Faster and Simpler Algorithms for Multicommo dity Flow and other
Fractional Packing Problems
Naveen Garg
Jochen Konemann
y
Abstract
This paper considers the problem of designing fast, approximate, combinatorial algorithms for
multicommodity ows and other fractional packing problems. We provide a dierent approach
to these problems which yields faster and much simpler algorithms. In particular we provide the
rst p olynomial-time, combinatorial approximation algorithm for the fractional packing problem;
in fact the running time of our algorithm is strongly polynomial. Our approach also allows us
to substitute shortest path computations for min-cost ow computations in computing maximum
concurrent ow and min-cost multicommodity ow; this yields much faster algorithms when the
number of commo dities is large.
Max-Planck-Institut f ur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. Supported by the EU ESPRIT
LTR Pro ject N. 20244 (ALCOM-IT).
y
Universitat des Saarlandes, Im Stadtwald, 66123 Saarbrucken, Germany.
0
1 Intro duction
Consider the problem of computing a maximum
s
-
t
ow in a graph with unit edge capacities. While
there are many dierent algorithms known for this problem we discuss one which views the problem
purely as one of packing
s
-
t
paths so that constraints imposed by edge-capacities are not violated. The
algorithm asso ciates a length with each edge and at any step it routes a unit ow along the shortest
s
-
t
path. It then multiplies the length of every edge on this path by 1 +
for a xed
. Thus the longer
an edge is the more is the ow through it. Since we always choose the shortest
s
-
t
path to route ow
along, we essentially try to balance the ow on all edges in the graph. One can argue that, if, after
suciently many steps,
M
is the maximum ow through an edge, then the ow computed is almost
M
times the maximum
s
-
t
ow. Therefore scaling the ow by
M
gives a feasible ow which is almost
maximum.
Note that the length of an edge at any step is exponential in the total ow going through the edge.
Such a length function was rst prop osed by Shahrokhi and Matula [12] who used it to compute the
throughput of a given multicommodity ow instance. While this problem (and all other problems
considered in this pap er) can be formulated as a linear program and solved to optimality using fast
matrix multiplication [15], [12] were mainly interested in providing fast, p ossibly approximate, com-
binatorial algorithms. Their procedure, which applied only to the case of uniform edge capacities,
computed a (1 +
!
)-approximation to the maximum throughput in time polynomial in
!
1
. The key
idea of their procedure, which was adopted in a lot of subsequent work, was to compute an initial ow
by disregarding edge capacities and then to reroute this, iteratively, along short paths so as to reduce
the maximum congestion on any edge.
The running time of [12] was improved signicantly by Klein
et.al.
[8]. It was then extended and
rened to the case of arbitrary edge capacities by Leighton
et.al.
[9], Goldb erg [4] and Radzik [11] to
obtain b etter running times; see Table 1 for the current best b ound.
Plotkin, Shmoys and Tardos [10] and Grigoriadis and Khachiyan [6] observed that a similar technique
could b e applied to solve any fractional packing or covering problem. Their approach, for packing
problems, starts with an infeasible solution. The amount by which a packing constraint is violated is
captured by a variable which is exp onential in the extent of this violation. At any step the packing is
modied by a
xed amount
in a direction determined by these variables. Hence, the running time of
the procedure depends upon the maximum extent to which any constraint could be violated; this is
referred to as the
width
of the problem [10]. The running time of their algorithm for packing problems
being only pseudo-p olynomial, [10] suggest dierent ways of reducing the width of the problem.
In a signicant departure from this line of research and motivated by ideas from randomized rounding,
Young [16] proposed an
oblivious rounding
approach to packing problems. Young's approach has the
essential ingredient of previous approaches | a length function which measures, and is exp onential
in, the extent to which each constraint is violated by a given solution. However, [16] builds the
solution from scratch and at each step adds to the packing a variable which violates only such packing
constraints that are not already too violated. In particular, for multicommodity ow, it implies a
procedure which does not involve rerouting ow (the ow is only scaled at the end) and which for the
case of maximum
s
-
t
ow reduces to the algorithm discussed at the beginning of this section.
Our Contributions.
In this pap er we provide a unied framework for a host of multicommodity ow
and packing problems which yields signicantly simpler and faster algorithms than previously known.
Our approach is similar to Young's approach for packing problems. However, we develop a new and
simple combinatorial analysis which has the added exibili ty that it allows us to make the greatest
1
possible advance at each step. Thus for the setting of maximum
s
-
t
ows with integral edge capacities,
Young's pro cedure routes a unit ow at each step while our procedure would route enough ow so as to
saturate the minimum capacity edge on the shortest
s
-
t
path. This simple mo dication is surprisingly
powerful and delivers b etter running times and simpler proofs. In particular, it lets us argue that
the
contribution of a constraint to the running time of the procedure cannot exceed a certain bound which
is independent of the width
. This yields the rst strongly-p olynomial combinatorial approximation
algorithm for the fractional packing problem (Section 3).
