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Chapter 11. A Min-Max Relation for Submodular Functions on Graphs

# Chapter 11. A Min-Max Relation for Submodular Functions on Graphs

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J . Edmonds, R. Giles

186

SnTEF,

SUTEF,

for any two sets S E F and T E F such that

SflT#8,

SUT#V.

For any family F of subsets of V, a real-valued function f ( S ) , S E F, is called

submodular on F if

(1.4)

u T)Sf(S)+f(T)

that S n T, S U T E F.

f ( S fl T ) + f ( S

for all S, T E F such

For any vector, x = (x, : e E E ) E RE,and any H C E, let

x ( H )=

2 (x. : e E H ) .

For any given graph G = (V, E ) , crossing family F on V, submodular function f

on F, and vectors a, d, c E ( R U { 2 w})", consider the linear program,

(1.5)

maximize

cx,

(1.6a)

where

dcxca,

VS E F , x ( S ( S ) ) - X ( S ( S ) ) Gf ( S ) .

(1.6b)

(1.7)

For y = (ys: S E F ) E R', let

c (ysf(S): S

F ( y , e ) = c (ys S

Yf =

:

EF),

E F, e E S ( S ) ) -

c (ys S

:

E F, e E

S(s)).

The linear programming dual of (1.5) is

(1.8)

(1.9)

+ za - wd

minimize

yf

where

y E RF, z E RE,and w E RE

satisfy

y 20,z 20,w 30,

V e E E, 2, - w,

+ F ( y , e ) = c,.

The 1.p. duality theorem says that:

(1.10) The maximum in (1.5) equals the minimum in (1.8), assuming either of

these optima exists.

Theorem. If c is integer-valued, and linear program (1.8) has un optimum

solution, then it has an integer-valued optimum solution. Hence, if c is integervalued, (1.10) holds even when restricted to integer-valued solutions [y, z , w ] of (1.9).

(1.11)

(1.12) Theorem. If a, d, and f are integer-valued, and linear program (1.5) has an

optimum solution, then it has an integer-valued optimum solution. Hence, if a, d , and

Min-max relation for submodular functions on graphs

187

f a r e integer-valued, (1.10) holds even when restricted to integer-valued solutions x of

(1.6).

Using a simple fact of linear programming, Theorem (1.12) is immediately

equivalent to:

(1.13) If a, d, and f are integer-valued, then every non-empty face of the polyhedron

P of the system (1.6) contains a n integer point. I n particular, if P has a vertex, then

every vertex of P is a n integer point.

2. Network Flows

(2.0) Let G = ( V ,E ) be a graph; let d, a E ( R U { * w } ) ~ , and let r,q E

(RU { m})". A feasible flow in network G in the classical sense of [lo] is a vector

x E R E which satisfies

*

(2.1)

d

S

x < a,

r, s x ( 6 ( v ) )- x(6(a)) s q.

for all v E V.

Let F, = { { v } :v E V } and F2 = {a: v E V } .Let F = F, U Fz. Let f ( ( v } )= q. and

f ( V - v ) = -ru.

Clearly, F is a crossing family, f is submodular on F, and (1.6) for this case is

(2.1). Theorems (1.11) and (1.12) for this case are well-known.

3. Polymatroids

(3.0) For a matroid M defined on the set E, the rank function of M is f ( S ) = IJI

for any maximal J S such that J is independent in M. (For example, where E is

the set of indices of the columns of a matrix A, and where J C E is independent in

M when the set of columns indexed by J is linearly independent.)

(3.1) The rank function f ( S ) , S C E, of a matroid on E is submodular; it is

non-decreasing: A C B C E implies f ( A )< f ( B ) ; f ( 0 ) = 0 ; and for each e E E,

f ( { e } ) s1. Such an f determines its matroid, say M, by the fact that J is

independent in M iff IJI = f ( J ) .

(3.2) Let f be any submodular function of all subsets of E. Let a E ( R U { 1+ m})".

The polyhedron,

P,, = {x E R E :0 < x

a ; x(S)

=Sf ( S ) , V S C

E},

known as a polymatroid, is much like the family of independent sets of a matroid.

J. Edmonds, R. Giles

in8

(3.3) Furthermore, Theorems (1.1 1)-(1.13) hold where the linear programs

(lSk(1.6and

) (1.8)-(1.9) are replaced by

(3.4)

maximize

and

{cx : x E Po}

the dual of (3.4).

