4
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HD28
M4 1
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no.2.o>4f-
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
FINDING MINIMUM-COST FLOWS
HY DOUBLE SCALING
Ravindra K. Ahuja
Andrew V. Goldberg
James B. Orlin
Robert E. Tarjan
Sloan W.P. No. 2047-88
August, 1988
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
FINDING MINIMUM-COST FLOWS
I$Y DOUBLE
SCALING
Ravindra K. Ahuja
Andrew V. Goldberg
James B. Orlin
Robert E. Tarjan
Sloan W.P. No. 2047-88
August, 1988
Finding Minimum-Cost Flows by Double Scaling
Ravindra K. Ahuja
Andrciv
V.
'"^
Goldberg
James B. Orlin
'
'
Robert E. Tarjan
June. 1988
ABSTRACT
Several researchers have recendy develof)ed
new
aJgorithms for the minimuin-cost flow problem.
several
0{nm
of
log log
these
techniques
U log(nC))
arc capacity U, and
yield
to
an
techniques that give fast
In this paper
algorithm
time on networks with n vertices,
maximum
arc cost
are the capacity-scaling approach of
m
we combine
running
arcs,
in
ma.ximum
magnitude C. TTie major techniques used
Edmonds and Karp,
the excess-scaling
approach of Ahuja and Orlin, the cost-scaling approach of Goldberg and Tarjan,
and the dynamic
with large
We
tree data structure of Sleator
maximum
also obtain a slightly better
problem.
In
we
arc capacity,
addition,
we
obtain a similar but slightly better bound.
bound
discuss
and Tarjan. For nonsparse graphs
a
for the (uncapaciiated) transportation
capacity-bounding
approach
to
the
minimum-cost flow problem.
Son
Yjxmg
ReseanA,
On
MA
Managemem, MJ.T., Cambridge,
02139. Reward) p»iti»Uy fupported by an NSF Presidcntia]
FeUowihip. Cooiraa M51517ECS, gram AFORS-88-0088 from the Aii Force Office of Scientific
granu from Analog Devices, Apple Computer IiK, and Prirae Computer.
Scbool of
Irrvesiigaior
aixJ
leave from Ijylian Institute of Technology, Kanpur, India.
Department of Computer Science, Stanford University, Stanford,
Presidential
NJ
Young
Investigator Grant
CCR-8g58097 and
an
IBM
CA
94305 Research
partially lupporled
by
NSF
Faculty Development Award.
Department of Computer Science, Princeton University, Princeton, NJ 08544 and ATAT Bell Laboratories, Murray Hill,
07974. Research partially fupponed by National Science Foundabon Grant DCR-8605962 and Office of Naval
Research Coniraci N00014-87-K-0467.
Finding Minimum-Cost Flows by Double Scaling
Ravindra K. Ahuja
Andrew
V.
2
1
'
Goldberg
'
James B. Orlin
Robert E. Tarjan
June. 1988
Introduction
1.
The minimum-cost
circulation
problem
calls for finding a circulation
network whose arcs have flow capacities and costs
this
problem
V and
tex set
assume
Graph
that
G is
arc set E.
G
arc capacity,
and
C
are integral.
For ease
f/
^
4.
e.g. [9,1 1,13.18,22].
maximum
Let
absolute value of an arc cost, respectively.
are subject to the assumption that
in stating time
bounds,
a network
all
(without loss of generality) that
two unless an
G is a real-valued
Time bounds con-
arc capacities, or all arc costs, respectively,
we assume
All logarithms in this paper are base
A pseudoflowfon
two
See
We require that the cost function be antisymmetric, i.e. c(v,w) = —c(w,v) for
We denote by n, m, U, and C the number of vertices, number of arcs, max-
taining t/ or
and
for studying
a network if each arc (v.w) has a rwnnegative real-valued capacity u(v,w) and a real-
arcs (v.w) e E.
imum
Our framework
contains no multiple arcs; our algorithms easily extend to allow multiple arcs.
valued cost c(v,H').
all
unit of flow.
in a
G = (V,£) be a directed graph with verWe require G to be symmetric, i.e. (v,>v) € E if and only if (w, v) € E. We
as follows.
is
p)er
of minimum cost
C ^2
explicit base is given.
function on the arcs satisfying the following
constraints:
f(v,w) ^ u (v,w) for
Sloui School
Young
(rf
MiiMjeraent, M.I.T., Cambridge,
Invertig«x» FdJowihip,
Re»carcfi, and gTxnU from
On lave from
all
MA
Contna 8451517ECS.
(v.w) e
AFCMiS 88-008S from
Inc.,
b)'
the Air Force
ui
(1)
NSF
Pretidentiil
Of6oe of
Scientific
aod Prime Compoier.
Indiui Institute of Technology, Kjuipur. India.
Department of Computer Science, Stanford University, Stanford,
Presidential
(capacity constiainl)
02139. Rete*rcl) p»rti*lly fupported
grant
AnaJog Devica, Apple Computer
E
Young
Investigator Grant
CCR-8858097 and
an
IBM
CA
94305. Research partially nipported by
Faculty Development
Department of Computer Science, Princeton University, PriiKCtoo, NJ 08544 and
N3 07974. Research
pairiaDy supported by National Science Foundation Grant
Research Contract N00014-87-K-O467.
NSF
Award
AT4T BeU
Laboratoiies. Murray Hill,
DCR-8605962 and
Office of Naval
-2-
f(y,w) = -f(w,v)
For a pseudoflow /and
for all (v.h-)
£
€
balance bjiv)
a vertex v, the
¥v)=
I
(2)
the net flow into v:
at v. is
/(u.v).
ii:(u,v)€
The
(antisymmetry constraint).
(3)
£
cost of a pseudoflow /is defined as follows;
cost if)
£
=
C(V.H')/(V,H')
(4)
/(v,»')>0
A
pseudoflow /is
a circulation if the
bf{\)
The minimum-cost
circulation
foIJowing constraint
=
is satisfied:
for every vertex v.
problem
is
(5)
that of finding a circulation
of minimum cost in a
given network..
The minimum-cost
circulation
Among
See e.g [9,11,13,18^2].
the
the best worst-case time bounds.
parameters
n.
[6], the
(m
log/i)
strongly polynomial
One
known
thirty years.
algorithms for this problem, there are three that have
Each of these algorithms
is
best for a different range of the
m, U, and C. The algorithms are the 0((,m log (/)(m
Edmonds and Karp
and the (9((m
problem has been intensively studied for over
+n
logn))-time method of
0(,nm log(n^/m) log (nC))-time method of Goldberg and Tarjan
-t-n
logn))-time method of Oiiin [16].
The
last
[11],
of these methods
is
*.
important idea
is
common
to all three of these algorithms, that
of scaling or successive
approximation. Scaling methods work by solving a sequence of more -and -more accurate approximations to the original problem. The approximations are obtained either by relaxing some of the
numerical constraints or by igrxjring some of the precision of the numeric parameters. Scaling
was introduced by Edmonds and Karp, whose algorithm
refinement of that of
A
network aJgorilhm
is
Edmonds and Karp
strongly polynomial
if its
metic oj>erations take unit lime, and also polynomial
that
/»
,
Oriin's algorithm
combines capacity scaling with repeated
running time
in
scales capacities.
m,
is
polynomia]
logt/, aixJ logC,
lake time polyiwmial in the number of bits of the operands. See [21).
in
n and m, assuming
arith-
assuming arithmetic operations
is
a
arc-
shrinking.
The algorithm of Goldberg and Tarjan
scales costs.
It
relies cojcially
by Tardos (21] and independently by Bertsekas
e-optiraality, intnxluced
on the notion of
[3,4].
Consideration of these algorithms suggests the question of whether capacity scaling and
combined
cost scaling can be
any algorithm obtainable using
to yield an algorithm faster than
A
cither technique alone, at least for a suitable range of n, m, U, and C.
was obtained by Gabow and Tarjan
lines
algorithm.
Although
this
time bound
[8],
who developed
an
0(nm
first
result along these
logn logt/ log(nC))-time
never less than that of Goldberg and Tarjan [11], the
is
algorithm does not require sophisticated data structures, whereas the Goldberg -Tarjan algorithm
uses both dynamic trees [19,20.22] and finger trees [14,23].
Our
0(nm
work
of
in
efforts
We
direction.
this
obtain
log logt/ log (nC))-time algorithm for the minimum-cost circulation problem.
combines four known
Himinalion of
(1)
continuation
a
is
an
Our
result
by transforming the minimum-cost ciroilation problem
into a
ideas:
arc capacities
transportation problem.
