A Rounding Technique For The Polymatroid Membership Problem

NORTH- HOLLAND
A Rounding Technique for the Polymatroid
Membership Problem
H. Narayanan
Electrical Engineering Department
Indian Institute of Technology
Bombay 400076, India
Submitted by Richard A. Brualdi
ABSTRACT
We present an efficient technique for finding a subset which maximizes w(X)
- p(X) over all subsets of a set E, where w and p are real modular and polyma-
troid functions respectively, using as a subroutine an algorithm which finds such
a set for functions a, p which are near w, p respectively. In particular we can
choose g,p to be rational with denominators equal to 121E13 if we can assume,
whenever p(X) + p(Y) > p(X U Y) + p(X n Y), that the difference between the
two sides is at least one. By applying our technique, we construct an 0(IE13r2)
algorithm for the case where p is a matroid rank function.
1. INTRODUCTION
Let w be a real vector on E = {el, . . . , e,}, and p a polymatroid rank
function on the subsets of E. The polyhedron Pp is the set of all non-
negative vectors 2 : E + 8 s.t. x(X) 5 p(X),X G E. The membership
problem is to determine if the vector (w(el), . . . , w(e,)) E Pp, and if not, to
find a subset of E that maximizes w(X) - p(X), X G E. It is well known
that the problem of minimizing a submodular function over subsets of E
reduces to the above problem.
LINEAR ALGEBRA AND ITS APPLICATIONS 221:41-57 (1995)
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42 H. NARAYANAN
At present, for solving the membership problem when p and w are
integral, there are available a polynomial algorithm based on the ellipsoid
method [3] and a combinatorial pseudopolynomial algorithm [polynomial
in ]E] and p(E)] [Z]. Our technique can use such methods as subroutines
and solve the problem when p and w are real. We assume that whenever
p(X) + p(Y) > p(X U Y) + p(X n Y), the difference between the two sides
is at least one. A simple example of such a function is pi + ppz, where
pr, pz are matroid rank functions and ,0 is a real number greater than or
equal to one.
Our method is as follows: we first construct a vector a, and a sub-
modular function p where 0 < (w - g)e < 1/(6]E12) Ve E E, and p - p
is monotone increasing with (p - p)(X) 5 1/(6]E]), X & E. We-then
raise the value of w at each e FE by a number not less than 1/(3]E]) and
solve in turn each of the resulting membership problems. If in the above
procedure, when the value of ei is raised in ~0, the minimal maximizing
set contains ez, then we say ei 2 ez. This defines a preorder on E, which
in turn induces a partial order on its equivalence classes. An appropriate
weight is assigned to each element in the partial order. It is shown that an
ideal of maximum weight in this partial order corresponds to the minimal
set that maximizes w(X) - p(X).
It can be shown that w, p may be chosen to be rational with denomi-
_
nators equal to 12]E13. Thus this method allows us to convert real mem-
bership problems to integral ones by scaling with a number (in this case
121E13) independent of the number of bits required to represent the values
of w and p.
As an application of our method we consider the matroid polyhedron
membership problem. Here p is already integral. So we need only approxi-
mate w by w and work with a reduced scaling-factor (=6]E12).
In this application we first construct an appropriate parallel copy EI,L~J
li
of E, each element e being replaced by kLw(e)jk parallel elements (k = 6
lE12) and use the simple and well-known matroid partition algorithm [6]
in stages to obtain the set that maximizes LwJI, - p. (The symbol LzJ~
represents the nearest lower integer multiple of l/k to z.) Next, at most
/El parallel branches are added to each element of Ek~~lb in turn, and the
corresponding maximizing sets obtained. This yields the partial order with
appropriate weights for each element. The procedure uses 0(IE131r12) calls
to the independence oracle. The second part of the algorithm, viz. finding
a maximum weight ideal of a partial order, involves finding a mincut and
requires 0( lE13) steps. Both the parts use only additions, subtractions, and
comparisons of real numbers. The overall space requirement is 0( IEj3r).
