A primal-dual algorithm for solving a maximal flow problem in a class of networks with gains

a class of networks with gains

Citation for published version (APA):

Koene, J. (1979). A primal-dual algorithm for solving a maximal flow problem in a class of networks with gains. (Memorandum COSOR; Vol. 7913). Technische Hogeschool Eindhoven.

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STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum-COSOR 79-13

Apr~mal-dual algorithm for solving a maximal flow problem in a class of networks with gains

by J. Koene

Eindhoven, November 1979 The Netherlands

A primal-dual algorithm for solving a maximal flow problem in a class of networks with gains.

J. Keene

Eindhoven University of Technology Department of Mathematics

Den Dolech 2 Eindhoven

Abstract

In a network with positive gains and without absorbing circuits

(direc-ted cycles) the problem of maximizing

inpu~floW

is considered. A

primal-dual algorithm to solve the problem is given. The restricted primal pro-blem has been chosen such that it is a propro-blem which can be transformed - by a scaling procedure - to a maximal flow problem in a pure network.

1. Introduction

Special llJ1g.orithms hcwe been developed to solve maximal flow problems in

generalised networks (or networks with gains) [5,8,9,10,14,15J. In all

cases, except Jewell [10J, positive gains are considered. Fujisawa [5J

and Grinold [8

J

handle the case of maximizing output¥low, whereas Jarvis

and Jezior [9J and Jewell [10J deal with the case of maximizing

input-I

flow. Onaga [14,15

J

determines the maximal outpu~flow in such a way that

inpu~low

is as small as possible. The problem of maximizing

inpu~flow

in a network with gains is denoted as the MFG-problem. In this paper the MFG-problem is considered in a network with positive gains and without absorbing circuits (directed cycles).

It has been observed in literature that under certain conditions a gene-ralised network (G-network) can be transformed to a pure network

(P-net-work) by way of a scaling operation [7,16J. If this is possible we say

that a G-network can be scaled. Obviously a MFG-problem then becomes a maximal flow problem in a pure network (MFP-problem). Reasons why one

would like to scale a G-network are [7J:

- increase of computational efficiency - elimination of round-off errors - simple check for infeasibility

- extending literature available for p-networks to G-networks.

In the following a G-network which can not be scaled is considered. The question we want to deal with is this: suppose the MFG-problem is solved by way of some primal-dual approach. Can the restricted primal problem be chosen in such a way that the underlying restricted G-network can be

scaled and thus the restricted MFG-problem becomes a MFP-problem. An

affirmative answer to this question can be given if networks with positive gains are considered, with the extra condition that the network does not contain any absorbing circuit.

The algorithm presented here can be seen as an extension and a

modifica-tion of Jarvis~ and Jeziors' algorithm [9J. An extension in the sense that ~

involved. A modification in the sense that the restricted primal problem is t~ansformedto a MFP-problem. The possibility of reducing the restric-ted primal problem to a MFP-problem makes it easy to prove finite

conver-96!nce of the algorithm. An advantage is that there is no need to

main-tain basic solutions for the restricted primal problem in order to assure

finite convergence, as in [9J. The notation of [9J is adopted; this

makes it easier to compare with Jarvis' and Jeziors' algorithm.

The rest of this paper is divided into 5 chapters. In chapter 2 the MFG-problem is stated and some terminology used in network MFG-problems is given. Chapter 3 handles the possibility of scaling a G-network. Necessary and sufficient conditions are stated. The primal-dual algorithm to solve the MFG-problem is given in chapter 4 and a proof of its finite convergence to the optimal solution is given in chapter 5. A simple example can be found in chapter 6. Finally some comments are given.

2. The MFG-problem. Some network terminology

We consider a directed connected graph or network G(V,E). V is the set

of nodes (i

=

s,t, 1,2, ••• ,n-2) and E the set of arcs (i,j), i,j € V.

Each arc has a positive capacity c

ij and a gain (or multiplier) kij• Two nodes have a special character: the source s and the sink t.

In the following statement of the MFG-problem the variables are x .. ,

~J 2.1 MFG: maximize v s 2.2 st.

l

x ..

-

1:

k .. x, . = v i

=

s j ~J j J~ J~ s 2.3

₌

0 i

=

s,t i € V 2.4

₌

-v t i

=

t 2.5 0 :s; x, . :s; _{c, . (i, j)} € E ~J ~J

The f\Ulction {x, .} defined on E is said to be a flow if it satisfies (2.3)

~J \ ~

and (2.5); v s is the inpu~flow , v t the outPu~flow. The G-network is com- ~

pletely described by relations (2.2) - (2.5).

