ON THE SIMPLEX ALGORITHM FOR NETWORKS ... - DSpace@MIT

ON THE SIMPLEX ALGORITHM FOR NETWORKS AND GENERALIZED NETWORKS by James B. Orlin Sloan School of Management MIT Working Paper #146783

I_

On The Simplex Algorithm for Networks and Generalized Networks by James B. Orlin Sloan School of Management MIT

Abstract We consider the simplex algorithm as applied to minimum cost network flows on a directed graph

G = (V, E).

First we consider the

strongly convergent pivot rule of Elam, Glover, and Klingman as applied We show that this pivot rule is equivalent to

to generalized networks.

lexicography in its choice of the variable to leave the basis.

We also

show the following monotonicity property that is satisfied by each basis B

of a generalized network flow problem. < B-lb', B-lb*


0 for all 0 > 0, and for g(O1 ) < g(02 ). (i)

A basis (B, N1, N2 )

small positive If a basis for

(B, N1 , N 2)

it follows that

is strongly feasible for (1) if and Pg(8)

for all sufficiently

.

(B, N1, N 2 )

is feasible (resp. strongly feasible)

Pg(el) and Pg(O2 )

feasible) for PROOF.

< 82

1

Then the following are true.

only if it is feasible for

(ii)

we define the parametric linear

Pg(0')

then it is feasible (resp., strongly for all

O'

with

01

0'

02.

Without loss of generality, let us assume that the basis is canonically oriented.

Let

Let us first consider property (ii). is feasible for Pg(Ol) and

Pg(82).

Then

b' = b -

(Ajj : Aj E N1).

Assume that the basis (B, N1 , N2)

11

8.

' B < B-lb

1 b' - g( - g(Ol)] , B-l

2)]

Moreover, by Lemma 1, it follows that for all B-1 l[b' - g(1)] By

Suppose now that


0

Then

RB
0, we obtain

for all

(5) O

and the continuity of

(4) and (5)

(B, N1, N2 )

that

fot all sufficiently small positive O.

Pg()

Then we can choose a

.

' -1(b ' - g(®) < B-l(b

and thus by the continuity of

g,

- g(O')) < UB

for all

Pg(0)

is feasible for

(B, N1 , N2 )

sufficiently small positive

O' > 0 for all

so that 0 < O
m Cmin.

(12)

Combining (6) and (12), we obtain the inequality k < [(m + l)m]-l (z

k+l

k),

-

and by Lemma 2 the number of consecutive degenerate pivots is O(m

2 log w ).

the

c'

Moreover, we may replace

described in Lemma 3.

w

by (c

1 < j

n).

: 1

i

Thus the number of pivots is

Next we consider the shortest path problem. that the problem is written as:

i

n)

for

O(m2 n log n).

In this case we assume

(min cx : Ax = 1, 0 < x

< m + 1

for

In this case, every feasible basis is strongly feasible so

that zk In addition, u

zk