In [l] Cunningham presents two algorithms for this problem. The first,
like our algorithm, uses only additions, subtractions, and comparisons and

POLYMATROID MEMBERSHIP PROBLEM 43
requires O(jEj8r) calls to the independence oracle and 0(lEj4r) space.
He also presents a variation which requires 0((E15r) time while requiring
0(lE12) space. But this latter uses real multiplications and divisions. One
order (of r) improvement in both of Cunningham’s algorithms has been
achieved by Tardos et al. by using the “layered augmenting path
approach” [4].
The outline of the paper is as follows: Section 2 deals with preliminaries.
Section 3 reduces the membership problem to that of maximizing an (ap-
propriately defined) weight function over the ideals of a partial order. The
partial order is built by repeatedly solving the membership problem over
we and p, which are suitable approximations of ‘w and p. Section 4 gives
an 0(jETr2) algorithm for solving the membership problem over matroid
polyhedra. Section 5 is on conclusions.
2. PRELIMINARIES
If X is a subset (a proper subset) of Y, we write X C Y (X c Y).
If z is a real number and k an integer, 1x1k ([xl k) denotes the greatest
(least) number m/k less (greater) than z with m an integer. We refer to
lxlk (rxlk) as th zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
e 1
ower-l/k (upper-l/k) rounding of 2. If 2 = [x:Jk, we
say that z is l/k-integral.
A set function p : 2E -+ % is a submodular [super-modular] function iff
p(X) + P(Y) 2 P(X uY) + P(X” Y)
[p(X) + P(Y) 5 P(XuY) + P(XnVI, X,Y C_E.
If further p(G) = 0 and p is monotone increasing, then it is a polymatroid
ran/c function. If in addition p is integral and p(A U e) 5 p(A) + 1, then
p is a matroid rank function. A pair of subsets (X, Y) is a modular pair
for a submodular (supermodular) function iff p(X) + p(Y) = p(X U Y) +
p(X n Y). Let the minimum nonzero magnitude of the difference between
left and right sides of the submodular inequality be denoted by 6,. We
deal only with polymatroid rank functions with 6, 2 1. Examples of such
functions are matroid rank functions, integral polymatroid rank functions,
and functions of the kind p1 + ppz, where ~1, pz are integral polymatroid
rank functions and p is a real number not less than 1. If f is a submodular
(supermodular) function, then the collection of all sets over which f reaches
its minimum (maximum) forms a lattice. The unique minimal and maximal
sets in this collection are denoted A(f), B(f) respectively.
The reader is assumed to be familiar-with the notions of matroid,

44 H.NARAYANAN
independent set, base, rank, nullity, circuit, fundamental circuit C(e, b) (the
unique circuit contained in e U b where e 4 b, b a base), coloop (an element
that is not present in any circuit), closure of a set in a matroid, etc. The
contraction M x T of a matroid M on a subset T of E is the matroid on
T with rank function p^where p^(X) s p(X u (E -T)) - ,o(E - T), X 5 T.
The restriction M.T of M to T has T as its underlying set but the same
rank function as M on subsets of T. The independence oracle of M will,
given an X C E, declare it as dependent or independent. (For definitions
and proofs regarding submodular functions and matroids see [6].)
Let Ml = (E,Zl), Mz = (E,&) be matroids. Let ZlV& be the col-
lection of all sets I12 which are unions of an independent set in Zi and
an independent set in Zz. Then (E,ZlV&) is a matroid, which we de-
note by MlVM2 and call the union of Ml and Mz. The union of M
with itself k times is denoted Mk. A set of k bases bl, , . , bk of M s.t.
Ubi is a base of Mk is called a set of k maximally distant bases of M.
Let rk, uk denote the rank and nullity functions of Mk respectively. Then
rk(X) = minycx[kr(Y) + (X - Y]] (Nash-Williams rank formula) and
Q(X) = max&[JYI - kr(Y)].