The dual of the MFG-problem is given by: 2.6 2.7 2.8 mn~mze \' c y L. i j i j (i,j) € E st.-1T ~ 1 s 1T ~ 0 t 2.9 ~ 0 (i,j) E: E 2.10 Y ij ~

a

(i,j) € E

The complementary slackness conditions are:

2.11 (-1T - l)v =

a

s s

2.13 2.14 ('rr. - k .. 1T. + y ..)xO'

=

0 ~ ~J J ~J ~J y .. (x .. - cO') = 0 ~J ~J ~J (i ,j) E E (i,j) E E

For our purpose it is more convenient to describe them in an other way: Put

2.15 Yi' = max [0, -(1T, - k .. 1T,)]

J ~ ~J J (i, j) E E

then the complementary slackness conditions become:

2.16 1T,

-

k .. 1T, > 0 -+ _{x ..}

₌

0 l. ~J _J ~J 2.17 0 < x ij < cij -+ 1T.~

-

k ..~J 1T._J

=

0 2.16 1T.

-

k .. 1T. < 0 -+ x ..

₌

c .. ~ ~J _J ~J ~J 2.19 v > 0 -+ 1T

=

;"1 s s 2.20 v t > 0 -+ 1Tt

=

0

A pure network is a G-network with k. . = 1 for all (i,j) E E, and as a

~J .

t

consequence of "conservation of flow" inpu1floW V

s always equals output

flow V

t.

A circuit is defined as a directed cycle. If the product of gains along a circuit C is less than (greater than) one, C is said to be an absorbing

3. Scaling of a G-network

In this chapter a G-network is considered which can be scaled. Truemper [16] gives necessary and sufficient conditions under which it is possible to scale a G-network to a P-network:

THEOREM 1. (Truemper) :

The following statements are equivalent:

(1) a G-network can be scaled to a P-network

(2) the rank of the incidencematrix of the G-network equals n-1 (n

is the number of nodes in the network)

(3) nonzero ~ (i = s,t, 1,2, ••• ,n-2) exist such that

k

ij := di kildj = 1 for all arcs (i,j) in the network.

Suppose d. ~ 0 (i

=

s,t,1,2, .•• ,n-2) are at hand; it is easy to transform

1.

the G-network into a P-network [16J:

Scalinq procedure : (a) multiply each column (i,j) of the incidence matrix

by d.•

1.

(b) divide each row j of the incidence matrix by d .• J

V

s and xij are transformed in the following way:

Ys := V

Id

s s

3.2 _y~J'... := x . ._J.J

/d.

l.

From now on it is assumed that k" > 0, V(i,j) E E

1.J

LEMMA 1. If k .. > 0, V(i,j)

E

E we can choose d. > 0, Vi

E'i

J.J ~

Proof: We have the freedom to choose d

=

+1. By induction it

s

follows that d. > 0, Vi E V since d. k . ./d. = 1,

1. 1. J.J J

kpplying the scalingprocedure to the MFG-problem the following problem (MFP-problem) is obtained: 3.3 _{MFP: maximize d s Ys} 3.4 fit·ly ..

-f

_Yji

=

_Ys i

₌

s , 1.J J 3.5

₌

0 i 'f: s,t 3.6

₌

_-Ys j,.

₌

_t c .. 3.7 0

s

y ..

s .2:L..

(i, j) E'E 1.J d_i

We note that the scaled network generally does not have integral arc

capacities even if the G-network has.

solutiontechni~are

available

for the MFa-problem which assure finite convergence even if the

arccapa-cities are irrational. Efficient (polynomially bounded) algorithms are: [1,2,3,6,11,12] •

Suppose the scaled problem (3.3) - (3.7) is solved. The result is an

op-timal ,solution

Y

ij'

Y

s and a min cut (Y, Y) •

An analogon of the max flow/min cut theorem is easy to give for a

G-network with positive gains, which can be scaled:

THEOREM 2. In a G-network with positive gains which can be scaled, the following is true:

max flow

=

in which d. 1.

c ..

l

.2:l.

= "weighted" min cut,

. d.

1.EY 1. j E

Y

are chosen such that d

=

1,d. k .. /d.