(3.5) This follows immediately from (1.11)-(1.13) by letting E of (3.2)be the

edge-sef of a graph G = (V, E ) such that the heads and the tails of the members of

E are all different;

(3.6) letting the F of (1.5) be

F

= { { t ( e ) :e E S } : S

E};

and letting the f of (1.5)be

f ( { t ( e ) :e E S } ) = f(S), for S

c E, as in (3.2).

Theorem (3.3) is especially simple when

(3.7) the vector a is all infinite, and when

(3.8) f(S), S C E, is a non-negative, non-decreasing submodular function.

The linear program (3.4)becomes

(3.9)

= C (c,x,

maximize

cx

where

Ve E E, x,

V S C E,

3

c

:e E E ) ,

0,

(x. : e E S ) S f(S).

The dual 1.p. is

(3.10) minimize

where

yf =

(f(S) . y ( S ) : S

c E),

VS C E, y ( S ) 3 0,

Ve E E,

2 ( y ( ~ ) e: E s c E ) 2 c,.

The so-called “Greedy Algorithm Theorem” says that:

(3.11) In the case of (3.7)-(3.9), and where the vector

c

is arranged so that

the following vectors x ” = ( x t ( , ) :i = 1,. . ., 1 E 1) and y o = ( y O ( S ) :S C E ) are optimum solutions, respectively, of (3.9)and (3.10).

Min-max relation for submodular functions on graphs

(3.12) Let S,

189

= {e(l), e ( 2 ) , . . ., e ( i ) } .

(3.13) Let X ! V )

k +l,...,lE/.

= f(Sl),x ! ( ~=

) f(S,)- f(S,-,) for

i

= 2 , . . ., k ;

and x t c t ,= 0 for i =

(3.14) Let yo(S,)= c e ( , ) -cec,-,),for i = 1 , . . ., k - 1; yo(&) = c,,,,; and yo(S) = 0

for other S C E .

That these are optimum solutions of (3.9) and (3.10) follows, using the weak 1.p.

duality theorem, by showing that cx" = yof, that y o is feasible for (3.10), and that x"

is feasible for (3.9).

It follows from the greedy algorithm theorem that:

(3.15) The vertices of PI = { x 3 0: x ( S )

form x " , as defined in (3.13).

f(S), VS C E } are the vectors of the

(3.16) In particular, where f is the rank function of a matroid, the vectors x o are

the (incidence) vectors of the independent sets of M. That is, x : = 1 for e E J and

x: = 0 for e E E - J, where J C E is an independent set of M.

(3.17) An interesting way to get a function f of the form (3.8) is to take a

non-negative linear combination of the rank functions of various matroids on E.

(3.18) Another way to get a very particular kind of f of the form (3.8) is to let

f(S) = g ( 1 S I), S C E, where g is a non-negative, non-decreasing, concave function.

That is, for i = 0,1,2,. . ., 1 E 1,

g ( i ) = g(O)+h(l)+h(2)+-*.+h(i),

(3.19) In particular, for the f of (3.18), where g(0) = 0, we have immediately from

(3.15), that a vector is a vertex of P , iff its components are any arrangement of

h ( 1 ) ,h ( 2 ) ,. . ., h ( k ) , and I E I - k zeroes for some k .

(3.20) Hence, the face P I n { x : x ( E ) = f ( E ) } of the P , of (3.19) is the convex hull

of the vectors which are the various permutations of the numbers h (l),. . ., h(l E I).

The greedy algorithm theorem, as presented here, and some other theory of

polymatroids, first appeared in [6]. Further, and better, treatments are [9] and [12].

Balas [ I ] recently presented a different derivation of a linear system defining the

convex hull of the vectors of all permutations of the numbers 1 , 2 , . . ., [El.We much

appreciate the thoughtfulness which Chvital devoted to bringing togethtr Balas'

work and ours.

190

J. Edmonds, R. Giles

4. Polymatroid intersection

(4.0) Let

a E (R U {

(4.1)

fl and

fi

*a})E.

be any two submodular functions of all subsets of E. Let

For i = 1,2, let

P, = { x E R E :0 s x s a ; x ( S ) s fi(S),VS C E } .

As in the last section, each P, ,is a polymatroid.

(4.2)

The polyhedron P I n P, is not generally a polymatroid.

(4.3) Nevertheless we do have the “Polymatroid Intersection Theorem” which is

(1.10)-(1.13) where linear programs (1.5) and (1.8) are replaced by

(4.4)

max {cx : x E P I n P 2 } ,

and its 1.p. dual.