An ideal of a preorder zp (partial order 2) is a set I^ s.t. if 2 E f and
z & y (z 2 y) then y E f. We can naturally associate with any ideal rof
>* an ideal I of the corresponding partial order L, I being composed of the
equivalence classes of elements of % We call I, I^corresponding ideals. An
element 2 is said to cover an element y of a partial order > iff, z 2 y, z # y,
and no z exists distinct from x, y s.t. x > z > y.
We abuse the notation in the following two instances: A modular func-
tion w with w(Q) = 0 is treated simultaneously as a vector w : E --+ !R as
well as a set function w : 2E -+ R with w(X) = CeEX w(e). In denoting
the complexity of algorithms we omit the modular sign-for instance, we
write 0(E3r) in place of 0((E13r).
3. MEMBERSHIP IN POLYMATROIDS
A stronger version of the membership problem is the following:
PROBLEM I. Let f - w -p, where w : E + !Ris a nonnegative vector
and p a polymatroid rank function on subsets of E. Find A(f).
In this section, we construct a preorder on the set E with an appropriate
weight function defined on its equivalence classes. Through a sequence
of lemmas we show that solving the membership problem (Problem I) is
equivalent to finding an ideal (minimal ideal) of maximum weight in the

partial order induced by the preorder on its equivalence classes.
LEMMA 3.1. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Let f be a supermodular set function.
(a) If a subset S c E is such that f(S) > f(Y) whenever Y c S, then
S C A(f).
(b) If a subset S C E is such that f(S) > f(T) whenever S 2 T, then
S 2 A(f).
Proof. (a): Let S g A(f). Then A(f) c S U A(f) and S n A(f) c S.
We have
f(S) + f(W)) I f(S U A(f)) + W n A(f)).
But f(S) > f(S n A(f)). So f(A(f)) < f(S U A(f)). This contradicts the
definition of A(f).
(b): Let S 2 A(f). Then S U A(f) > S. We have
f(s) + f(A(f)) L f(s uA(f)) + f(s ” A(f)).
But
f(S) 2 f(S u 4.0).
So f(4.0) L f(4.0 n S), w K contradicts the definition of A(f).
h’ h ??
LEMMA 3.2. Let f,g be supermodular with f 2 g, f - g monotone
increasing. Then A(f) > A(g).
proof. Let S maximize f. Then f(S) > f (S U A(g)). However,
f(S) + f (A(g)) 5 f (S ” A(g)) + f (S ” A(g)).
Hence f (S n A(g)) 2 f (A(g)). From monotonicity we have
(9- f)(S” A(g)) 2 (g - f)(A(g)).
Adding the two inequalities, we get
ds” A(g))2 dA(g)).
So S > A(g). In particular S can be taken to be A(f).
Lemma 3.2 helps in relating A(f) and A(g) when g (= f) is a super-
modular function derived from f by approximation as described below in
Definition 3.1.

46 H.NAR.AYANAN
DEFINITION 3.1. Let w be a nonnegative vector on E. Let E,QI be
positive numbers. We denote by w a vector which satisfies (w - w)(e)
< E, e E E. We denote by we, e E E, the function which satisfies we(e)
= w(e) + cr and w”(Z) = w(a), Z # e. We denote by p a polymatroid
rank function such that p - p is a monotone increasing & function and
0 2 (e--p)(X) 5 EIEJ,XC E. If f = w-p, thenfdenotesu-pand fz
denotes we - p.
We now outline our method and point out the role played by the lem-
mas that follow. The value of f is at most 24EI more than f at any subset
of E. So, if we choose o to be not less than 2cIEI, we must, have A(fL)
nonnull whenever e E A(f). Now A(f) can be easily shown to be con-
tained in A(f) (L emma 3.3). Further, it can be seen (Lemma 3.6) that
if e2 E A(f”$) then A(f”‘) C A(f:). Th’is suggests the construction of
a preorder on E in wh% ei 2 e2 iff es E A(f”‘). The set A(f) would
be an ideal in this preorder. Now if we choose-: and CYsufficiently small
[E = M61E12), o = 1/(3Wl, ‘t ’ P
1 1s ossible to assign weights to equivalence
classes in the preorder such that the value of f on any ideal is simply the
sum of the weights of the equivalence classes contained in it (Theorem 3.1).
