The Hirsch Conjecture for Dual Transportation Polyhedra

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

THE HIRSCH CONJECTURE FOR DUAL TRANSPORTATION POLYHEDPA

M.L. B a l i n s k i

F e S r u a r y 1 9 8 3 CP- 8 3- 9

C o Z Z a b o r a t i v e P a p e r s r e p o r t work w h i c h h a s n o t b e e n p e r f o r m e d s o l e l y a t t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s a n d w h i c h h a s r e c e i v e d o n l y l i m i t e d r e v i e w . V i e w s o r o p i n i o n s e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e s e n t t h o s e o f t h e I n s t i t u t e ,

i t s N a t i o n a l Member O r g a n i z a t i o n s , o r o t h e r o r g a n i - z a t i o n s s u p p o r t i n g t h e w o r k .

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-236 1 L a x e n b u r g , A u s t r i a

ABSTRACT

An algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most (m-1 ) (n-1 ) extreme edges.

THE HIRSCH CONJECTURE

FOR DUAL TRANSPORTATION POLYHEDRA M.L. Balinski

Laboratoire d'~conom6trie de 1'Ecole Polytechnique, Paris, France

The d i s t a n c e between a pair of extreme points of a convex

polyhedron P is the number of extreme edges in the shortest path that joins them. The d i a m e t e r of P is the greatest distance between any pair of extreme points of P. The Hirsch conjecture

(see [5], pp. 160 and 168, [7]) is that the diameter of a convex polyhedron defined by q halfspaces in p-dimensional space is at most q-p. In linear programming jargon it is that given r lin- early independent equations in nonnegative variables it is

possible to go from one feasible basis to any other in at most r pivots all the while staying feasible.

For unbounded polyhedra in dimension 4 or more the Hirsch conjecture is false [7], so it is false in general. However, it has been proven to be true for certain special cases: the poly- topes arising from the shortest path problem [8], Leontief sub- stitution systems [6], the assignment problem [4]

,

and certain classes of transportation problems [I]. I will show it is true for the unbounded polyhedra arising from dual transportation

problems. The approach will also establish a combinatorial char- acterization of extreme points that has proven to be very useful.

The d u a l t r a n s p o r t a t i o n p o l y h e d r o n for the m by n matrix c

w

is

where

I R (

^{= m and}

I c ~

^{= n .} S e t t i n g u l = 0 i s an a r b i t r a r y c h o i c e t h a t r u l e s o u t l i n e s o f s o l u t i o n s (ui+6) I ( v j - 6 )

.

I t i s n a t u r a l t o s t u d y t h e e x t r e m e p o i n t s of D i n t e r m s of a b i p a r t i t e g r a p h model. L e t t h e s e t of m nodes R s t a n d f o r t h e rows of c and t h e s e t of n n o d e s C f o r t h e columns. The e q u a t i o n s

N

-

u 1 = o , u i + v

-

j ' i j f o r ( i , j ) E T , i E R , j E C , form a maximal l i n e a r l y i n d e p e n d e n t s e t i f and o n l y i f T i s a s p a n n i n g t r e e . The u n i q u e s o l u t i o n u , v t o

." -

T i s an e x t r e m e p o i n t o f D i f and

o n l y i f ui

+

v j 2 c i j f o r i , j

4

T : I w i l l s a y T h a s t h e e x t r e m e p o i n t u , v and f o r c l a r i t y w i l l sometimes w r i t e T ( u , v ) . Given a

U N U U

s p a n n i n g t r e e T t h e u n i q u e s o l u t i o n t o

U1 u3 U4 R: Row Nodes

1 v2 v3 v4 v5 ^V6 C: Column Nodes

F i g u r e 1 . Spanning t r e e . S i g n a t u r e ( 3 , 2 , 1 , 3 )

i t s e q u a t i o n s i s immediate. I n t h e s e q u e l o n l y s p a n n i n g t r e e s t h a t have e x t r e m e p o i n t s u , v a r e c o n s i d e r e d :

-

^-4 t o a n y T t h e r e c o r r e s p o n d s e x a c t l y one e x t r e m e p o i n t .