Where we take the intersection of three polymatroids, P , flPz f l Ps, in place

of two, the (1.11)-(1.13) part of (4.3) is generally not true. Of course, the (1.10) part

still holds, it being merely an instance of the 1.p. duality theorem.

(4.5)

We get (4.3) as a special case of (1.10)-(1.13) by letting the E of (4.0) be the

edge-set of the same graph G = (V, E ) as in ( 3 3 , that is, such that each e E E and

its end-nodes comprise a separate component of G ; letting d = 0;

(4.6)

-

(4.7)

letting F = { t ( S ) :S C E } U { h ( S ) :S

CE}

where t ( S ) = { t ( e ) : e E S } and

__

h ( S )= V - { h ( e ) :e

(4.8)

E S } = { t ( e ) :e €

letting f(t(S))= min [ f , ( S ) , k ]

-

E } U { h ( e ) : e\$Z S } ;

for S C E,

f ( h ( S ) ) = min [f2(S),k ] for S C E,

where k = min [ f l ( E )f, 2 ( E ) ] .

It is straightforward to verify that F is a crossing family of V, that f is a

submodular function of F, and that for this F, f, and d, the system (1.6) is equivalent

to the system

(4.9)

0s x s

a;

vs c E, x(S)

fl(S),

x(S)

f2(S).

Min-max relation for submodular functions on graphs

191

(4.10) Where f l and f 2 are the rank functions of any two matroids on E, say MI

and M a , the polymatroid intersection theorem becomes the “matroid intersection

theorem”:

The (1.13) part immediately implies that:

Where P, is the polyhedron of matroid M , on set E, i = 1,2, the vertices of

P , n P2 are precisely the vectors of subsets of E which are independent in both MI

and M2,that is, they are precisely the points which are vertices of both PI and P,!

(4.11)

Likewise the (1.12) aspect of the matroid intersection theorem (when a = 00)

gives us that:

(4.12) The maximum weight, 2 (c, : e E J ) , of a set J

both matroids, MI and M , , equals

(4.13)

c E which is independent in

(fl(S)* y l ( S ) + fi(S). yz(S) : s C E )

min

where

vs c E, y , ( S ) 3 0, y 2 ( S ) 3 0;

Ve E E,

( y l ( S ) + y2(S):e E S

E ) 3 c,.

And the (1.11) part gives us that:

If c is integer-valued then the y , ( S ) , S C E, i = 1,2, of (4.13) may be

restricted to integers.

For the case where c is all ones, equation (4.12)-(4.13) reduces to:

(4.14)

(4.15)

m a x { ( J I : J, independent in MI and M z )

=

min {f,(S)+ f 2 ( E- S ) : S

c E}.

The polymatroid intersection theorem, where the f are non-decreasing and

without the constraint x a, and the matroid instances of it, first appear in [6].

Algorithmic proofs of matroid instances were obtained and published earlier, [ 5 ,

81. The theorem, with the constraint x a and without the restriction on 6, as well

as the main generalization (l.lOk(1.13) being presented here, first appears in [12].

(4.16)

5. Directed cut k-packings

Let G

(5.0)

=

(V, E ) be an acyclic graph and let

D ( G )= { S C V : 0 # S #

v,s(s)= 01.

J. Edmonds, R. Giles

192

(5.1) Clearly, D ( G ) is a crossing family on V. A set of edges of the form 6(S) for

some S E D ( G ) is called a directed cut of G.

(5.2)

For a given integer-valued function f(S), S E D ( G ) ,a set H C E such that

Iff n W)ls f(S),

VS E W G ) ,

is called a directed cut f-packing of G. The incidence vectors of the directed cut

f-packing of G are precisely the integer solutions of the system

(5.3) Ve E E, O s x ,

S

1,

VS E D ( G ) ,x(6(S)) S f(S).

(5.4) When f ( S ) is submodular, in particular when f(S) is a constant integer k,

system (5.3) is of the form (1.6) and so theorems (l.lOk(1.13) apply.

For a constant k, directed cut k-packings are easily treated without the present

theory .

The theorem of Dilworth o n the maximum number of incomparable elements in

a partial order immediately implies that:

(5.5) A subset H of the edges of an acyclic graph G is contained in the edge-set of

as few as k directed paths in G iff 1 TI =S k for any T C H such that

(5.6) n o directed path of G contains more than one member of T.

It can be shown that

(5.7) a set T C_ H has property (5.6) iff T is contained in some member of D ( G ) .