So to find A(f) we merely find the minimal ideal of maximum weight in
this preorder. In this value is positive, the corresponding ideal is A(f).
Otherwise A(f) is null.
LEMMA 3.3. Let f = w - p, fi, fz be as in Definition 3.1. Let
e 6 A(f).
Then
(9 XL3 = ALOl
(ii) A(fJ G 4.0.
The routine proof is omitted.
The following lemma, which is immediate from Lemmas 3.1 and 3.2, is
relevant when there are elements which are already known either to belong
to or to not belong to A(f).
LEMMA_3.4. Let a E A(f) and c $ A(f). Let f : 2E;c-a --f !J?be
defined byf(X) = f(Xua)-f(a), X C E-c-a. Then A(f) = A(f)-a.
LEMMA 3.5. Let (Y> 2cIEI. If A(f) = ip then e $ A(f).

Proof. Let e E A(f). Since f(A(f)) > 0, we have, by the definition of
f, f(A(f)) > -2elEI. Hence ff(A(f)) > 0. Thus A(fz) cannot be void.
??
Lemmas 3.4 and 3.5 above allow us, without loss of generality, to make
the following assumption.
ASSUMPTION 1
(i) A(f) = a.
(ii) A(JJ # Cpfor each e E E.
REMARK 3.1. If the first part of Assumption 1 is violated, Lemma 3.4
permits us to work with the new function f(X) = f(X U A(f)) - f(A(f)),
X C E - A(f). If the second part of Assumption 1 is violated for any
e E E, by L&ma 3.5 we know that e $ A(f). So we can restrict the un-
derlying set to E - e.
LEMMA 3.6. Let d E A@:). Then A@ C A(fQ
Proof. Since d,e E A(!:) we have f:(Y) = f:(Y) for all Y > A(!:).
Hence fd(Y) < fd(A(ffJ), Y > A(fz). The result follows now from the
second part of Lemma 3.1, taking S to be A(fz) and f to be fd. ??
Lemma 3.6 permits the definition of the following preorder.
DEFINITION 3.2. We define the preorder & on the elements of E as
follows: Let zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
e,d E E. Then e rp d iff A(l.z) > A@. The partial order
induced by & on its equivalence classes will be denoted by 2. The set of
elements of E equivalent to e under & will be denoted by (e). We will
refer to zp and 2 respectively as the preorder and partial order of (f,o),
or of f if (Yis clear from the context.
-
We now state our main theorem, which reduces Problem I to that of
finding the minimal ideal of maximum weight in the above preorder. First
we define the required weight function.
DEFINITION 3.3. Let zp be the preorder of (f,a), and let II be the
set of equivalence classes of &. Then the function h on II is defined as
follows:
V(e)) = f(A(fj) - f(A(f”,) - (e)).

48 H. NAFUYANAN
For X C II we define h(X) E Cc,, e x h((e)), h(a) E 0.
REMARK. We will henceforth denote fz by f” if o is clear from the
-
context.
THEOREM 3.1._ Let Assumption 1 hold. Let c 5 l/(S~E]“),o > 24E]
[= 1/(3]E])]. Let I be an ideal in the preorder Zp . Then f(I) = Ccelcl h((e)).
To prove Theorem 3.1 we need two lemmas.
LEMMA 3.7. Let Assumption 1 hold. Let a > 2cIEI. Then we have:
(9 f(A(f?) > -24EI.
(ii) f(A(J
” )) - f(A(f” ) - (e)) > -2fIEI.
(iii) If I’ is an ideal of (>p) then f(I’) > -2cIE12.
(iv) Let (e) be a maximal element in the ideal z Then
f(A(f” )) - f(A(f” ) - (e)) 2 f(T) - f(r- (e)) - 4W2.
Proof. (i): By Assumption 1, A(f” ) # Q, and A(f) = @. Hence
f” (A(f” )) > 0 and f(A(f” )) > --Q = -24EI.