To o n e e x t r e m e p o i n t , however, t h e r e c a n c o r r e s p o n d many t r e e s T : t h i s happens when ui

+

v

-

j

-

' i j f o r ( i , j )

4

^{T and i s}

c a l l e d " d e g e n e r a c y " . I n t h i s c a s e any s p a n n i n g t r e e c h o s e n from { ( i , j ) ; u i + v j = c i j } h a s t h e same e x t r e m e p o i n t .

The (row) s i g n a t u r e o f a t r e e T i s t h e u n i q u e l y d e f i n e d v e c t o r o f t h e d e g r e e s of i t s row n o d e s a -4 = ( a l ,

...,

^a m ^{I l a i = m+n-1, ai = > 1 .}

Lemma. Two d i f f e r e n t t r e e s T , T 1 w i t h t h e same s i g n a t u r e s have o n e and t h e same e x t r e m e p o i n t .

1 1 P r o o f . L e t t h e e x t r e m e p o i n t s of T and T ' be u , v and u

,y .

4 _I ._I l U

-

^.v

I w i l l show t h a t u = u , v = v

.

T # ^T 1 means t h e r e i s some node i l E R I ( i l , j l ) E T b u t

( i l t j l ) $ T 1 ( s e e F i g u r e 2 f o r what f o l l o w s ) . I n T 1 l e t ( i 2 , j l ) w i t h i2 # il be t h e edge on t h e u n i q u e p a t h t h a t j o i n s j l t o i l , and c o n s i d e r node i 2 .

F i g u r e 2. S o l i d l i n e s i n T ; d a s h e d l i n e s i n T 1

.

I t must have d e g r e e a t l e a s t 2 i n T 1 and s o i n T . T h e r e f o r e t h e r e e x i s t s a n e d g e ( i 2 , j 2 ) E T , j 2 # j l . NOW, l e t ( i j t j 2 ) be t h e edge on t h e u n i q u e p a t h t h a t j o i n s j 2 t o i l i n T 1

.

C o n t i n u e t o b u i l d t h i s p a t h u n t i l a node a l r e a d y on i t i s e n c o u n t e r e d a g a i n , f o r m i n g a c y c l e : ( i h , j h ) , ( i h + l t j h ) I . . - t ( i R I j R ) ' ( i h t j R ) - C a l l t h e e d g e s o f t y p e ( i k , j k ) of t h e c y c l e odd, and of t y p e

and ( i h , j R ) e v e n . Then ( i k + 1 ¹ l k )

+ = ' i j f o r ( i , j ) o d d , < c . f o r ( i , j ) e v e n

ui ui + v j ⁼ lj

j and

1 < c . f o r ( i , j ) o d d , ¹ 1

Ui + V j = lj ^ui

+

v j = c i j f o r ( i , j ) even.

Summing,

i m p l y i n g e q u a l i t y h o l d s t h r o u g h o u t and s o

u + v = = ^U1

+

^V' for all (it j) in the cycle- i j 'ij i j

Transform T' by taking from it all even edges and putting in it all odd edges. The new T' has the same signature and the same u1 ,vl but more edges in common with T. Repeat until T = T 1

,

..) U 4

I I

showing u _..) = u _.y

,

v _.y = v _U

.

Given T(u,v), let (kt&) be one of its edges with k and R

U U

both of degree at least 2. A pivot on (ktk) obtains T (ul 1 ,vl )

N U

as follows (see Figure 3) : drop (k,R) from T to obtain two connected components, T containing k and Tk ' containing 2.

Let E = m i n { c i j - u i - v ~ E T ' , ~ E T k 1 2 0 and (g,h) be some

j ' ^- _{1 k} R

edge at which this minimum is obtained. Set T = T U T

u

(g,h)

( (g, h) is the "incoming" edge)

.