Hence, we have that

(5.8) a set H E is a directed cut k-packing in G, for constant integer k, if and

only if H is contained in the edge-set of some k or fewer directed paths in G.

(5.9) Corollary. A set H C E is a directed cut k-packing in G, for constant integer

k, if and only if H can be partitioned into some k or fewer 1-packings of the directed

cuts in G.

It follows directly from (5.8) that:

(5.10) For a given acyclic graph G ‘ = (V’, E’), a given integer k, and given

edge-weighting c = (ce: e E E’), t h e maximum weight directed cut k-packings of

Min-max relation for submodular functions on graphs

193

G' can be realized as the optimum integer flows of the optimum network flow

problem described in Section 2,

(5.11) where the G of Section 2 is the G' of (5.10), with the same edge-weighting,

together with, for each e E E', k extra edges in parallel with e and each having

weight of zero; also let G have a new node S, k new zero-weighted edges going

from S to each u E V', a new node t, and k new zero-weighted edges going from

each u E V' to t. Let d be all zeroes, a be all ones, rs = qs = k, r, = q, = - k, and

r, = 4. = 0 for u E V'.

For the case k = 1, and c all ones, the subject of this section is treated by

Vidyasankar and Younger [16].

6. Directed cut k -coverings

Let G and D ( G ) be as in Section 5.

Where g(S) is a non-negative integer valued function of S E D ( G ) , a set

C E such that 1 C n S ( S ) l 5 g(S) for every S E D ( G ) is called a directed cut

g-covering of G.

(6.0)

The incidence vectors of the directed cut g-coverings of G are precisely the

integer solutions of the system

(6.1) V e E E, O S x ,

- x(S(S))

S

1,

c f(S) = - g ( S )

for every S E F = ( S

c V:

E D(G)}.

(6.2) A function g(S) is called supermodular when - g(S) is submodular.

(6.3). When g(S) is supermodular, in particular a constant k, the system (6.1) is of

the form (1.6) and so Theorems (l.lOb(1.13) apply. The integer min-max relation of

(1.10)-(1.12) becomes:

(6.4)

Where g(S), S E D ( G ) , is any integer supermodular function such that

O=zg(S)s16(S)I for every S E D ( G ) ,

where c,, e E E, are integers, and C is a direct-cut g-covering of G, we have

(6.5)

min

(cp: e E C )

J. Edmonds, R. Giles

194

(6.6)

=maxz(y,.g(S):SED(G))-x(ze

(6.7)

- m a x ( C ( y , . g ( S ) : S€ D ( G ) )

-

c (max [o,

- c,

+

x

: e EE)

(ys : e E f i ( ~ ) ) ]:

over integers y s 3 0 and ze 2 0 such that,

tle E E,

-

ze

+ 2 (ys : e E 6(~))s c,.

In particular, where the c, are all ones, formula (6.5)-(6.7) becomes

(6.8) Theorem. The minimum cardinality o f a directed-cut g-covering of G equals

the maximum over all

(6.9)

Y C D ( G ) of

1 u ( 6 ( S ) :s E Y)I +

c (g(S)

I

- S(S)l:

s E Y),

Where g ( S ) is all ones, (6.8) implies the theorem of Lucchesi and Younger [I41

that:

(6.10) The minimum cardinality of a 1-covering of the directed cuts of G equals

the maximum cardinality of a family of mutually disjoint directed cuts of G.

(6.11) A graph G = (V, E ) is called strongly connected when, for every u, v E V,

there is a directed path in G from u to v. A connected graph G is strongly

connected if and only if every e E E is contained in a directed polygon (directed

cycle) in G.

(6.12) It is easy to show that c C E is a 1-covering of the directed cuts of a

connected graph G if and only if the graph obtained from G by “shrinking” the

members of C is strongly connected - equivalently, if and only if the graph

obtained from G by adjoining to G, for each e E C, an edge e’ such that

h ( e ’ )= t ( e ) and t ( e ’ )= h ( e ) , is strongly connected.

W e hope t o be able t o prove the following conjecture:

(6.13) For any constant integer k > 0, C C E is a k-covering of the directed cuts

of G = (V, E ) if and only if C can be partitioned into k I-coverings of the directed

cuts of G.