(ii): Follows from the above when we use the fact that f(A(f” ) - (e))
< f(A(f)) = 0.
(iii): Clearly the ideal I’ can be expressed as the union of principal
ideals A(f” ” ), i = 1, . . . , t say. By the supermodularity of f we have
-
f(A(f” ‘)) + ... +f(A(f” “ )) 5 f
- + xf(xj)T
where the Xj are obtained by appropriate unions and intersections of
A(f” “ ). By the first part of the present lemma we have
-
&A(f)) > -lE1(2W) = -24E12.
i=l
By Assumption 1, f(Xj) IO for all J’. Hence f(U,, A@“‘)) > --2dE12.
(iv): Let I’ be obtained from F by deleting the equivalence classes of
its maximal elements, say el, . . , et. Then
f(T) = f(T) + [f(I’ u (el)) - f(P)] + .. + If@)- f(I’ U . . U (et-d>l.

Hence, using the supermodularity off, if the statement is violated at some
et, we have
f(r) > f(I’) + &(A(L’.Y) - f(A(f”“) - (et))1 + 4cIE12.
-
i=l
Using the previous two parts of the present lemma, we have
f(T) > -2eIE12 - JEl(2eIEI) + 4cJE12= 0,
which violates Assumption 1.
LEMMA 3.8. Let Assumption 1 hold. Let IEl > 2 and let2 5 1/(6jE12).
Let I^be an ideal in &, and let e be a maximal element in I. Then
fMf”H - f(A(f”) - (e))= f(T)- f(I^-(e)).
Proof If equality does not hold, then by the supermodularity off and
since 6f 2 1, we must have
f(A(f”)) - f(A(f”) - (e))I f(F)- f(f- (e))- 1.
But o 2 f(X) - f(X) 5 2eIEI and f(x) < 0, X G E. Hence,
- -
.&J(f)) - f(4f”) - (4) I f@ (f)) - f(4f”) - (4) + 24EI.
and
f(?) - f(I^- (e)) + 243 5 f(r) - f(r- (e))+ 2+7 + 244.
Hence
f(A(f”)) - f(A(f”) - (e )) I f(F) - f(T
- (e )) + 4W - 1.
It is easy to see that for E 5 l/(61E12) and /El > 2, we have 1 - 4cIEj
> 4cIE12. Hence
f(A(f”)) - f(A(f”) - (e )>< f(T
) - f(? - (e )) + 4W2,
which contradicts the last part of Lemma 3.7. R

50 H.NAFt.AYANAN
Proof of Theorem 3.1. Let ? = (ei) U. . . U (ek) , where ei &, ej if i < j.
Then
f(F) = f ((el)) + [f ((el) U (e2)) - f (41 + ...
+ [f(I) - f((el) U . . U (ek-I))].
Observe that (ei), (ei) U (ez), . . . , (el) U.. .U (ek_1) are ideals of &. Hence
by Lemma 3.8 we can write
f(T) = f (A(f”‘)) - f(a) + [f(A(f”‘)) - f(A(f”‘) - &41+ ...
+ [f(A(f”“)) - f(A(f”“) - (ek))l,
- -
which proves the theorem. ??
The following easy lemma allows a restatement of Problem I.
LEMMA 3.9. The ideals of the preorder & of (f, a) of maximum weight
correspond to sets over which f reaches its moximum. These ideals form a
distributive lattice with unique minimal and maximal elements. In particu-
lar A(f) corresponds to the minimal ideal of moximum weight of &.
The preorder Lp naturally induces a partial order 2 on the set of its
equivalence classes. Each element (e) in the partial order can be assigned
the weight h((e)). To find the minimal ideal of maximum weight we can
solve a minimum cut problem as in [5]. This algorithm is O(V3), where V
is the set of elements of the partial order.
Let us now consider the problem of constructing suitable w and p in
some important special cases. In all these, by a naive application of&r
technique, the complexity of finding A(f) in terms of number of calls to
the oracle turns out to be O(Er), if r is the complexity of finding A(fe)
for each e E E. In addition, finding the minimal ideal of maximum weight
in the partial order requires 0(E3) elementary steps. We further note
that these problems cannot be converted into integral problems by scaling
directly with an integer without using the techniques of this paper.