If row node 1 E T~ define 1

-

ui

-

^ui

+

E, i E T

,

_ui ¹ _{= ui} ^otherwise

,

1 1

v = v

-

^{E j E T}

^,

^{v = v otherwise ;}

j j j j

and if row node 1 E T' define

ui 1 = ui

-

^{E, i E T k}

,

^{ui 1} = ui otherwise

,

1 k 1

v _j = v _j

+

E, j E T

,

^v _j = v _j otherwise

.

E 2 - 0 because u,v satisfies all inequalities. The choice of E

- 1 1

I I

guarantees that u

,!

satisfies them all as well and that it 1 "

belongs to T

.

T': a ' = ( I , 3, I , 2, 3)

C

Figure 3. Pivot from T to T I

.

1 1

If E: = 0 then (u,v) = u

,

) and we have two different

Y -4 -4

trees having the same extreme point (degeneracy). If E > 0 then u,v and u1

..

^{Y 4} ,vl are ^- n e i g h b o r s , connected by an extreme edge of D.

In either case, if a is the signature of T then the signature a '

-4 1

-

of T' is the same except that a: = ak

-

^{1 and a} = a

+

^1.

g g

T h e o r e m I . T h e d i a m e t e r o f D (c) is a t m o s t (m-1) (n-1).

m,n

-

T h i s b o u n d is t h e b e s z p o s s i b l e ,

Proof. I give a method that constructs a path of at most

(m-l)(n-1) extreme edges between any pair of extreme points.

The idea is to begin at one extreme point (the "initial" one) and to pivot in order to obtain a tree T whose signature is equal to that of the other extreme point (the "destination") : for then, by the lemma, T has as its extreme point the desired one.

Let a be the signature of the current tree T (e.g., the

U J

initial one), and aT the signature of the destination extreme point. If ai < a i l i is a

*-

d e f i c i t node. If there are d deficit nodes, m-d is the number of nondeficit nodes. If a, > a,, i is

*

* *

^{I ~}

a s u r p l u s node. The n e t d e f i c i t is {E(ai- ai) ; a i > ail. The method has the property that the number of surplus nodes never increases and within at most m-d pivots the net deficit must decrease by 1.

Choose some surplus node and designate it the s o u r c e s and some deficit node and designate it the t a r g e t t. Pivot on the edge (s,R) incident to the source s that is on the unique path joining s to the target t (see Figure 4). Call Q the set of row nodes of the component of T - (s,R) that contains t. S ~ Q . The degree of some g E Q increases by 1: either (i) it was not a deficit node of T or (ii) it was.

Fiaure 4.

( i ) I f n o t , name it t h e new s o u r c e s1 and r e p e a t : p i v o t on

1 1 1

( s

,

^l? ) t h e e d g e on t h e p a t h j o i n i n g s' t o t i n T

.

^{The s e t o f}

1 1 1

row n o d e s Q1 o f t h e component T

..

^{( s}

^,

^{l? )} c o n t a i n i n g t b e l o n g s t o Q b u t must b e s m a l l e r : s 1 Q

.

Each t i m e a nondef i c i t node s d e g r e e g o e s up it i s i m m e d i a t e l y b r o u g h t down and c a n n o t a g a i n

i n c r e a s e u n l e s s t h e t a r g e t node i s changed. T h e r e f o r e , i n a t most m-d p i v o t s a c a s e ( i i ) must o c c u r .

( i i ) The n e t d e f i c i t d e c r e a s e s by 1 . I f t h e n e t d e f i c i t i s z e r o , t h e d e s i r e d t r e e i s f o u n d . O t h e r w i s e , name new s o u r c e and t a r g e t n o d e s and c o n t i n u e .

The n e t d e f i c i t c a n b e a t most n-1; t h e number o f n o n d e f i c i t n o d e s a t most m-1: t h i s g i v e s t h e u p p e r bound ( m - 1 ) (n-1) on t h e number o f p i v o t s and s o on t h e d i s t a n c e .