(2)
Cost scaling within the e-optimality framework as proposed by Goldberg and Tarjaa
(3)
A variant
(4)
The dynamic
A
Oinm
of the Edmonds-Kaip approach
rel>'ing
on excess
scaling, as developed
by
Oriin.
tree data structure of Sleator and Tarjan.
simpler version of our algorithm that does not use the dynamic tree data structure runs in
+ log(nC)/log
logt/ (1
We obtain
logt/)) time.
with very large arc capacities.
We
also obtain
slightly better
improved bounds
bounds for nonsparse graphs
for the (uncapacitated) transpor-
tation problem.
The eliminauon of
tive
approach
suggested by
is
to
arc capacities is crucial to the efficiency of our algorithms.
bound the
Gabow
arc capacities
and Tarjan
[8].
by adding an
interest.
Our
Our explorations of this approach
results using this
lead to algorithms with
the transportation problem and discuss
In Section 3
we develop
scaling and e-opUmality.
we
are of
that
uses a classical net-
inside a scaling loc^.
This paper consists of six sections
lem.
we develop
approach are described toward the end of the paper.
For example, we obtain a polynomial-time algorithm for the problem
work simplex algorithm
alterna-
extra outer capadty-scaling loop, as
time bounds worse than those mentioned above, but the analytical methods
independent
An
its
a generic
In Section 4
in addition to the introduction.
relationship with the
In Section 2
minimum-cost
we
define
circulation prob-
algorithm for the transportation problem based on cost
we
refine the generic algorithm to use excess scaling,
analyze the resulting method. In Section 5
we add
the use of
dynamic
trees. In
Section 6
and
we
4-
way of
c»nsider the use of capacity bounding as an alternative
Section 7
we summarize
our results,
comment on
dealing with arc capacities.
In
the possible practicality of our algorithms, and
mention some open problems.
The Transportation Problem
2.
The minimum-cost
circulation algorithms
we develop in
Sections 3-5 will be stated in terms
of a related problem, the transportation problem. In order to discuss
We
terminology.
(5
u r = V,
tices in
We
nr=
5
4»)
call a bipartite
if
V €
T sinix, we
in
(d(v)
numbers such
demand
the
is
demand). Given an uncapacitated
sportation problem
is
can
be
denote by n
network uncapacitated
to the real
T
V
if
for each arc (v,w) with v € 7.
mapping from V
^
bipartite
partitioned
two
into
such that every arc has exactly one vertex in 5 and one
S sources and those
u(v,w) =
d(v)
G
call
problem, we need some
this
at
A
and «2
i
if «(v.h')
=
oo for
sizes
<
if
vertex v), and
network
G
v
e 5
We call ver-
each arc (y,w) with v e S and
(-d(,v)
J^ d(v)
V6 V
=
a bipartite
is
network
is
a
the supply at vertex v),
(total
supply equals
and a supply-demand vector
minimum-cost pseudoflow /satisfying
that of finding a
in T.
S and T
of S and T, respectively.
supply-demand vector d on
that div)
bipartite
^^
sets
total
d, the tran-
the following con-
straint:
bj (v)
We
it
call a
= d(y)
for all V
pseudoflow /<?aj»fe/e
if
it
e
V (supply-demand
satisfies (6).
We call
constraint)
(6)
a transportation
ptxMtm feasible
if
has some feasible pseudoflow. Checking the feasibility of a trarisportation problem can be done
using any
maximum
There
tion
is
a
flow algorithm, e.g. [1,2,9,10].
well-known, simple transformation that
will convert
any minimum-cost
problem into an equivalent transportation problem [18^4]. Given a network
construct another network
G' = (V u£,i4), where A
The
reversals for every arc (v.w) e E.
reversals
have zero capacity. Arc
define a supply-demand vector
rf
has cost
={V,E) we
contains arcs ((v,h'),v), ((v,h'),h') and their
arcs ((v,h'),v)
((v,m'),v)
G
circula-
and
((v,h'),h')
have
infinite capacity; their
zerc> aiMl arc {(y,w),yv)
has cost c(v,h').
We
on G' by
<'((v,h'))
diy)
=
= -u(v,h')
u{v,w) for
5^
u:(v,w)i
for all (v.w)
E
all
e E,
V G V.
(7)
,
-5-
Any
circulation
cost(f), given
/on G
corresponds
pseudoflow
to a feasible
f
onG'
is
of sources of G' and Vas being the
set
3.
A
minimum-cost
circulation
we
regard
E as
being the
set i4
has size
arcs.)
bounds for the transportation problem and
shall derive time
for the
if
of sinks, G' has n\ = m, and 02 =n; arc
4m. (Each source has two incoming and two outgoing
bounds
problem on G' gives a
invertible. TTius a solution to the transportation
solution to the minimum-cost circulation problem on G. Observe that
We
=
by f((.v,w),w) = f(v,w), f(iv,w),v) = u(y,w)-fiv,w) for each arc (v.w) g E.
This correspondence
set
such that cost{f)
translate
them into time
problem based on the above transformation.
Generic Algorithm for the Transportation Problem
We
obtain a generic algorithm for the transportation problem by translating the
minimum-
cost circulation algorithm of Goldberg and Tarjan [9.11] into the setting of the transportation
problem.
We
depend on
modify the algorithm
We
this modification.
to
be an augmenting path method; the time bounds
omit proofs of
many of
we
derive
the basic results, since they are direct
translations of the proofs of Goldberg and Tarjan.
Let
sink set
V,
i.e.,
max
7"
G=(V = 5ur, i4)bean
of size n2, arc
nj + «2. by
[-d(.v)
I
n,
uncapacitated bipartite network with source set
A of size m, and supply-demand
set
and rain [rtx.ni) by
V e 5), and by
riQ.
C the maximum
We
\tciOT d.
denote by
We
S of
denote the
U the maximum
total size
supply,
absolute value of an arc cost Note that
size n
i.e.,
1
of
U=
U and C are
defined so that the transformation of Section 2 from a minimum-cost circulation problem to a
transportation problem preserves the values of
For a pseudoflow /on G, we define
tation
a pseudoflow
problem
Given
is
to be solved
is feasible, i.e.,
is
by
= fr/v)^(v).
feasible if every vertex has zero excess.
there
is
some
(8)
We
shall
assume
that the transpor-
feasible pseudoflow.
a pseudoflow /. the residual capacity of an arc (v.h') with respect to /is
«/(v,w)
Arc (y,w)
C.
the excess ey(y) of a vertex v
e/(v)
Thus
U and
unsaturated
if
Uf(v,w)
A price function /? on G
is
the reduced cost of an arc (v,^)
>
= u(v,w) -f(y,w).
and jafura/erf otherwise.
a real-valued function
is
(9)
on the
vertices.
Given
a price function p,
-6
Cp(v,w)
Let e
>
0, let
/be
a pseudoflow, and let
with respect to price function p
Cp (v,w)
^ -e
p be
(10)
a price function.
Pseudoflow /is e-optimal
if
for every unsaturated arc (v,w) (e-oplimality constraint).
Pseudoflow fis cptimal with re^ct
classical result
= c(v,w)+p(,v)-p(w).
to
p
if
e-optimal for e
is
it
of iietwork flow theory and follows from
flie
= 0. The
(1 1)
following theorem
is
a
duality theorem of linear program-
ming.
Theorem
3.1 [11].
respect to
As
some
A
feasible pseudoflow
is
of
minimum
cost if and only if
is
optimal with
price function p.
Bertsekas [4] discovered, a weaker condition suffices
Theorem 32.
it
If all arc costs are integers
-
and e <
—
if all arc
costs are integers:
then a feasible flow
,
is
of minimum cost
if
IriQ
and only
if
it is
e-optimal with respea to
some
price furKtionp.
Proof. Analogous to the proof of Theorem 2.3 of
any simple cycle contains
at
most 2«o
shall
assume
It
and hence
some of the concluding remarks of Section
applies cost scaling based
maintains a price function
cally zero.
G is bipartite
7),
we
that all arc costs are integers; thus TTieorem 3.2 applies.
Our algorithm
k'Z.l.
using the faa that
D
vertices.
In the remainder of this paper (except in
[1 1],
The algorithm
p and an
on Theorem
3.2.
error parameter
e.
It
uses a cost-scaling factor
Irutially
t-C and p
is identi-
consists of repeating the following step until the termination condition
is satisfied.
Cost-Scaling Step. Let
ble pseudoflow
—
e<-
.stop.
2/1
replace e by tik.
f
/be
the identically zero pseudoflow.
and a price function p' such
Otherwise,
let
p
that
f
is
By modifying /and p,
find a feasi-
e-optimal with respect to p'.
be defined by p(v)=p'(v) + e
if
V G 5, p(v)
= p'(v)
if
v
e
If
7";
-7
Note
is reset
that
only the price function p
is
carried over from iteration to iteration; the j)seudoflow
to zero after each iteration.