Case 1: w,p real; 6, 2 1. In this case we could add to p a submodular
function g that grows strictly with the size of the set and lower round the
result. (Addition of g to p is necessary, since direct rounding of p might
result in submodularity being violated when the two sides of a modular
equality are rounded. This would not happen with pfg, since this function

has no modular pairs.) For instance, we could take
1
s(X) = 6,E,
- - &ClXl - lEl12.
We can then round (p + g)(X) to th e nearest lower 1/(121E(3)-integer to
yield the polymatroid rank function p. It can be verified that p-p is mono-
tone increasing, (e - p)(X) 5 1/(6pj). The function w can be rounded
below to the nearest 1/(121E13)-integer to yield y. Clearly in this case we
can take E = 1/(6jE12) an Q = 1/(3lEl). Scaling p and we by 121E13
d
converts the finding of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE
A(f” ) into an integral problem.
-
Case 2: w real, p = p1 + ppz, where ~1, p2 are matroid rank functions
and fl 2 1. In this case we take g(e) = LwJk(e) and Q to be p1 + [/3lkp2,
where k = 6lEI 2. We can take E = 1/(61Ej2) and o = 1/(3lEI). Now to
find A&” - p) we need only find A[k$ - (kpl + [kP]p2)], which again
is an integral-problem. This set can be found working only with matroids
as follows: Let Mi, IMz be matroids with ~1, pz as their rank functions.
Replace each element z in each matroid by 61E12$(z) parallel elements.
Let the resultingmatroic& be zi, Gz. Then the desired set is the set of
noncoloops of (Mr )aV(Mz )b, where a = 61E12 and b = [6(~71~Pj.
4. MEMBERSHIP IN MATROID POLYHEDRA
From the two cases considered in the previous section, it appears that
in order to convert the given membership problem to integral problems we
have to increase the size of the underlying set by scaling it by 0(E2) or
0(E3). However, if applied after more careful analysis of the underlying
integral membership algorithm, the rounding technique will usually yield
an algorithm of substantially lower complexity. We carry out this exercise
for the case of matroid polyhedra in the present section.
Let us consider the problem of finding A(w - p), where w is a real
nonnegative vector and p a matroid rank function. By the results of the
previous section this problem can be solved by constructing the preorder
zp of (u - p, cy) by finding A(f) for each e E E. Here w,&,fe are as in
Definition 3.1 (p can be taken as p itself), E = 1/(61E12), and (Y= 1/(3lEj).
We will solve the maximization problem for f” by maximizing instead
the function kf” , which is integral but reaches its maximum at the same
sets as f”. Now maximizing kf” can be interpreted as maximizing a more
conveni&t function defined u&g a matroid derived from M by adding
parallel edges suitably as described in Definition 5.1 and Lemma 5.1 below.

52 H. NAFlAYANAN
DEFINITION 5.1. Let M be a matroid on E with p as its rank function.
Let k be an integer. Let w be a nonnegative weight vector on E with its
values l/k-integral. Then the matroid Mk, on Ekw is constructed by
replacing each e E E in the matroid M with kw(e) parallel elements. The
set E can be identified with an appropriate subset of Ekw. For each element
e’ of ,?&,, there is a corresponding element, say R(e’), in E. If K & E, then
Kk, will denote the subset of Ek, built by replacing each e E K with
kw(e) parallel elements. If L C Ekw then LRk = {R(e’),e’ E L}. The rank
function of Mk, is denoted pk,,, and defined by p&(X) E p(X~k).
The following elementary lemma allows us to reformulate our problem.
LEMMA 5.1. Let f = w - p, where w is a real vector and p a matroid
rank function.
(i) A(w - p) = A(kw - kp).
(ii) Let the values of w be l/k-integral. Then A(kw - kp) = (A(1 . 1
- kpkw))Rkr where (I . 1 - kpk,) is defined on the subsets of Ek,,
and ,&w is the rank function of the matroid MkW.
