Chapter 11. A MinMax 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 realvalued 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 integervalued, and linear program (1.8) has un optimum
solution, then it has an integervalued optimum solution. Hence, if c is integervalued, (1.10) holds even when restricted to integervalued solutions [y, z , w ] of (1.9).
(1.11)
(1.12) Theorem. If a, d, and f are integervalued, and linear program (1.5) has an
optimum solution, then it has an integervalued optimum solution. Hence, if a, d , and
Minmax relation for submodular functions on graphs
187
f a r e integervalued, (1.10) holds even when restricted to integervalued 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 integervalued, then every nonempty 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 wellknown.
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
nondecreasing: 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
edgesef 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 nonnegative, nondecreasing 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 socalled “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).
Minmax 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
nonnegative 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 nonnegative, nondecreasing, 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
edgeset of the same graph G = (V, E ) as in ( 3 3 , that is, such that each e E E and
its endnodes 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).
Minmax 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 integervalued 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 nondecreasing 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 kpackings
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 integervalued 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 fpacking of G. The incidence vectors of the directed cut
fpacking 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 kpackings 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 edgeset 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 kpacking in G, for constant integer k, if and
only if H is contained in the edgeset of some k or fewer directed paths in G.
(5.9) Corollary. A set H C E is a directed cut kpacking in G, for constant integer
k, if and only if H can be partitioned into some k or fewer 1packings 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
edgeweighting c = (ce: e E E’), t h e maximum weight directed cut kpackings of
Minmax 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 edgeweighting,
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 zeroweighted edges going
from S to each u E V', a new node t, and k new zeroweighted 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 nonnegative 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
gcovering of G.
(6.0)
The incidence vectors of the directed cut gcoverings 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 minmax 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 directcut gcovering 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 directedcut gcovering 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 1covering 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 1covering 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 kcovering of the directed cuts
of G = (V, E ) if and only if C can be partitioned into k Icoverings 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 directedcut fpackings, for submodular f, and directedcut gcoverings, for supermodular g, as different special
195
Minmax relation for submodular funcrions on graphs
cases of a more general system, in fact the two are equivalent: H is an fpacking for
G if and only if E  H is a gcovering 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 integervalued optimum solution for every integervalued c such
that it has an optimum solution. We say that a polyhedron is totally dual integral if
it is the solutionset of a totally dual integral system.
Theorem. If a polyhedron P is the solutionset of a totally dual integral
system which has integer righthand sides, then every nonempty 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 integervalued c such that the
minimum exists, then for any c such that max{cx: Ax s b } exists there is an
integervalued 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 integervalued solution x if and only if there is a vector T such that TAO is
integervalued, 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 integervalued 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 nonempty 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
nonempty face of P, and hence over all members of some minimal nonempty face
of P. Thus it suffices to show that every minimal nonempty face of P, say
Po = { x : A o x = b"},has an integervalued member. Suppose not. Then, by (7.3), let
rr be such that r A o is an integervalued vector and rb " is a noninteger.
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 integervalued. Then
c ' = (A + r ) A " is integervalued. 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 crossfree 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 crossfree family on V if n o two
members of F cross.
(8.1) A tree T, with nodeset V ( T ) ,and with directed edgeset E ( T ) , together
with a function 1 from a set V to V ( T ) ,is called a Vlabelled tree T.
(8.2) For any Vlabelled 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 crossfree family if and only i f
there exists a Vlabelled tree T such that
F
= { S ,: i E E ( T ) } .
Proof. It is easy to check that (8.4) implies F is crossfree
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
Vlabelled tree T satisfying (8.4).
If F' is a crossfree 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 Vlabelled tree T with
labelling function 1, which satisfies (8.4).