Lemma 33.
At the beginning of a cost-scaling
step,
/(the zero pseudoflow)
is
e-oplimum with
respect to p.
Proof.
Any
arc (v.h-) that
cunent cost-scaling step
hand
the
is
first.
Then (v.w)
in the previous step.
but any pseudoflow has
all
Ixif
3.4.
e
Suppose
T.
arc flows finite.
Thus CpCv.w) =
TTie transportation algorithm
Cp-{v,M>)
+
jte
f
the other
,
since u(y,M')
^ -kt ^-kttO,
= <=°,
since
/
D
/te,p(>v)=p'(H').
correct and terminates after
is
thai the
be the pseudoflow and p' the price function
unsaturated with respect lo
is
optimal with resf)ecttop',andp(v)=p'(v) +
Theorem
w
/has v e S and
Then Cp(y,w) = c(y,w)'^-C = -t. SuRX)se on
that the cost scaling step is not the first
computed
is /te
unsaturated with respect to
is
O
(1
+ logi(no C))
iterations.
Proof.
ous.
Correctness follows from TTieorem 3.2. The bound on the number of iterations
is
obvi-
D
The
heart of the algorithm
is
the conversion of an e-optimal pseudoflow into an e-opiimal
feasible p)seudoflow and the corresponding modification of the price function.
We
call this the
refinement computation. Our generic refinement algorithm consists of a sequence of two kinds of
local transformations,
price function.
active if cj (v)
To
>
0.
one of which modifies the pseudoflow and the other of which modifies the
define the transformations,
An
unsaturated arc (v.h-)
we
is
use the following terminology.
eligible if Cp (v.w)
<
0.
consists of repjeating the following steps, in any order, until no vertex
is
A
vertex v
is
The refinement algorithm
active,
and then defining
push (y,w):
Applicability: Vertex v
Action: Push up lo 5
is
active, u/ (v.w)
= min
by an amount up
{
>
e/ (v), m^ (v.w)
0,
}
and Cp (v,^)
units
£ e.
of flow fiom v to
w by increasing / (v.w)
to 6.
relabel (v).
Applicability: Vertex v
is
reachable from some active vertex by a path of eligible arcs, and
there
no
is
eligible arc (v.w).
Action: Replace p(v) by
Lemma
Any pushing
3.5.
decreasesp(v) by
max
{
p(w) -c(v,w) -e).
The
3.6.
relabeling of a vertex v
at least e.
Proof. Analogous to the pnx)fs of Lemmas 5.2 and 5.3 of [ 11
Lemma
A
or relabeling step preserves e-optimality.
price of a vertex v decreases
O
by
D
].
{biQC) during refinement. Hence v
is rela-
beled Oikno) times.
Proof. Analogous to the proof of
in
Lemma
5.7 of [11]
O
is
Lemmas
where the cost scaling
(ne),
to include a cost scaling factor of k yields an
ment
and hence that any simple path
Now we
in
each arc
out of V
first
One such
along
it.
During
To
^cs, reduces the bound
list
is
/(v) for each vertex
The algorithm repeatedly attempts
When
based on the idea of finding
of
Step } (start
V and push
it
empty and repeating the following
to be
new
such a path
is
found, flow
is
pushed
onto
5.
Go to
Step
2.
by the next
onto S and repeat Step
v,
arc
on
/(v). If the
an eligible arc (v,w)
2; if not,
Step 3 (relabel). Relabel
/(v). If V is not the
If
1.
Otherwise, select some active vertex
replace the current arc
to Step 3.
S.
consists of
steps until termination occurs in Step
If there are tk) active vertices, stop.
path).
The algorithm
Let v be the top vertex on S. While the current arc of v
go
con-
Initially the current arc
Step 2 (extend path).
gible arc,
list
such paths, the algorithm uses depth-first search, implemented using a stack
find
5
v.
This
v.
to find a path of eligible arcs
a search, vertices are relabeled as necessary to extend the path.
initializing
G is bipartite,
to O (hjoz). D
{knt) bound. Observing that
a fixed incidence
from an active vertex to a vertex of negative excess.
price changes
by a factor of two. Revising the argu-
are is designated the current arc out
arc on /(v).
The bound on
[11).
describe a version of the refinement algorithm that
(v.h-).
the
is
O
is
G contains at most 2«o
augmenting paths. The algorithm uses
tains
and 5.9 of
5.7, 5.8,
go
is
end of /(v)
is
is
not eligible,
reached without finding an
found, test whether e/yv)
1
0.
If so,
eli-
push
w
to Step 4.
the top vertex
only vertex on 5, pop
it
on
5.
from
Reset the current arc of v to be the
S.
Go to Step
2.
first
arc
on
Step 4 (augment). Let
and min [uj{x,y)
that
X
on
is
We
at
I
is
6
on
S,
method
by
quantity not
the current arc out
is
Empty 5 and go
5.
short.
The augmenting
3.7.
positive
more than
minimum
the
of ej (v)
of x). For each current arc {x,y) such
to Step
1.
augmenting path version of the refinement algorithm, or the aug-
the
menting path algorithm for
Lemma
and {x,y)
5, increase /(j.y)
cal] this
any
be
We
caU an execution of Step 4 an augmentation.
path algorithm maintains the invariant that there
is
no cycle of
eli-
gible arcs.
Proof. Analogous to the proof of Corollary 5.6 of [11].
Remark. The proof of Lemma 3.7 uses the
Lemma
3.8.
Proof.
The
ple.
The
The maximum
vertices
fact that
Theorem
3.9.
size of
S
G is bipartite
a
pushes take place only along eligible
fact that
arcs.
D
most 2no.
is at
on S always define
D
path of eligible arcs.
By Lemma
O(knom) time
correct and runs in
is
sim-
is
D
gives the claimed bound on the size of 5.
The augmenting path algorithm
3.7 such a path
plus 0(«o)
time per augmentation.
Correctness follows from
Proof.
Lemma
We
3.5.
bound
number of additions
to
pcT augmentation by
Lemma
0(
also the time spent in Step 2 changing current arcs of v
I
/(v)
which
I),
By Lemma
tices, is
The time
for (he
3.8 plus at
most one per
Observe
to
we
that there are
first
relabeling.
The time
changing current
is
O(nQ)
to relabel a vertex v is
between
summed
arcs,
relabelings.
over
ver-
all
0(no). The claimed time bound follows.
D
derive time bounds for various versions of the augmenting path algo-
augmenting paths
Let us
is
The number of pops from S
does not change the current arc of v causes an
that
do an augmentation
5.
Augmenting Path Algorithm
In this secti'on
tices for
number of pops from
Oiknom). An execution of Step 2
Bounds
rithm.
the
3.6, the relabeling time and time spent
addition to S.
4.
is
S equals
The
the running time as follows.
two kinds of freedom
in Step 1,
and
in the
in this algorithm, in ihe
amount by which
the flow is
analyze the simple method in which each augmentation
as large as possible; that
is,
in Step 4,
S
is
selected as follows: 5
choice of starting ver-
= min
{
is
augmented
in
by an amount
ej(,v),
min
Step
4.
that is
{uj{x,y)
I
x
is
-\0-
onS,x * w, and
(x,y)
Lemma
saturates an arc.
5.10 of
0{nlm)
or
k
if
maximum
the
With
).
active vertices by one, reduces the
number of
Lemma
a current arc)
is
[1 1]);
is
number of
hence the
problem
total
arc saturations
O(knQm)
is
(see
running time of the augmenting path algorithm
is
O(knlm),
analogous to Dinic's bound of
O
(n^m)
to two. TTiis
[5];
by one, or
vertices of negative excess
number of
3.6 implies that the total
chosen equal
flovt'
method, each augmentation either reduces the
this
bound
is
for
indeed, the augmenting path algorithm itself can be viewed as an
analogue of Ahuja and Orlin's variant [17] of Dinic's algorithm.
We obtain a better bound
scaling.
This method
Edmonds and Karp
maximum
all
is
arc capacities are integral and not loo large)
(if all
based on Orlin's variant [16] of the capacity-scaling algorithm of
[6] for the
flow algorithm of
minimum-cost circulation problem and
Gabow
also analogous to the
is
Henceforth (except in Section 7)
[7].
we
shall
assume
that
arc capacities are integral.
The excess-scaling algorithm maintains an
is
by using excess
the largest
estimate
A
of the
maximum
power of two not exceeding U. The algorithm maintains two
0)
The sum of
(ii)
The
all
positive excesses
residual capacity of any arc
is at
is
excess. Initially
A
invariants:
most 2nA;
either infinity or an integer (possibly zero) multiple of
A.