=

s 1. 1.J J l,V(i,j)EE

Proof: The maximal flow in the scaled problem is:

iEY j E Y c ..

.2:l.

d. 1. and v equals d y • s s s

The value of the dual objective function is

L

i E yo j EY 'IT

=

1/ i d. ~

=

0 Yij = 1/ d. ~ = 0 i E Y

-i E Y (i,j) E (Y,Y) otherwise c ..

...=J.

d. ~

So we have a feasible primal solution x ..

=

~J V(i,j) E E, a feasible dual solution 'IT" Vi

~

Y_ij, V(i,j} E E, such that the values of their

tive function are equal. So they are optimal.

4. A primal-dual algorithm for the MFG-problem

In this chapter the following assumptions hold: 4.1

4.2

1. k

ij > Ot V(i,j) E E .

2. there are no absorbing circuits in the G-network A primal-dual algorithm is presented to solve the MFG-problem

for networks for which assumptions 4.1 and 4.2 hold. It has the following structure:

1. Start with x .. = 0, V(i,j) E E, v = 0 and 'IT., y .. such that the dual

~J s ~ ~J

constraints and complementary slackness conditions are satisfied

except for (2.8) and (2.20).

2. Solve restricted primal problem (essentially a MFP-problem). Adjust

vaxiables v and x .. '

S 1.J

3. The current solution is optimal or new values for 'IT. (and y .. ) can be 1. l.J

given and step 2 repeated.

Because v = O,x .. '" O,V(i,j) E E is a primal feasible solution there

s ~J

is no possibility of infeasibility of the MFG-problem. The set of arcs E is partioned into three subsets:

4.3 E

1 := { (i, j) TTi - kij 'IT._J > 0, Yij

=

O}

4.4 E

2 := { (i,j) TTi - k .._1.J 'IT._J 0, Yij '" O}

4.5 E

3 := { (i,j) TT.~ - kij 'IT._J < 0, Yij > O}

Arcs in E₁ are called inactive arcs, in E₂ active arcs, in E₃ hyperactive

arcs.

Suppose that as restricted primal problem the MFG-problem is considered with arcs restricted to the set E

2• The following lemma is stated:

LEMMA 2. If 'IT. < 0, V. E V throughout the algorithm then the restricted

1. l.

1

Vi E V. In - k .. 0

Proof. Take d. = - / ., the set E₂ 1T. _'rl'.

1. _'rl'~ 1. 1.) )

and with the choice of d. it follows:

1.

k..

:= d. k . .Id. = 1 fQr all (i,j ) E E_Z

1.) 1. l.J J

Theorem 1 says that the network with arcs in E

Z can be

scaled

0

What we search for are feasible 'rl'. < 0 such that the primal relations

1.

and complementary slackness relations hold except for (Z.8) and (Z.20).

If X

ij = 0, V(i,j) € E is chosen initially we need 'rl'i < 0 such that 'rl'1 - k

ij 1Tj ~ 0, V(i,j) € E. Restricting ourselves to generalised

net-works without absorbing circuits such '11'. can easily be found with the

1.

help of a shortest path routine. Whenever absorbing circuits are present,

with a similar argument as Grinold gives in [8,p. 53ZJ, it can be shown

that 1f

i

=

0 might occur for some i € V. Next the algorithm is stated. A

j~.tification is given afterwards. MiG-algorithm

(4.6)

1. Initialisation

Determine the shortest distance A. from s to all nodes i € V,

1.

in which the length of an arc (i,j) is defined as:

d

iJ. := ln k. .l.J A. (i,j) € E

- /.11.

Put 1Ti :

= -

e (1Ts

=

-1) i € V

(4.7)

2. Consider the MFG-problem with arcs only in E 2

Put d. := _ 1/

1. 1T • 1.

i € V

Perform scaling procedure

Solve the obtained MFP-problem (by one of the procedures in

[1,2,3,6, 11 , 12

J)

Determine new v , xi" V(i,j) E:: E Z

s _ J

3. Consider sets:

{ (i , j )

-4.8 A

1 := i E 'I., j E 'I., 'IT.~ - k, ,_~J 'IT,_J > a}

{ (i,j)

-

< a}

4.9 A

2 ;= i E 'I., j E 'I., 'IT.~

-

kij 'IT,_J Determine 4.10 6 1 := max [-00 _('IT, - k .. 'lTj)/k_ij 'ITj , (i,j) E A₁

J

,

~ ~J 4.11 6

2 := max [- 00

,

- {'IT,~ - k .. 'lT~J j)/1Ti

,

(i,j) E A2

J

4.12 6 := max [-1, 6

1,62

J

If 6 = -1 the current solution v_s' x .. is optimal

~J Otherwise put:

4.13 1T, = 1T, _if i E 'I.

