Ahuja, Orlin
the
maximum
flow
and Magnanti[1988]). Out
paper demonstrate that the ideas of scaling are useful not only in purely
combiiwtorial algorithms, but also in a more general perspective.
22
The time bounds
problems are both
m
CXVn
Ahuja
logC).
This time bound
log(nC)) achieved by the algorithms of
however,
[1988];
shortest
0(nm
of our algorithms for the assignment
it is
nearly as fast as the
problem and the 0(n(m + n log
is
and shortest path
not as attractive as the time
Gabow and
0(nm)
bound
of
Tarjan [1987] and Orlin and
label correcting algorithm for the
n)) successive shortest
path algorithm for the
assignment problem (see Engquist [1982] and Fredman and Tarjan
[1984]).
In addition,
they dominate the runiung times of a nimiber of algorithms that are used in practice for
solving the shortest path and the assignment problems (see for example. Glover
[1985],
Glover
etal. [1986]).
Perhaps the greatest advantage
problems
is
et al.
that they
do not require
v^ith respect to the
a special
assignment and shortest path
purpose algorithm in order to have the
advantage of a good worst-case performance. The algorithms suggested here are general
purpose network simplex algorithms.
The
rules are simple to state
implement. In addition, they allow a reasonable amount of
the entering vauiable.
ways without
increasing the computational complexity.
to
flexibility in the selection of
one can search the node
In particular,
and simple
list
S
in several different
This flexibility contrasts with
the inflexibility of the existing polynomial time primal simplex algorithm for these
problems, which are very restrictive in the selection of the entering
Another possible
ratio
by which A
scale factor of
reduced
at the
our algorithms
the choice of the scale factor, the
However, the algorithms can use any arbitrary
2.
and require 0(P
is
end of a scaKng phase. The algorithms as stated use a
network simplex algorithm can be shown
case, the
number
is
flexibility in
m^ U
of pivots
vciriables.
logpC) time.
to
sccile factor p.
perform 0(P
nm U
In this
lognC) pivots
For the assignment and shortest path problems, the
would be 0(P n^ logpC) and the time would be 0(p
nm
logpC).
Thus
from the worst-Ceise point of view, any constant value of P is optimal; empirically the
best value may be determined through computational testing.
We
have also provided a robust rule
shortest path
for the label correcting algorithm for the
problem whose worst-Ceise running time
is
only a factor logC greater than
the best other implementation.
Again, the effectiveness of our approach in practice
awaits computational testing.
Simultaneously,
we have improved
the worst-case
running time of implementing Dantzig's pivot rule by nearly a factor of
anticipate that this result
is
primarily of theoretical interest,
it
may be
n.
practical
implementation of the network simplex algorithm requires that the arc v^th
violation
is
selected.
While
if
we
some
maximum
23
In passing,
we
again mention the problem of finding a polynomial time primal
network simplex algorithm. The algorithm presented here
and
is
exponential time in the worst-case.
algorithm
is
due
is
pseudo -polynomial time,
The only known polynomial time simplex
to Orlin [1984], but this algorithm is a
dual simplex algorithm and
is
not
a "natural" implementation.
Acknowledgement
This research was supported in part by the Presidential
Young
Investigator Grant
8451517-ECS of the National Science Foundation, by Grant AFORS-88-0088 from the Air
Force Office of Scientific Research, and by grants from Analog Devices, Apple Computer,
Inc.,
and Prime Computer.
24
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