In Step
1
,
the algorithm always chooses a starting vertex v with ej (v)
exists, the algorithm replaces
units of flow along the
(i) is
and
tries again.
maintained; immediately after
A
invariant
changes, the
(ii),
which
no such vertex
In Step 4, the algorithm always pushes
We call
a
sum of
When A =
1
A
most 2nA.
can actu-
units of flow
/2, all excesses are zero,
maximal period of time during which
A
A
that invariant
positive excesses is at
all
in turn guarantees that
be puslied each time an augmentation occurs.
algorithm terminates.
if
augmenting path. The choice of starling vertices guarantees
Augmenting by A preserves
ally
A by A/2
^ A;
slays constant i
and the
phase of
the algorithm.
Lemma
0{n
4.1.
The
total
number of augmentations done by
the excess-scaling
algorithm
is
log U).
Proof. Each augmentation either reduces the
reduces the
ing a phase.
sum of positive
excesses by A.
The number of phases
is
number of
By
(i),
vertices with negative excess
the latter case can occur only
0(log U). The bound follows.
D
by one or
0(n) limes dur-
-11-
Theorem 42.
The
0{nQ{hn + nlog
IJ)) time.
the transportation
version
excess-scaling
Using
method
this
augmenting
of the
algorithm
runs
in
in the transportation algorithm gives a
bound
for
problem of O(no(hn + n log U) logi (nC)) time
—
log U, nC
m
Choosing k = min {2+
)
path
for
any k such
bounds
yields the following time
that
l^k^nC.
for the transportation
problem:
O(nom
O(non ]ogU
Proof. Immediate from
transformation
Lemma 4.1
Section
of
0{nQ \ogU ¥ n
the
solve
will
(n logn)
We
bound on
shall
assume
exchange the source
set
we
'.
If there are
no
call this a
log
Vtlmln.
D
the transportation algorithm
minimum-cost
a
combined with
the
problem
in
bound
of
circtilation
the
that
2.
S -£ and by
a
if
{/grows nonpolynomi-
tni,
and the sink
i.e.
set
«o = "2. which
> 7nA,
it
suggests the possibility of obtaining
some
for
suitable modification of the algo-
without loss of generality: ifn\
is
<
ni,
and negate the supply-demand vector.
by changing Step
active vertices, stop. Otherwise, if
increase the flow
Otherwise, replace
actually a push, since ilf(v,w)
Cp (h'.v)
obtain
(/= (o((/j/«o)'og/i), which only holds
number of augmentations
/ij
we can
slightly,
an improvement on the
>
some
some
1
to the following:
active vertex v
on (v.w) by ey
(v)
and repeat Step
in detail,
we make
then Uf(w,v)
cost antisymmetry that Cp (v,^) <e.
>
0,
A
1
e 5 has an outgo-
and repeat Step
active vertex v has ey (v)
A by A/2
Before analyzing the modified algorithm
is
3.4.
if
is
special push.) Otherwise, if
onto S and go to Step
push
log U)))
shall present the result, since
ing arc (v.H-) such thai f(y,w)
(We
(—
m
number of augmentations. This
We modify the excess-scaling algorithm
Step 1
U <2mln\
log
algorithm
excess-scaling
4.1 only if log
ally with n. Nevertheless,
rithm.
(/jC)/log
and Theorem
log min{ n,t/)) on the
bound of Lemma
O
2
if
log t/ (l-H log (nC)/log log U)) time.
By changing
an
+ log
The excess-scaling version of
Corollary 43.
0(wj
(1
log (nC))
k A, push
'.
vertex v
'.
several observations.
A
special
which implies by e-optimality
special
1
that
push maintains the invariant
12
that there is
no cycle of
Once
eligible arcs.
a vertex v
€ 5 has zero excess,
zero until the end of the algorithm. The excess on any vertex v e
moved by
flow
total
special pushes
tations that decrease the
is at
number of
2nA. The
is at
most 2nA. Thus the
total
vertices with negative excess
other augmentations during a single phase
moved by
which
thus less than
is
5
total
most 3n A + J^ 2nA/2' = In A.
1-0
moved from
flow
It
is at
is strictly
ils
excess remains
less than 2A.
moved during augmen-
flow
most nA. The
is at
a given time until the
end of
the algorithm
its
We call
an
The modified algorithm maintains
(ii)
Every arc
'
most 2n A) and,
is at
(v,h') is either
flow
follows that once an arc (y,w) has flow exceeding In A,
arc (v.w) that can never be saturated open.
positive excesses
total
most the sum of the positive excesses,
flow remains positive until the end of the algorithm, and (w.v) can never be saturated.
sum of all
The
open or has
in place
of invariant
a residual capacity that
(ii),
is
invariant
(i)
(the
the following:
an integer
(
possibly zero)
multiple of A.
We
can verify invariant
(ii)
by induction on the number of steps taken by the algorithm,
'
simultaneously showing that every augmentation in Step 4 can actually
Lemma
i&O
4.4.
log f/
(/Jo
Suppose
+
number of augmentations made by
first
Henceforth
+
until the
to
O Oogmin{
the modified excess-scaling algorithm
zero
by
a
most one augmentation
is at
is
during phase
end of the algorithm there
more phases,
3
TTiere
augmentation from v
arc (v.w) with fiv,w)
reduced
units of flow.
n logmin{ n,U]).
that the
After 2 log «
some
total
Corisider a vertex v e S.
Proof.
flow.
The
move A
is
/.
A'
special
.
When
push.
from v per phase.
This augmentation moves
A
units of
always an arc (v.w) with /(v,w) ^ AIn.
the current value of the excess estimate
t%n
starting
such an arc exists,
if
Hence v can have
A'
= A/8«^, and
there
is
not before, the excess at v
is
is
positive
excess
n,U]) phases, and there are 0(n log min{ n,U]) augmentations
only
starting
during
from ver-
tices in 5.
Now
consider a vertex v e 7. If v does not receive additional flow from special pushes dur-
ing a phase, there can be
at
moves
2A
push
to v
starting
less than
most one augmentation
units of flow to v,
from v during the phase.
We
starting
from v during the phase.
which can account
for at
A
special
most two augmentations
charge such augmentations to the corresponding special
push. Since there are only n special pushes, the number of augmentations starting from vertices
in r,
summed over all
phases,
is
O
(n
+ no log
U). This gives the desired bound.
D
13
log
U = n((n/no) log
CoroUary
Theorem
bounds
time
presenting
In
/i),
for
since otherwise the bounds are the
same
assume
Theorem
as those in
that
4.2 and
4.3.
Assume
4.5.
U = n((n/«o) 'og^).
that log
transportation
0(nQ(km + no]ogU)logt(noC))
—m
any
for
bound
time
a
gives
algorithm
Then
0(nQ (km +
of the augmenting path algorithm runs in
the
we
method,
modified
the
such
*
the modified excess-scaling version
'og U))
/Iq
for
the
time.
Using
method
problem
transportation
l^k^rtoC.
that
this
The
choice
in
of
of
"0
k =
min
{
2
+
log U,
noC]
yields the following time
Oinom
Oinl log
Corollary' 4.6.
+
1/ (1
U log U = CI ( —
log (noC))
log (noC)/ log
if
log
U
bounds
for the transportation problem:
< 2m/no\
(— log U)))
m
if
log
U
^ 2m/ no.
logn), the modified excess-scaling version of the transportation
n
algorithm combined with the transfonmation of Section 2 will solve a minimum-cost circulation
problem
in
O
(n^ log U{\
-^
log (nC)/ log logt/)) time.
Remark. Every bound derived
is
U*
replaced
=4+
by
in this Section
smaller
aixjther
parameter
For
the
U*
the
number of augmentations
sum of
4.4 can be reduced to
bounds of Theorem
in
Lemma
4.1
positive excesses is initially at
number of augmentations during phases
O
in
(«o logl7*
which
+n
4.2, Corollary 4.3,
improved bounds are analogous
[6].
^
=4+
to the
log
A>
U*
can be reduced to
u(v,w)/m.
£
(n
\ogU') by
most nU', which implies
is
min [n.U*
Theorem
O
4.5,
O
)).
(n).
U
problem,
transportation
(v,»')«
observing that the
algorithm
.
each occurrence of the parameter
€ S
The bound on
in the
U'
if
Y, (-^(v))/n. For the minimum-cost circulation problem,
V
Lemma
remains valid
Similarly, the
that the
bound
in
Corresponding improvements
and Corollary 4.6 follow. These
bound EdmcHids and Karp obtained
for their transportation
-14-
5.