(iii) A(1 . 1- kpk,) is the set of noncoloops of (Mhi)k.
We now present a simple variation of the basic matroid partition algo-
rithm (see for instance [S]) which we need in order to find A([ . 1- kpk,)
in the matroid Mk,.
ALGORITHM MP (Basic matroid partition algorithm). To construct k
maximally distant bases of a matroid M defined on ,!? and the set Ak of
noncoloops of the matroid M”.
Initialization. Select k bases of M, say by,. . ,bE. Set bl = by,. . . , bk
= bi. Set C = ,!? - lJf=, b,.
Step 1. Construct the auxiliary digraph Gg for the set B = {bl, . . . , bk}
as follows: GB has one vertex for each element in ,!?. A directed edge (ei, ej)
marked b, is added whenever ei 6 b,, ej E b,, and ej E C(e,, b,).
Step 2. Find the shortest path in Gg from an element e E C to an
element e’ belonging to more than one of the bases. If no such path exists
GO TO STEP 6.
Step 3. Let the shortest path from e to e’ be (edge marking shown
in parentheses) e = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC
ec(b,a)el(bil) . . . et(bit)et+l = e’, where e, ei, . . . , et, e’

are distinct but the sequence (bio, bil, . . . , bit) has (not necessarily distinct)
elements from { bl, . . . , blc}. Repeat the following for j = 1,. . . , k: If the
elements equal to bj in the sequence (bio, . . , bit) are bij,, . . , bij,, jl <
J
’Z < < .A-, set bj = (bj - {ej,+l,. . . ,ej,.+~}) u {ej,, . , ej?}.
Step 4. Set C = E - Ui=,bj.
Step 5. GO TO STEP 1.
Step 6. Output bl, . . . , bk as the k maximally distant bases. Output
the set of all vertices in the auxiliary graph Gs reachable by paths from
elements in G, as Ak. STOP.
Since the algorithm is well known, we omit its justification.
REMARK
(i) It is easy to derive, using the Nash-Williams rank formula, that A() .I
- kp) = Ak.
(ii) To find A(I.j-(mpl+npz)), where pi, pz are rank functions of A4i, n/ r,,
one can similarly build M;“ VMF and take the reachable set.
Our aim is to use Algorithm MP in stages to find A( k [wJ k - kp), where
k = 61E12. In order to obtain the complexity of the resulting algorithm we
first state routine results about the complexity of Algorithm MP.
LEMMA 5.2. Let M be a matroid on g.
(i) In Algorithm MP the construction of Gg requires O(kr(E - r)) calls
to the oracle, where r is the rank of the matroid M.
(ii) The overall complexity of Algorithm MP is O(kr(E - r)C) calls to
the oracle. In addition it requires 0(C(E)2) elementary steps (for
shortest path computations), where C = II3 - (by U U b;) 1.
Now let M be a matroid on E. Consider the situation where the new
matroid on g is obtained by replacing each edge in M with p, parallel
branches. In this case we can work with a reduced auxiliary graph for
the new matroid which is essentially the auxiliary graph of M. Thus the
computation of the shortest path in this reduced auxiliary graph is O(E2)
even though the size of the underlying set in the new matroid is CeEEpe.
The length of the shortest path between any pair of vertices in the auxiliary

54 H. NARAYANAN
graph can be seen to be bounded by r(M). From these observations it is
easy to derive the following lemma.
LEMMA 5.3. In Algorithm MP, let C be the set ,!?- (by U.. ’ u b:), and
let E be a set such that every branch in I? is parallel to some branch in E.
Then
(i) the complexity of updating the auxiliary digraph once in Algorithm
MP is O(r”(IEI - r)) calls to the independence oracle;
(ii) finding the shor es path in the auxiliary digraph is 0(E2);
t t
(iii) the (calls to the oracle) complexity of Algorithm MP is min(O(lcr( IEl
- r)C),O(kr(lEl - r) + r2(IEJ - r)C)).