(6.14) The function I S(S)l, S E D ( G ) ,is modular - that is, it is both submodular

and supermodular. Hence, though we derived directed-cut f-packings, for submodular f, and directed-cut g-coverings, for supermodular g, as different special

195

Min-max relation for submodular funcrions on graphs

cases of a more general system, in fact the two are equivalent: H is an f-packing for

G if and only if E - H is a g-covering for G, where

7. Total dual integrality

(7.0) We say that a system, A x s b, of linear inequalities in x, with rational A and

b, is totally dual integral when the dual of the linear program max{cx :

Ax s b} has an integer-valued optimum solution for every integer-valued c such

that it has an optimum solution. We say that a polyhedron is totally dual integral if

it is the solution-set of a totally dual integral system.

Theorem. If a polyhedron P is the solution-set of a totally dual integral

system which has integer right-hand sides, then every non-empty face of P contains

an integer point - in particular, any vertex of P is an integer point.

(7.1)

Or, stated another way:

Theorem. For any finite linear system, Ax S b, having rational coefficients,

if min{yb: y 3 0 , y A = c } is an integer for any integer-valued c such that the

minimum exists, then for any c such that max{cx: Ax s b } exists there is an

integer-valued optimum x.

(7.1')

(7.2)

Using Theorem (7.1) we can conclude (1.12) immediately from (1.11).

To prove (7.1) we use the following lemma which we presume to be classical.

(7.3) A finite system of linear equations, A o x = bo, having rational coefficients,

has no integer-valued solution x if and only if there is a vector T such that TAO is

integer-valued, and r b 0 is not an integer.

Proof of (7.1). Assume the hypothesis of (7.1) for t h e system Ax

P = {x: Ax S b}. By the 1.p. duality theorem we have immediately that

b. Let

max{cx : x E P } is an integer for any integer-valued c such that the

maximum exists.

A face of P is any subset of the form P o = {x E P : A o x = bo}where Aox =sbo is a

subsystem of A x s b. It is easy to show that

(7.4)

(7.5) if P o is a minimal non-empty face of P, then P o = {x: A Ox = b'}. By the

complementary slackness theorem of linear programming, for any c such that

J. Edmonds, R. Giles

196

max { c x : x E P } exists, the maximum is achieved over all members of some

non-empty face of P, and hence over all members of some minimal non-empty face

of P. Thus it suffices to show that every minimal non-empty face of P, say

Po = { x : A o x = b"},has an integer-valued member. Suppose not. Then, by (7.3), let

rr be such that r A o is an integer-valued vector and rb " is a non-integer.

Any c = AA", for a vector A 3 0, is such that cx is maximized over P by any

member of Po, since for x E Po we have cx = A A o x = Ab", and for x E P we have

c x = AAx S Ab".

Choose A 2 0 such that A + r 2 0 and such that c o = AAo is integer-valued. Then

c ' = (A + r ) A " is integer-valued. By (7.4), for i = 0,1, d ' = max{c'x: x E P } is an

integer. By (7.5), for i = 0,1, we have c ' x = d' for every x satisfying AOx = b".

Hence, d ' - d" = c ' x - c o x = r A " x = r b o is an integer. Contradiction. 17

8. Tree representation of cross-free families

(8.0) Two sets S, T C V are said to cross if S r

l T # 0, S U T # V, S g T, and

T C S. A family F of subsets of V is called a cross-free family on V if n o two

members of F cross.

(8.1) A tree T, with node-set V ( T ) ,and with directed edge-set E ( T ) , together

with a function 1 from a set V to V ( T ) ,is called a V-labelled tree T.

(8.2) For any V-labelled tree T, we have a family {S,:i E E ( T ) } of subsets of V

determined as follows: for each i E E ( T ) , there

is a unique T ( i )C V(T) such that,

with respect to graph T, 6 ( T ( i ) )= { i } , 6 ( T ( i ) )= 0; T ( i ) is the set of nodes u

(including the node u = t ( i ) )such that the unique path in T from u t o t ( i )does not

contain i. W e let

S, = { u E V: l ( u ) E T ( i ) } .

(8.3) Theorem.

(8.4)

A family F on set V is a cross-free family if and only i f

there exists a V-labelled tree T such that

F

= { S ,: i E E ( T ) } .

Proof. It is easy to check that (8.4) implies F is cross-free

If F consists of just one set S, then let T consist of a single e q e i and, for each

u E V, let l ( u ) = t ( i ) if ZI E S, and l ( u ) = h ( i ) if ZIES. Clearly T and 1 are a

V-labelled tree T satisfying (8.4).

If F' is a cross-free family on V, such that I F'l 3 2 , choose some S E F' and let

F = F ' - {S}. Assume, by induction on 1 FI, that we have a V-labelled tree T with

labelling function 1, which satisfies (8.4).

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