~ ~

-4.14 = (1 + _6)1Ti i"f i E 'I.

5. Proof of the MFG-algorithm

LEMMA 3. Initially wi < 0, Vi ~ V and wi - k .. w. ~ 0, V(i,j) ~ E

~J J

Proof. There are no absorbing circuits in the network. So for the product of gains along a circuit (directed cycle)Cwe have:

IT k

ij ~ 1 or with the definition of dij also

(i,j) ~ C

r

(i,j) ~ C

d .. ~ 0 along a circuit. That is to say that there are

~J

no negative cycles in the shortest path problem described in the

-initi~li.;Clti;~-step:--so4~ is-finite-and in accordance with (4.6):

wi < 0 initially for all i ~

v.

BecaU6e ~i is·the shortest distance

from s to i we have

=

-10 (-WI) and d .. J ~J wi - k .. w. ~ 0 ~J J V(i,j) ~ E = In k ij, so it follows: V(i,j) ~ E

o

o

LEMMA 4. At any iteration of the algor!thm either -1 <

a

< 0 and the new

wi < 0 or -1

=

a

and the optimal solution is obtained.

Proof. Inspection of (4.8) - (4.12) shows that always -1 S

a

<

o.

If

a

=

-1 then;

=

0 and conditions (2.8) and (2.20) are

satis-t

fied. In lemma's 5,6,7 it will be shown that all primal condi-tions, all dual conditions except (2.8) and complementary slack-ness relations except (2.20) are maintained throughout the

al-gorithm. This proves that if

a

=

-1 the current solution will

be optimal. I f -1 <

a

< 0 from lemma 3 and (4.13) and (4.14) it follows that wi < 0 at any iteration of the algorithm for all

i ~ V.

THEOREM 3. Under conditions 4.1 and 4.2 the restricted primal problem

..

can be transformed to a MFP-problem by way of scaling.

LEMMA 5.

LEMMA 6.

All primal constraints are maintained throughout the algorithm.

Proof. Conservation of flow is maintained in all algorithms in [1,2,3,6,11,12J. Initially x

ij = O,V(i,j) € E. By induction

tb6 statement follows

0

All dual constraints except (2.8) are maintained throughout the algorithm.

Proof. 1T

S = -1 in every step (always s € Y). What ever 1Ti might be, with the choice:

the dual constraints are maintained.

o

LEMMA 7. All complementary slackness conditions except (2.20) are

main-tained throughout the algorithm.

-Proof. We define E:.. :

=

1T. - k. . 1T., E:.. :

=

1T. - k. . 'IT. (

~J ~_ ~J J ~J ~ _~J_J

V(i,j) € E, and determine E: •• for arcs in (y,y),(y.y},(Y,Y) ~J (Y, Y) : (Y, Y) E: . . = E: •• ~J ~J (Y,

Y)

= E: ij

-

8k .. 1T,~J _J

(y,

Y) = E: •. + 81T . ~J ~ (Y, Y) = (1 _{+ 8)}E: •• ~J

o

and remains inactive (e

ij > 0) it becomes active (see (4.10». An inactive arc has e . > 0

iJ unless 6 = e . ./k. .

1T.,

then

~J ~J J

An active arc has e .. = O. It remains active

(€ ..

= 0) or it

~J _ ~J _

becomes hyperactive if (i,j) € (Y,Y) or inactive if (i,j) € (y,Y).

A hyperactive arc has e .. < O. It remains hyperactive

(€ ..

< 0)

~J ~J

unless 6

=

-e.

'/1T.,

then it becomes active (see 4.11».

~J ~

So the complementary slackness conditions are maintained.

We have proved sofar that the algorithm converges to the optimal solution. It will be shown that it does so in a finite number of steps.

T~ORE~~4.. The MFG-algorithm reaches the optimal solution in a finite number of steps.

Proof. The following three arguments provide the proof: 1. there are finitely many restricted primal problems.

2. after each dual variable change at least one more node is

attainable in a labeling proce~ So after at most (n-1) dual

variable changes breakthrough will occur, i.e. the objective value strictly increases. Therefore a restricted primal pro-blem is repeated not more than (n-1)times.