Use of Dynamic Trees
The algorithms discussed
By
complicated data structures.
bounds derived
in Section
in Section
4 are quite simple and do not require the use of any
adding the use of dynamic trees [19^0,22],
4 by almost a logarithmic
factor.
Our use of dynamic
the
trees is analo-
to their use in other networic flow algorithms [2,9,10,1], 12,19^2].
gous
The dynamic
tree data structure allows the
maintenance of a collection of vertex-disjoint
rooted trees, each arc of which has an associated value. Each tree
is
we can improve
a child of vertex w, there
is
a tree arc from v to w.
both an ancestor and a descendant of
itself.
is
Each vertex
The data
an
in-iree; that is, if vertex v
in a tree is regarded as being
structure supports the following seven
operations:
fmd-root{y):
Fmd
find-size{v}: Find
and retiim the root of the tree containing vertex
and return the number of vertices
in the tree
find-value(vy. Find and return the value of the tree arc leaving
value returned
fi.nd-Tnin{\):
of a
tie,
Fmd
to
choose the vertex
Add
real
w closest to the
number x
Combine
If v is a tree root, the
*i\\i\
find-\alue(yv)
minimum.
tree root.
to the value
of every arc along the path from v
V
is
w by making w the parent of v and gjving
This operation does nothing if v and w are in the same tree or
not a tree root
Break the
tree containing v into
two
trees
by deleting the edge joining
parent; return the value of the deleted edge. This operation breaks
infinity if v is a tree
A
sequence of
C>(/log(z
We
arc is
its
+
In case
the trees containing v and
the arc (v.w) the value x.
cutiv):
v.
v.
find-rootiv).
link(v,w,xy.
if
containing vertex
is infinity.
and return the ancestor w of v
change-value(v,x)'
v.
1))
/ tree
time ifr
is
v to its
no edge and
returns
root
operations, starting with an initial collection of singleton trees, takes
the
maximum
tree size [10,19,20,22].
use this data structure to represent a subset of the eligible current arcs. The value of an
residual capacity.
The
data structure allows flow to be
moved along an
entire path at
15-
once, rather than along one arc
at
a time.
In applying this data structure to the transportation problem,
time bounds
if
we
we can improve
take advantage of the special structure of the problem, specifically the fact that
G is bipartite, and hence so is every dynamic tree. Let us assume that
We redefine the size of a dynamic tree to be the number of vertices of T
the semantics of \hc find size operation, but does not aiTect
also
modify
the data structure so that
we
its size.
To do
dynamic
trees (the ones manipulated
of the actual
this
tree child in T.
maximum
tree contains at
it
tree size
/
trees translates into
new
according to the
of a dynamic
find-root(y) followed by the arc (find-rootiy),w)
a one-vertex
(start
at least
new
path). If
A, replace
to
Step
and repeat Step
bound
a stack
tree.
next arc on / {w). If the end of / (w)
(h',x)
is
(2
S has
a virtual
virtual trees.
+
It
1)) time,
where
dynamic
trees.
1
Sr $no, on
that defines a path
w on 5,
the
max-
of eligible
then the tree path from v to
Initially
5
is
consists of repeating the
1.
1.
stop.
Otherwise,
let
Otherwise,
no vertex has excess
if
v be a vertex of excess at least A.
2.
Step 4. Otherwise, while the current arc of
arc
S
z,
The algorithm
Step 2 (extend path). Let v be the top vertex on 5. Compute
eligible
We
virtual tree con-
path of eligible current arcs.
no vertex has positive excess,
A by A/2
Push V onto S and go
is a
dynamic
following steps until termination occurs in Step
Step 1
Every
the excess-scaling algorithm uses these modified
current arcs as follows: if vertex v appears just below vertex
empty and each vertex forms
I.
definition of size.
The algorithm maintains
tree.
T
I
a pointer to its parent
0(1) operations on
In addition to an excess estimate A, the algorithm uses a fixed
size
^
which consists
tree,
since every virtual tree vertex in
its size,
I
represent each of the actual
operations on actual dynamic trees lakes 0(1 log
The following version of
imum
We
the value of the outgoing tree arc.
most twice
at
S
I
most twice as many vertices as
Each of the deleted leaves has
deleted.
Every operation on actual
follows that a sequence of
2 is the
S
i.e.
contains. This changes
it
by the operations) by a virtual dynamic
and has stored with
number of vertices
tains a
any dynamic
n\ ^ ^2,
implementation significantly.
its
introduce an extra layer of abstraction.
tree with all leaves in
in the actual tree,
the resulting
found,
is
test
w
is
w = find root{v).
If ej
(w) <
not eligible, replace the cunent arc of
z.
If
go
w by
reached without finding an eligible arc, go to Step
whether find-siieiy) + find-sizeix) <
0,
3.
to
the
If
an
perform
so.
link(w,x, u(y,w)-f(v,w)). If not. push x onto 5. Repeat Step 2.
Step 3 (relabel). Relabel w. For each tree arc (y,w), perform cut(y). If v
only vertex on S, pop v from
S.
Go to
Step
2.
= w and
v
is
not the
16
Add A
Step 4 (augment).
Go to
cut (find-min(v)).
Pop
Step 4b.
let
change-value(v, -A).
Perform
Slep 4a.
^/w).
to
find-value(find-min(v))
= 0,
perform
Step 4b.
V from S. If
5
X = V and replace v by the
and go
While
is
empty, subtract
new
A from
ej (v) and
top vertex on 5. Let
go to Step
w = frnd-rootiy).
Otherwise,
1.
Add A
to
/ (w.x)
Step 4a.
to
This algorithm stores flow explicitly for arcs that are not in dynamic trees and implicitly for
Whenever a cut
tree arcs.
When
current value.
is
performed, the arc cut must have
the algorithm terminates, every arc
its
in a
still
flow restored lo
dynamic
tree
its
correct
must have
its
correa flow computed. These compulations have been omitted from the description above.
The
that use
analysis of this algorithm
dynamic
k=2. The
Oinom
log (z
rithm, which
+
total
Since with this method the time bound for the
not improved by using a non-constant cost-scaling factor,
number of
if v
links and cuts
0(.,nQm)
is
(see
The proof of Lemma
1)).
means
guarantees that
finds izeiv) +
is
algorithm
excess-scaling
similar lo the analysis of other r^twork flow algorithms
trees, e.g. [2,9,10,11,12,19,22].
transportation problem
choose
is
that there are 0(.n
and
w
find-sizeiyv)
z.
Thus
in the
Lemma
e.g.
7.2
dynamic
of
shall
tree version of the
[11]),
time
taking
4.1 is valid for this version of the excess-scaling algo-
logU) augmentations. The definition of
are consecutive vertices
>
performed
we
on
S,
and
w is
the algorithm
not the top vertex on S
,
then
either the tree containing v or the tre« containing >v has size
exceeding zll. Since every vertex on 5
is
O(no/z), and the time per augmentation
in a different
is
dynamic
Oi(nofz) log
(z
+
tree, the
1)).
maximum
Thus we obtain
height of S is
the following
result.
Theorem
0(no {m-¥
sportation
2
= min
{ 1
5.1.
The
dynamic
vereion
tree
of
— \ogU) log (z + 1)) time, for any z satisfying
algorithm
+
runs
—
\ogU, no
m
)
O(nom
in
C>(fio
(m +
algorithm
excess-scaling
the
1
^ z^ no. With
this
— logt/)log(z + l)log(/zoC))
runs
method, the tran-
time.
Choosing
gives the follovting time bounds for the transportation problem:
log (2
+ -^
m
logt/) log (noC)) if log
U <nom/n:
in
17
0(n ]ogU logno
Remark. The bound
better
in
Theorem
log (noC))
if
U
5.1 for the case log
U
log
S rtom/n.
^riQm/nis not an
interesting one, since a
bound of O(nom log no log ("oC)) can be obtained by implementing
ing path algorithm (without excess scaling) using dynamic trees.
the generic
augment-
D
Corollary 52. The dynamic tree implemenution of excess scaling can be used to solve the
minimum-cost
circulation
problem
Oijvn log logf/ log (jiqC)) time
in
a bound of 0(jim \ogn log (jiqC))
OiriQlogU
can
-^
reduce
U <n.
If log t/
>
n,
scaling.
bound
the
log
obtainable with a dynamic tre« implementation of the aug-
is
menting path algorithm without excess
We
if
on
number
the
augmentations
of
to
n logmin{n,t/)} by modifying the excess-scaling method as in Section 4. This
leads to the following results.
Theorem S3. Assume
that
\ogU =
0. (in/ no) logn).