Our approach yields a new, very natural, algorithm for maximum concurrent ow (Section 5) which
extends in a straightforward manner to min-cost multicommodity ows (Section 6). Both these algo-
rithms use a min-cost ow computation as a subroutine as do all earlier algorithms. Contradicting
popular belief that using min-cost ow as a subroutine is better, we provide algorithms for these two
problems which use shortest path computations as a subroutine and are faster than previous algo-
rithms by at least a factor min
n
n;
nk
log
k
m
o
where
k; m; n
are the number of commo dities, edges and
vertices respectively.
Table 1 summarizes our results.
T
sp
and
T
mcf
are the times to compute single-source shortest paths
and single-commo dity min-cost ow in a graph with p ositive edge lengths and costs while
T
orc
is the
time taken for each call to an oracle as in [10]. All our algorithms are deterministic and compute
a (1 +
!
)-approximation to the optimum solution. For brevity we dene
C
1
def
=
d
1
1
log
1+
1
m
e
where
(1
1
)
2
= 1 +
!
and
C
2
def
=
d
1
2
log
1+
2
m
1
2
e
where (1
2
)
3
= 1 +
!
.
Problem Previous Best Our running time Improvement
Max. multicomm. O(
!
3
m
2
log
m
) [13]
mC
1
(
kT
sp
)
!
1
ow
Fractional Packing Pseudo-polynomial
mC
1
T
orc
Strongly poly.
running time [10, 16] running time
Spreading metrics
O
(
!
3
nm
log
nT
sp
) [3]
mC
1
(
nT
sp
)
!
1
Maximum
O
(
k
(
!
2
+ log
k
) log
nT
mcf
) (2
k
log
k
)
C
2
T
mcf
In constants
concurrent ow [11, 9] (2
k
log
k
+
m
)
C
2
T
sp
For
k
m=n
Max. cost-bounded
O
(
k
(
!
2
+ log
k
) log
n
log(
!
1
k
) (2
k
log
k
+ 1)
C
2
T
mcf
log(
!
1
k
)
concurrent ow
T
mcf
) [5] (2
k
log
k
+
m
+ 1)
C
2
T
sp
For
k
m=n
Table 1
: A summary of our results
Note that in the running time of our algorithms for concurrent ow problems we can replace
C
2
log
k
by
O
(log
n
(
!
2
+ log
k
)) using a trick from earlier papers; we remark on this in Section 5.
2 Maximum multicommodity ow
Given a graph
G
= (
V ; E
) with edge capacities
c
:
E
!
R
+
and
k
pairs of terminals (
s
i
; t
i
), with one
commodity associated with each pair, we want to nd a multicommodity ow such that the sum of
the ows of all commo dities is maximized. The dual of the maximum multicommodity ow problem
is an assignment of lengths
l
:
E
!
R
+
to the edges such that
D
(
l
)
def
=
P
e
l
(
e
)
c
(
e
) is minimized. This
is sub ject to the constraint that the shortest path between any pair
s
i
; t
i
under the length function
2
l
, which we denote by
dist
i
(
l
), is at least one. Let
(
l
)
def
= min
i
dist
i
(
l
) b e the minimum length path
between any pair of terminals. Then the dual problem is equivalent to nding a length function
l
:
E
!
R
+
such that
D
(
l
)
(
l
)
is minimized. Let
def
= min
l
D
(
l
)
=
(
l
).
The algorithm proceeds in iterations. Let
l
i
1
be the length function at the beginning of the
i
th
iteration and
f
i
1
be the total ow routed in iterations 1
: : : i
1. Let
P
be a path of length
(
l
i
1
)
between a pair of terminals and let
c
be the capacity of the minimum capacity edge on
P
. In the
i
th
iteration we route
c
units of ow along
P
. Thus
f
i
=
f
i
1
+
c
. The function
l
i
diers from
l
i
1
only
in the lengths of the edges along
P
; these are mo died as
l
i
(
e
) =
l
i
1
(
e
)(1 +
c=c
(
e
)), where
is a
constant to be chosen later.
Initially every edge
e
has length
,
ie.
,
l
0
(
e
) =
for some constant
to b e chosen later. For brevity
we denote
(
l
i
)
; D
(
l
i
) by
(
i
)
; D
(
i
) respectively. The pro cedure stops after
t
iterations where
t
is the
smallest number such that
(
t
)
1.