3. the restricted primal problem is equivalent to a MFP-problem

6. Example

Fig. 6.1.

Consider the network as specified in table 6.1. Table 6.1 Arcs C.. k. , ~J J.J ·l 51 200 2 s2 200 3

2

s3 150 2 12 225 1 1t 500 1 23 175 3 2 2t 50 3 2 31 300 1 3t 100 1 6 0, 6 1 ln2, 62 3 6 3 ln2, 6t In2

=

=

=

In -

=

₌

iii 2

=

-1 1 2 1 1 'II' _'11'1

=

- 2'

'11'2

=

-

,

11'3

=

_{- 2'}

11't

=

-

-iil 3 2

a

1,

a

1 2, d2 3 d 3 2, 2

=

=

=

•

=

d_t

=

iil 2

Calculate 'I1'i - kij '11', _{for all arcs. The set E}

2 is denoted in fig. 6.2.

J

Determine c

1:l

.1

d , for arcs in E1- 2• Determine y, .1.) and xij for arcs in E2• See table 6.2.

table 6.2

Fig. 6.2.

Arcs 1f, - k. , 1T, _{ci/di Yij} X

ij

=.

di Yi j l. l.J J 51 0 200 200 200 s2 0 200 0 0 53 0 150 100 100 12 1 0 0 6 1t 0 250 250 500 23 1 0 0 2 2t ₂1 0 0 31 0 150 50 100 3t 0 50 50 100

The saturated arcs are denoted with heavy lines in fig. 6.2.

{s,1,2,3},

{t},

A 1 {(2,t)},

=

~, 81 1 8 y

=

y

=

:::: _A 2

=

- -9 = 1T

=

_-1, 1T 1 2 1 4 1

=

- 2'

1T2

= - 3'

1T3 =

- 2'

1T

=

-9"

s t d 1, d 1 2, d2 3 d 3 2, dt 9

=

::::

=

2'

:::: :::: S ₄

Calculate 1T - k, , 1T _{for all arcs. The set E}

2 is denoted in fig. 6.3.

i l.J j

Detendne _Ci/d

Table 6.3. Fig. 6.3.

.

Arcs n. - k .. n, c . ./d. _Yij X ij = dij Yij ~ ~J J ~J l-sl

a

200 200 200 s2

a

200 100 100 3 3 s3

a

150 100 100 12 1

a

a

6 It 1 250 500 18 23 1 0 0 12 2t

a

100 100 50 3 3 31

a

150 50 100 3t 1 50 100 18

y

=

{s,1,2,3},

Y

=

{t}, A =~, A =~,

e

=

-1. 1 2

The maximal inputflow has been reached: v

s 1000 . =

-:r- '

see f~g. 6.4. 1000 3 -650 Fig. 6.4.

Some comments

We observe the following: in order to prove finite convergence for their

algorithm Jarvis and Jezior [9J perform in the restricted primal problem

a labeling technique in such a way that the primal solution remains basic. In the algorithm presented here there is no need to demand this.

A good computational efficiency of the algorithm might be expected., since the restricted primal problem can be solved by polynomially bounded al-gorithms.

In [17J Truemper emphasizes the basic relationship between the

MFG-pro-blem and the minimum cost proMFG-pro-blem in pure ~etworks.The structure of this

algorithm makes that even more clear: both problems can be solved by a pri-mal-dual approach in which the restricted primal is a MFP-problem.

Finally we remark that the MFG-algoritm can be converted to an algorithm for solving the problem of maximizing outputflow v

t • This can be done

along the same lines as in [9J. The algorithm is then applicable to

net-works with positive gains and without generating circuits if the follo-wing quantities are redefined:

d ..:= -In k ..

~J ~J

/:, _{:"'" shortest distance from i E V to sink t}

i 'lI'. := eili ~ d. := 1/'lI'. ~ 1. 8 1 := max [-00, - (1T •~ - k, ,~J 1T.)/1T.,J ~ (i,j) E A1

J

8

2 := max [..co, (1Ti - k, .~J 1T , )J /k ..~J 1Tj , (i,j) E A2

J

1T

i := (1 + 8)1T.,~ i E Y

-'lI', :=

Acknowledgement.

I would like to thank Prof. Dr. J.F. Benders for his v.aluable comments in writing this paper.

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