Then
the
—
dynamic
no
the modified excess scaling algorithm runs in
l<z<no-
satisfying
OiriQ
Oino
(m +
—
«o
log
With
U) log
(z
+
method,
this
1) log
O(no (m +
log
U)
log
tree
Choos
(noC)) time. Choosing
z
= min
Ume bounds
log (2
0{no
0{nm
log (2
-•-
If log t/
=n
I/*,
log
m
is
in
log U,
no]
gives
—m
U
U)
log (/loC)) if log t/
log «o log (/»oC))
if
log
1/
< m;
^ m.
minimum-cost circulation problem can be solved in
time,
in this Section
as defined in the
corresponding improvements
log
n
-^ log V) log (nC))
where U'
^-
(— k)gn), the
Remark. Every bound derived
by
runs
z
for the transportation problem:
O(nom
Corollary 5.4.
—m
any
"0
{ 1 -^
2
the following
1)) time, for
algorithm
transportation
the
-^
(z
implementation of
in the
remains valid
remark
at
if
each occurrence of
the end of Section 4.
bounds of Lemmas
4.1
V
is
replaced
This follows from the
and 4.4 discussed
in that remaiic.
D
-
6,
18-
The Capacity Bounding Technique
depend cnjcially on the elimination of
TT»e results derived in Sections 3-5
the transformation to a transporution problem discussed in Section 2.
there
is
some more
direct,
turns out to be related
at least alternative,
wih
way
One may
ask whether
A
question that
to deal with arc capacities.
maximum
whether the Ahuja-Orlin excess-scaling algorithm for the
is
[1,2] generalizes in a natural
problem
analogy
or
arc capacities via
way
minimum-cost flow problem via cost
maximum Bow
the generalization of other
we show
to the
flow
scaling, in
algorithms to this problem [9,10].
We
answer to both of these questions
is
a qualified "yes."
consider the minimum-cost circulation problem as defined in Section
1,
with integer capacities
In this section
and
We propose
costs.
effect
is
a
that the
way of solving
to convert the original
problem into a sequence of
a standard "vertex-splitting" transformation.
bounded problems, we propose
+nm) \ogU
problems
in
To
each of which
The
[8].
The
solve the resulting capacity-
a modification of the Ahuja-Oriin excess-scaling
algorithm nested inside an e-scaling loop.
Oi{n^\ogm
OQog U)
by m. This idea was used by Gabow and Taijan
the arc capacities are integers bounded
method requires
problem using an outer capacity-scaling loop whose
this
resulting triple scaling
maximum
algorithm
log(nC)) time using no fancy data stnictures. Although
flow
mns
in
algorithm
this
has an inferior complexity bound as compared to the bounds obtained in previous sections, the
method and
its
Now we
lation
analysis have independent interest
give details.
problems
each
at
The outer
capacity-scaling loop constructs
iteration, a target
First, the
as follows.
to the arc capacities by doubling the current capacity and adding
The
target
problem
The
restricted
is
(Cu'.c), where
problem
is
is 1.
cost c(vi,V2)
c'(v2,wi)
=
=
for each V
e
€
two
into
V, and
Vand
More
The
restricted
lem inner and a vertex V2
Observe
that
problem
outer,
one
is
added
to the capacity of each arc
denote the restilting capacity function by «'.
graph and c
is
the original cost function.
we
formally, the restriaed problem
to obtain the graph
£' contains an
C=
arc (vj.vj)
is
call
Networic
(G',m",c')-
is
(V',£'), where
obtained by
V contains
of capacity «(vi,V2) =
an arc (v2,h'i) of capacity «"(v2,h'i)
c(y,w) for each arc (v.w) e £.
capacity zero.
next bit of precision
obtained from the target problem by bounding the flow through
each vertex v of G = (V.£)
vertices vj and V2 for each v
We
G is the original
every vertex of the target problem by m.
splitting
circu-
problem and a restricted (capacity-bounded) problem.
These are obtained from the current network
(v,w) such that the current bit of u(v,w)
two minimum-cost
= m'(v,h')
m
and
and cost
G'
also contains opposite-direaed arcs of
We
call a
vertex v; in the restricted prob-
an arc (vi,V2) a split arc.
every ciroilation /* in the restricted netwoiic corresponds to a circulation /in
the target network given by /(v.w) =/'(v2,h'i).
By
construction, the costs of
same. Conversely, for every circulation /in the target network, there
tion/ given by /(v2,h',)=/(v,w), /'(v,,V2)=
£
(X.V)6£
/(x.v).
is
/and
/* are the
a corresponding arc func-
Function/is
a circulation if
and
-19
only
if /'(v
We
i.e..
i
,
V2) < /n for every
assume (without
split arc (v
u"
,V2).
loss of generality) that the original
the identically zero arc function
initializing /aixl
i
to be zero
on
all
At
a circulatioa
is
problem has no negative
capacities;
a high level, the algorithm consists of
arcs and repeating the following steps for each bit of preci-
sion in the capacities, proceeding left-to-right through the bits:
Step
(Construct the
I.
new
target
problem by introducing the next capacity
(v.H')e E. replace «'(v.h') by 2u'(v,h')
current bit of «(v,h')
is
if the
current bit of tt(v,H')
is
zero,
For each
bit.)
by 2u'(y,w)+
1
if the
one.
Step 2. Construct the restricted problem
{G\u" ,c').
Step 3. Find an optimal solution /* to the problem {G' ,u" ,c').
Step4. Construct the circulation /in (G,«',c') corresponding to /*.
Step 5. (Modify the target problem so that the zero circulation
is
optimal.) For each (v,^) e £,
replace u\v,yv) by m'(v,h') -/(v.w).
The algorithm maintains
for the old target
Gabow and
Lemma
problem
Tarjan
6.1.
The
the invariant that
on entry
to Step
1,
The following
(as modified in Step 5).
the zero circulation
result is similar to a
is
optimal
lemma of
[8].
circulation
/computed
in Step
4
is
an optimal solution to the target problem
(G.m'.c').
Proof.
By
culation
is
induction on the
number of iterations. On
optimal for the old target problem. This
tialization,
the
is
most recent entry to Step
1
,
the zero cir-
true for the initial entry because of the ini-
and true for each subsequent entry by the induction hypothesis. Thus there
price function
p
for
old target problem.
which
By
is
some
the reduced costs of all positive -capadty arcs are non-negative in the
the construction in Step
problem have capacity one or
zero.
1. all
negative-reduced-cost arcs in the
Let/* be an optimal solution to the new
target
new
target
problem such
that/* contains no zero-cost cycle of positive flow (any such cycle can be eliminated by reducing
its
flow). Grculation /* can be
(There
is at
restricted
decomposed
most one such cycle
network
into at
most
m simple cycles, each of flow value one.
for each negative reduced-cost arc).
that corresponds
to/* thus
hdsf(y\,V2)<m
The
arc function
f on the
for each split arc (vi,v2). x.t.f
-20-
is a circulation.
lem.
Since
f
is
optimal for the target problem,
f
optimal for the restricted prob-
is
follows that any optimal solution to the rcstriaed problem corresponds to an optimal
It
lemma
solution to the target problem, and the
We
shall describe an
is true.
implementation of Step 4
and Tarjan [9,11] (already discussed
D
that uses the e-scaling
in Section 3 in the context
approach of Goldberg
of the transportation problem)
maximum
[1,2].
The
£-scaIing loop starts with a zero circulation and a zero price function; the zero circulation
is e-
with an inner loop thai
is a
modification of the Ahuja-Oriin
optimal with respect to the zero price function for e
=
C.
Then
refinement subroutine that halves e and pnoduces a circtilation
f
is
e-optimal with respect top.
When
(Recall that the restriaed network has
e
In
<
—
,
the
vertices.)
the
f
flow algorithm
method
iteratively applies a
and a price function p such
method terminates with an optimal
The formal
that
solution.
definition of the e-scaling loop is
as follows:
Step 4.1.
Let/=0.
e
=
C. and
p=0.
Step42. Foreachoutervertex V2.1etp(v2) = max {c'(v2.h'j) (v2,h',)€ £').
I
Step 43. While E
t
—
.
perform (t,f,p) <- refine {t,f ,p).
2n
The
special structure of the restriaed network allows the maintenance of the following
invariant in Step 4.3: for any residual arc (v.w) having negative reduced cost, v
We
is
an inner vertex.
say that a pseudoflow/has the inner vertex property with respect to a price function
invariant holds. Step 4.2 guarantees that this invariant holds
The
correctness of the e-scaling loop and the fact that
tions of refine follow
from the
results of
The following implementation of
described by Goldberg and Tarjan [9,1
maximum
on entry
Goldberg and Tarjan
refine is
1],
it
p
if this
to Step 4.3.
terminates after
OOog(nC))
itera-
[9,11].
based on the generic implementation of refine
specialized to use excess scaling as in the Ahuja-Orlin
flow algorithm [1^].
refine it,f.p).