2.1 Analysis
For every iteration
i
1
D
(
i
) =
X
e
l
i
(
e
)
c
(
e
) =
X
e
l
i
1
(
e
)
c
(
e
) +
X
e
2
P
l
i
1
(
e
)
c
=
D
(
i
1) +
(
f
i
f
i
1
)
(
i
1)
which implies that
D
(
i
) =
D
(0) +
i
X
j
=1
(
f
j
f
j
1
)
(
j
1) (1)
Consider the length function
l
i
l
0
. Note that
D
(
l
i
l
0
) =
D
(
i
)
D
(0) and
(
l
i
l
0
)
(
i
)
L
where
L
is the longest path along which ow is routed. Hence
D
(
l
i
l
0
)
(
l
i
l
0
)
D
(
i
)
D
(0)
(
i
)
L
Substituting this bound on
D
(
i
)
D
(0) in equation 1 we get
(
i
)
L
+
i
X
j
=1
(
f
j
f
j
1
)
(
j
1)
which implies that
(
i
)
Le
f
i
=
By our stopping condition
1
(
t
)
Le
f
t
=
and hence
f
t
ln(
L
)
1
(2)
Claim 2.1
There is a feasible ow of value
f
t
ln
1+
1+
3
Proof: Consider an edge
e
. For every
c
(
e
) units of ow routed through
e
the length of
e
increases by
a factor of at least 1 +
. The last time its length was increased,
e
was on a path of length strictly less
than 1. Since every increase in edge-length is by a factor of at most 1 +
,
l
t
(
e
)
<
1 +
. Since
l
0
(
e
) =
it follows that the total ow through
e
is at most
c
(
e
) ln
1+
1+
. Scaling the ow,
f
t
, by ln
1+
1+
then
gives a feasible ow of claimed value.
Thus the ratio of the values of the dual and the primal solutions,
, is
f
t
ln
1+
1+
. By substituting
the b ound on
=f
t
from (2) we obtain
ln
1+
1+
ln(
L
)
1
=
ln(1 +
)
ln
1+
ln(
L
)
1
The ratio
ln(1+
)
1
ln(
L
)
1
equals (1
)
1
for
= (1 +
)((1 +
)
L
)
1
=
. Hence with this choice of
we have
(1
) ln(1 +
)
(1
)(
2
=
2)
(1
)
2
Since this quantity should be no more than our approximation ratio (1 +
w
) we choose
appropriately.
2.2 Running time
In the
i
th
iteration we increase the length of the minimum capacity edge along
P
by a factor of 1 +
.
Since for any edge
e
,
l
0
(
e
) =
and
l
t
(
e
)
<
1 +
and there are
m
edges in all, the total number of
iterations is at most
m
ln
1+
1+
=
m
ln
1+
((1 +
)
L
)
1
=
=
m
(1 + ln
1+
L
)
=
.
3 Packing LP
A packing
LP
is a linear program of the kind max
n
c
T
x
j
Ax
b; x
0
o
where
A; b
and
c
are (
m
n
)
;
(
m
1) and (
n
1) matrices all of whose entries are positive. We also assume that for all
i; j
, the
(
i; j
)
th
entry of
A
,
A
(
i; j
), is at most
b
(
i
). The dual of this
LP
is min
n
b
T
y
j
A
T
y
c; y
0
o
.
We view the rows of
A
as edges and the columns as paths.
b
(
i
) is the capacity of edge
i
and every
unit of ow routed along the
j
th
column consumes
A
(
i; j
) units of capacity of edge
i
while providing
a benet of
c
(
j
) units.
The dual variable
y
(
i
) corresponds to the length of edge
i
. Dene the
length
of a column
j
with
respect to the dual variables
y
as
length
y
(
j
)
def
=
P
i
A
(
i; j
)
y
(
i
)
=c
(
j
). Finding a shortest path now
corresponds to nding a column whose length is minimum; dene
(
y
)
def
= min
j
length
y
(
j
). Also
dene
D
(
y
)
def
=
b
T
y
. Then the dual program is equivalent to nding a variable assignment
y
such that
D
(
y
)
=
(
y
) is minimized.
Once again our procedure will be iterative. Let
y
k
1
be the dual variables and
f
k
1
the value of
the primal solution at the beginning of the
k
th
iteration. Let
q
be the minimum length column of
A
ie.
,
(
y
k
1
) =
length
y
k
1
(
q
) | this corresponds to the path along which we route ow in this
iteration. The minimum capacity edge is the row for which
b
(
i
)
=A
(
i; q
) is minimum; let this be row
p
. Thus in this iteration we will increase the primal variable
x
(
q
) by an amount
b
(
p
)
=A
(
p; q
) so that
f
k
=
f
k
1
+
c
(
q
)
b
(
p
)
=A
(
p; q
). The dual variables are modied as
y
k
(
i
) =
y
k
1
(
i
)(1 +
b
(
p
)
=A
(
p; q
)
b
(
i
)
=A
(
i; q
)
)
4