The bound i s b e s t p o s s i b l e . C o n s i d e r t h e p o l y h e d r o n D _m,n ( c )

.Y

w i t h c i j = m i ) ( - 1 Suppose i l < i 2 and j l < j 2 : it i s i m - p o s s i b l e t o have b o t h ( i l

,

j 2 ) and ( i 2 , j ) i n a t r e e T ( u , v ) _{.y a,}

.

^{T h i s}

i m p l i e s t h a t t h e t r e e s ^T o f t h i s D ( c ) a r e c h a r a c t e r i z e d a s a l l m,n

t h o s e t h a t have " n o c r o s s i n g s " (see F i g u r e C)

--

i t b e i n g u n d e r - s t o o d t h a t t h e row and column n o d e s a r e drawn i n t h e i r n a t u r a l o r d e r s . I n p a r t i c u l a r , t h i s p o l y h e d r o n a d m i t s n o d e g e n e r a c y . I n p i v o t i n g from o n e t r e e t o a n e i g h b o r i f node i t s d e g r e e de- c r e a s e s by 1 t h e n t h e d e g r e e o f e i t h e r node i + l o r node i-1 must i n c r e a s e by 1 . T h e r e f o r e , t o d e c r e a s e t h e d e g r e e o f node 1 by 1 and i n c r e a s e t h a t o f node m by 1 it t a k e s m-1 s t e p s . T h i s shows t h a t t o go from t h e e x t r e m e p o i n t w i t h s i g n a t u r e ( n , l ,

...,

1 ) t o t h a t w i t h ( 1 , .

. .

, l , n ) it t a k e s (m-1 ) (n-1) s t e p s .

F i g u r e 5. A t r e e w i t h " n o c r o s s i n g s " .

An immediate result of the foregoing is:

T h e o r e m 2 . To e v e r y i n t e g e r v e c t o r a, ai

2

1, lai = m+n-1

L.

t h e r e c o r r e s p o n d s a n e x t r e m e p o i n t u,v.

..

^L.

P r o o f . Given any such vector a the method of the above

- * ^*

proof finds an extreme point having a as its signature.

L.

So for nondegenerate polyhedra D (c) there is a one-to-one m,n

..

correspondence between extreme points and signatures. This char- acterization enables one to describe and count all faces of

D (c) [3]

.

It has also motivated a new algorithm for the as- m,n

-

signment problem that is guided entirely by the signatures and terminates in at most (n-1) (n-2) /2 pivots [ 2 1

.

Acknowledament

It is a pleasure to acknowledge the fact that part of this work was done during my visit to I.I.A.S.A. during the summer of

1982.

References

[I] M.L. Balinski, "On two special classes of transportation polytopes," in M.L. Balinski (ed.), Pivoting and Extensions, Mathematical Programming Study 1, North-Holland Publishing

Co., Amsterdam, (1 974)

,

^43-58.

[2] M.L. Balinski, "Signature methods for the assiqnment problem,"

research report, Laboratoire d1~conom6trie de 1'Ecole Poly- technique, (November, 1982)

.

[3] M.L. Balinski and A. Russakoff, "Faces of dual transporta- tion polyhedra," research report, Laboratoire d1Econom6trie de llEcole Polytechnique, (November, 1982).

[4] M.L. Balinski and A. Russakoff, "On the assignment polytope,"

SIAM Review, 16 (1974), 516-525.

[5] G.B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, N.J. (1963).

[61 R.C. Grinold, "The Hirsch conjecture in Leontief substitution systems," working paper, Center for Research in Management Science, University of California at Berkeley, (March, 1970).

[7] V. Klee and D.W. Walkup, "The d-step conjecture for poly- hedra of dimension d < 6," Acta Mathematics, 117(1967), 53-78.

[8] R. Saigal, "A proof of the Hirsch conjecture on the polyhedra of the shortest route problem," SIAM JournaZ of Applied

Mathematics 17 (19691, 1232-1238.