Step
R.l.
(Saturate
negative-cost
arcs.)
C';,(^1,V2)<0, let/(Vi,V2) = «"(Vl.V2).
For each
split
arc
(v,,V2)
such
that
-21-
SiepR.2. Gnitialize A,e.) Let
A = 2* where
Step R.3. (Inner loop.) While
A1
Step R.3.1. While there
is
a
1
it
= flog ml
.
Replace e by e/2.
repeat the following steps:
push or relabel operatiOT
that applies, perform such an
operation.
Step
R J. 2.
A by A/2.
Replace
Step R.4. (Restore the inner vertex property.) For each inner vertex vi. replace p(vi) by
p(vi)-€/2.
Step
R J.
The push and
(cf ,p).
Return
relabel operations are defined as follows:
push(v,w).
ApplicabiliTy: Cfiv)
> A/2.
<^'(h')< A/2,
u'/'
(v.w)
>
0,
and Cp(v,w)
<
-e/4.
Action: Send min {A/2.uy-'' (v.w)) units of flow from v to w.
relabel(v).
Applicability: C/'(v)
>
Action: Replace p(v) by
Some remarks
making
moves
the
is
Cp(v,w) t -e/4 for each residual arc (v.w).
max {p(w)-c'(v,w)-e/2].
are in order here. Step R.1 saturates
all
negative reduced-cost arcs, thereby
f into an optimal pseudoflow but introducing excesses and deficits at vertices.
the excess flow
new
arxl
value of
amounts
Step R.3
to the vertices with deficits while maintaining e/2-optimality (for
Step R.4 restores the inner vertex property by changing p; after this
e).
step,
f
no longer E/2-optijna] but only £-optimal.
Step 4.3 maintains a value
exceeds A/2.
A
is
A
that is an
upper bouiyJ on the largest excess.
halved. All excesses are integers; by the time
A<
1,
aD
When no
deficits
excess
have been can-
celed.
Each pushing
step
moves excess from
a vertex with excess
exceeding A/2 to a vertex with
excess not exceeding A/2, and through an edge of cost between -e/2 and -e/4. Thus each push
is either
by
saturating or
at least
Ae/8.
it
moves
We shall
at least
A/2
units of flow; in the latter case
use this cost reduction to bound the
it
reduces the cost
off
number of nonsaturating pushes.
22
We
implied.
have omitted
a description of
These
can be found
details
how
push and relabel operations
to determine
in [2]; the
that
can be
implementation uses the analogue to the stack-
push/relabel step described there. The total time spent in such overhead in a single execution of
refine
0(nm).
is
The proofs of
the following
lemmas
are easy modifications of proofs
of analogous lemmas
in[9,!0]:
Lemma 62. The push
and re/a^e/ operations preserve e-optimality.
Lemma
op»eration relabel(v) decreases
63. 0) Each
tion of refine, the
The
number of
total
Lemma
We
push.
6.3 implies that the running time of refine
bound of 0(n^ log
shall establish a
To bound
the
+ nm)
log
X
costp(f')=
is
p
to
/
-
+ nm) on
0(nm)
the
During an execuv.
(iii)
O(n^), taking 0(nm)
is
is
0(nm).
plus 0(1) per nonsaturating
number of nonsaturating pushes,
triple
scaling algorithm.
define the cost of a pseudoflow
f
with
be
£
c;(v,w)/'(v,w)=
X
c'(v.>v)/(v,w)-
does not depend on p,
c'(y,w) f(y,w) (as defined in Section
ef'(y)p(y).
v€V
/'(v,w)>0
is a circulation its cost
21
is
(ii)
6en, for every vertex
refine
U log(nC)) bound on the
/'(>,»')>
that if
m
number of nonsaturating pushes, we
respect to a price function
costif)
can decrease
relabel operations during an execution of refine
thereby obtaining an Oi(n^ logm
Observe
at least e/4.
The number of saturating pushes during an execution of
(iv)
time,
maximum amount by which p{v)
p(v) by
i.e.
cost^C/")
= cost(/') where
1).
/'(v,>»')>0
As noted above,
a nonsaturating
vertex V that decreases p»(v) by an
We
by
want
to study
push decreases costpif) by
how much
6.4.
Let
/ be
a circulation,
capacity and cost functions u and
A
relabeling of a
costp(f) can vary during an execution of refine.
states a general result about circulations
Lemma
Ae/8.
amounts increases cost^CT) by «/'(v).
relating cosXpif) to the cost of an optimal circulation.
which
at least
c,
f
We
do
this
This requires the following lemma,
and pseudoflows.
a pseudoflow,
respectively.
and p a price function on a network with
Then
-23
C0Stp(f)-COSl(f)<
-Cp(v,H')Uy'(v,H').
5;
<
(v.w): Cf(y,w)
Proof. Lei
f be
f
the pseudoflow obtained from
by saturating
all
negauve reduced-cost
arcs.
Then
costp(f)
£
= costp(f')+
-Cp(y,w)uf'(v,w).
(v.w): e,(v,H')
<
But /can be obtained from /" by increasing flow along
a coUeciion
of paths and cycles of arcs in
Cf„, each of which has nonnegative reduced cost Thus cost(f) ^ costp(f"), and the lemma
fol-
D
lows.
Consider a time during the execution of
function p,
we
define a potential
<!>
by
<I>
=
(cosipif)
The following lemma bounding
circulation.
For the current pseudoflow
refine.
is
<I>
-
cost (/*)) /(Ae),
where
f
is
f
and price
any optimal
the heart of the analysis of nonsaturating
pushes.
Lemma 65. -^Sn^ <'^<6nm/A.
Proof.
we
We
note
e/2,
we
shall
first
that
prove that -48A£/»^ < cosip(f)
f
is
-
cost(/*)
e/2-optimal with respect to p.
By
obtain a price function p' with respect to which
^ 6cnm. To
obtain the upper bound,
decreasing prices on inner vertices by
f
is
e-opiimal and such that
inner vertex property with respect top'. Since no vertex has excess exceeding
^cosip(f) + tnm. By
Lemma
£
of
all
of all positive excesses
is at
Hence
the total
nm + 4nm = 5nm. By
vertex.
Hence
if)
residual
<0
most 2n A ^ 4nm; hence so
negative excesses. For any inner vertex
m-€f(v).
cosip-
-C^' (V,H')U/'(V.H').
(y,w):<:^'(v,w)
sura
has the
6.4,
COStp' (f)- cost (/)<
The
A < 2m,
f
v,
is
the negative of the
the total residual capacity of
capacity of
all
all
sum
arcs leaving
arcs leaving inner vertices
is
at
is
most
the inner vertex property, every negative reduced-cost arc leaves an inner
24
-c'p-
J^
(v,w): c^' (y.w)
Combining
To
inequalities gives costp
obtain the lower bound,
with respect to which/*
is
we
(v.w)
uf
< Sam.
(v.w)
<
- cost(J*) < 6tnm.
{f)
note
first
thai
we
can assume
optimal and such that lp(v) -f *(v)
I
that there is a price function
< \2tn
since the repeated executions of refine in Step 4.3 of the e-scaLng
optimal circulation /* and a
final price
change of any vertex over
total price
function
p* such that/*
the iterations
all
of
have costif) -costp-i/) <0, since /* can be converted
arcs of
Gy
,
all
/
method
will
v.
This
is
produce an
optimal with respect to p*; the
is
refine is at
into
for every vertex
p*
most
^ 6nt/2' = Men. We
by increasing flow on residual
of which have nonnegative reduced cost. But
costp-if)
- costp(f) <
X
Ip'(v) -p(v)
1
1
e/(v) ^ 48 Aen^
I
vev
sum of the
since the
excesses.
Lemma
positive excesses
Combining
6.7.
0(ji} log/«
The
is at
most 2nA, as
inequalities gives cost(f*)
A
— costp(f) ^ 48A£«^.
number of nonsaturating pushes during
an
D
execution
<I>
by
at least 1/8.
relabeling that decreases the price of a vertex
There are
at
pose that during
most log
m
+
1
iterations
Any
by x increases
of Step R.3 in
refine.
saturating
<I>
by
at
most
phases and applying
A between two
Lemma
6.5,
iterations
Consider the
= Oin\
it
can increase
<I>
by
at
j'* iteration.
<,
follows that J^p,
tnm + 48n^ + n^Oog
= 0(n^
log
m+
m + nm). D
Sup-
number of non<1)
of
at least
most n^. Summing over
we have
5;(p,/8-Xi)
J^i
is
x.
this iteration the total decrease in vertex prices is x,. If p, is the
Pil%-Xi. Changing
^
refine
push also decreases
saturating pushes during this iteration, then the iteration causes a net decrease in
Since
of
+ nm).
Proof. Each nonsaturating push decreases
<I>.
the negative of the stmi of the negative
is
1).
all
-25-
Theorem
The
6.8.
m
0((n^ log
+ nm)
for finding a
minimum-cost
circulation
runs in
logt/ log(nC)) time.
Using dynamic
way
scaling algorithm
triple
trees in the inner
they are used in
loop of the
triple scaling algorithm,
reduces the time bound to
[2],
0(nm
log (1
+
—
m
Further minor improvemenls might be possible using additional id&as in
analogously to the
\ogU
log/n)
We
[2].
log (nC)).
shall not pursue
however, since in any case the approach of Sections 3-5 produces better
this possibility further,
bounds.
We
conclude
restricted
by noting
this section
problems generated
way
that there is an alternative
minimum
(mUC)) on
log
capacity-scaling
0{nm^
U log
log
number of pivot
the
method with Step
(mC)) time,
if
in
the time to
G. This time bound
is
obtained a bound of
U = m,
and the
implemented using the network simplex method runs
3
do one pivot step
necessary to transform the graph by splitting vertices;
on every arc
who
In a restriaed problem,
steps.
a simple
reduced-cost pivot rule with
lexicography to avoid cycling. This rule was studied by Oriin [15],
0(nmU
sequence of
Namely, we can use
in the outer capacity-scaling loop.
version of the network simplex rule, specifically DanLzig's
to solve the
it
not noteworthy;
is
0(jn). In this algorithm,
suffices to
more
impose
interesting
is
a capacity
the
mere
it is
in
not
bound of m
fact that
com-
bining one scaling loop with a standard version of the network simplex algorithm gives a
polynomial-time algorithm. Without the scaling loop, the same version of the r^rwork simplex
algorithm can take exponentially
7.
many
pivot steps [24].
Remarks
We
have
0(jnm log log
log 1/
=
n
(
shown
U log («C))
—n log
n).
the
that
We
time,
minimum-cost
even
and
in
problem
circulation
0(nm
log (2
-•
can
— log U) log
m
be
solved
(/»C))
time
in
if
have derived analogous bounds for the transportation problem. Our
algorithms use scaling of both costs and capacities, combined with an augmenting path method
and an implementation based on d>Tiamic
the
minimum-cost
Oin^
log t/
(7],
(1
circulation
» log (rtC)/
0{nm
log/j loglogn),
problem
log log t/))
namely \o%U = 0(\o%n) and
trees.
if
log
log£/
If
is
dynamic
0{nm
= fJ(
—n
C = 0(log«),
trees are not used, the time bourxl for
log t/ (1
logn).
+
log (nC)/ log log (/)).
Under
the similarity assumption
our time bound with dynamic trees
which beats the best previous bound of 0(,nm logn log(
for very dense graphs (/n
=
Q
(,-
logn
)).
or
—m
is
n}
))
[11] except
26
We
tice
expect that
some version of our
algorithm, probably without
dynamic
be competitive with or superior to previously existing algorithms.
required experiments to confirm or refute this hypothesis.
We
trees, will in prac-
have no{ yet done the
Our experiments with
rithms (see e.g. [9]) suggest thai periodic scans to tighten prices
may
similar algo-
increase the practical,
though not the theoretical, speed of the algorithm.
We
have discussed
the Ahuja-Orlin
A
The
flow algorithm in the inner loop of the Goldberg-Tarjan
makes
open question
minimum-cost
minimum-
analysis of this technique uses the cost of the current jjseudoflow as a
quality. This
tantalizing
for the
capacity-bounding technique, which allows us to use a modification of
maximum
cost flow method.
measure of its
a
the analysis very intuitive.
whether there
is
circulation problem.
is
We
an
O
(nm log logn log
(r»C))-time algorithm
believe that the answer
is
yes and that a
modification of our methods will lead to such a bound. Such a result would probably give a time
bound of
(.run
over known
results.
8.
[1]
log logn) for the
maximum
flow problem, which would also be an improvement
(See [2,10].)
References
R. K. Ahuja and
J.
B. Orlin,
"A
fast
and simple algorithm for the
maximum
flow prob-
lem." Technical Report 1905-87. Sloan School of Management. M.I.T., Cambridge.
MA,
1987; Operations Research, to appear.
12]
R.K. Ahuja, J.B. Oriin. and R.E. Taijan, "Improved time bounds for the
problem."
[3]
to
"A
distributed algorithm
working paper. Laboratory
[4]
flow
^pear.
D.P. Bertsekas,
MA,
maximum
for Information
problem." unpublished
for the assignment
and EJecision Sciences, M.I.T., Cambridge,
1979.
D.P. Bertsekas, "Distributed asynchrorwus relaxation methods for linear network flow
problems," Technical Repon LIDS-P-1606, Laboratory for Information and Decision Sciences. M.I.T.. 1986.
[5]
E.A. Dinic, "Algorithm for solution of a problem of
maximum
flow in networlcs with
l)OweT estimation," Soviet Math. Dokl. 11 (1970). 1277-1280.
[6]
J.
Edmonds and R.M. Karp.
"Theoretical improvements in algorithmic efficiency for net-
woric flow problems." y. Assoc. Comput.
I?]
Mack. 19 (1972). 248-264.
H.N. Gabow. "Scaling algorithms for network problems."
(1985). 148-168.
J.
Comput. System
Sci.
31
27-
[8]
H.N.
Gabow
and R.E. Tarjan. "Faster scaling algorithms fornetworic problems," SIAM
J.
Comput., submitted.
[9]
A.V. Goldberg, "Efficient graph algorithms
for sequential and parallel
computers," Ph.D.
maximum
flow problem," J.
Thesis, M.I.T.. 1987.
[10]
"A new
A.V. Goldberg and R.E. Tarjan,
approach to the
Assoc. Comput. Mach., to appear.
[11]
A.V. Goldberg and R.E. Tarjan, "Fmding minimum-cost circulations by successive
approximation,"
[12]
A/ar/i.
qfOper. Res.,
to appear.
A.V. Goldberg and R.E. Tarjan, "Finding minimum-cost circulations by canceling negative cycles," J. Assoc.
Comput. Mach., submitted; also Proc. Twentieth Annual
ACM
Symp. on Theory of Computing (1988), 388-397.
[13]
E.L. Lawler, Combinatorial Optimization: Networks
Winston.
[14]
New
Yorlc,
NY,
Mehlhom, Data
R.
and Matroids,
Holt, Reinhart, and
1976.
Structures
and Algorithms. Volume
1:
Sorting
and Searching,
Springer- Veriag, Berlin, 1984.
[15]
J.
B. Orlin,
"On
Programming 24
[16]
J.
B. Orlin,
tieth
[17]
J.
(1985), 166-178.
faster strongly
B. Orlin and R. K. Ahuja,
maximum
Steiglitz,
Twen-
maximum
flow and
1988.
Combinatorial Optimization: Algorithms and Com-
Englewood Oiffs, NJ. 1982.
D.D. Sleator and R.E. Tarjan,
"A
data structure for
dynamic
trees," J.
Comput. System
26 (1983). 362-391.
D.D. Sleator and R.E. Tarjan, "Self-adjusting binary search trees,"
J.
Assoc. Comput.
(1985), 652-686.
E. Tardos,
ica.
[22]
cost flow algorithm," Proc.
flow problems," Technical Report No. 192, Operations Research
C.H. Papadimitriou and K.
Mach. 32
[21]
minimum
distance-directed algorithms for
MA,
5ci.
[20]
"New
Center, M.I.T., Cambridge,
plexity, Prentice-Hall.
[19]
polynomial
Annual ACM Symp. on Theory of Computing (1988), 377-387.
parametric
[18]
"A
the simplex algorithm for networks and generalized networks," Math.
"A
strongly polynomial
minimum
cost circulation algorithm," Combinator-
5 (1985), 247-255.
R.E. Tarjan, Data Structures and Network Algorithms, Society for Industrial and Applied
Mathematics, Philadelphia,
PA
1983.
28-
[23]
R.E. Tarjan and C.J. van
simple polygon."
[24]
Wyk. "An
SIAMJ. Comput.,
H.M. Wagner, "On
O
(n loglogn)-time algorithm for triangulating a
17 (1988). 143-178.
a class of capacitated transportation
problems." Management Science
5(1959). 304-318.
(25]
N. Zadeh.
"A
bad network flow problem
for the
simplex method and other
flow algorithms." Math. Programming 5 (1973). 255-266.
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