REMARK. Observe that kr((El -r)lCl wi usually be less than ICl(lEl
‘11
- r)r2, since k will usually be less than r. In the present case, however,
k = 6jE12, and so the latter term will be much less. As described earlier, in
order to maximize w - p, (p the rank function of matroid it4 on E), we need
to find A(I.l-kp~Lw~~k) for each e E E with k = 61E12 and cz = 1/(3lEI). By
Lemma 5.1 this is equivalent to finding the set of noncoloops of the matroid
(M~~,,,“J,J~. We will now describe informally how this may be done.
Stage 1. We find, using Algorithm MP, the set of noncoloops of IV, lvll ,~,
where m = 6lEJ, and a set by”,. . . , bz of m maximally distant bases of
M m LWJ
l,&
. Since ICI could be as large as O(IE12 - r) but the initial set of
bases could be identical, the complexity of the algorithm for achieving this
(using Lemma 5.3) is O(r(lEl - r)E + r2(IEI - r)(E2 - r)) calls to the
independence oracle plus 0( E2( E2 - r)) e ementary steps for shortest path
1
computations. We may, by Assumption 1, take the set of noncoloops of
(M,lwl,n)m to be void. (Otherwise we could contract this corresponding
set in M and work with the resulting matroid.)
Stage 2. M~L~J~is obtained from M_lvll,_ by first replacing each el-
ement with IEl parallel branches and then adding k([w(e)Jk - [w(e)Jm)
parallel branches for each e E E C E,L~~ ,,,. We then construct the set
of noncoloops of M~L,,,J~on EklWlk as follows: The initial set of k bases
in Algorithm MP is obtained by taking [El parallel copies of each of the
6lEI bases (which cover E,L~J,~) that we get as the output of Stage 1.
This would mean that the initial set C in Algorithm MP would have
W12bJ@) - lwlm(E)I 1
eements. Clearly ICI < 6(E12[1/(61El)]lEl =
IE12. The initial reduced auxiliary graph is derived directly from that
in Stage 1. The complexity of Algorithm MP at this stage is therefore
0(r2(IEI - r)E2) calls to the oracle plus 0(E4) elementary steps. As in

the previous stage, we assume that the matroid (M~L~J~)~ has only coloops,
so that the k maximally distant bases cover E~L,,,J
k.
Stage 3. We obtain MklWr.jr from M~L~J
b by adding to element e of
E C Eklw~b additional parallel branches equal in number to a(61Ej2) =
21EI. We then construct the set of noncoloops of MklWc:,, on E~L~~J~.
The initial set of k bases in Algorithm MP are the k maximal bases of
M,+,,,, obtained at the previous stage. Since these cover Eklwl k, we may
take the initial set C for Algorithm MP to be of size 21EI. The initial
reduced auxiliary graph is available from Stage 2. Complexity at this stage
is O(r2(IEI -r)E) calls to the oracle plus 0(E3) elementary steps. We have
to repeat this IEl times. So overall complexity at this stage is O(r2(IEl
- r)E2) calls to oracle plus 0(E4) elementary steps.
We thus see that all the stages together require 0(E3r2) calls to ora-
cle plus 0(E4) elementary steps. This is the complexity for building the
partial order 2. We saw earlier that an algorithm is available for finding
a maximum weight ideal in a partial order that is 0(V3) where V is the
set of elements in the partial order. [In the matroid case it is not difficult
to see that IV1 2 min(r(M),v(M)). For, if I is any ideal of 2, we have
r(l) = CcejEI %e), where p(e) = r(A(f”)) - r(A(f”) - (e)). Now (e) can-
not lie in the closure of A(f”) - (e), and hence p(e) > 1. Thus no ideal can
have cardinality more thanits rank, and in particular this is true of the set
of all (ei). A similar argument with the dual partial order shows that this
cardinality cannot exceed v(M).] We thus see that the overall complexity
of our algorithm for maximizing w-p is 0( E3r2) calls to oracle plus O(E4)
elementary steps.
The space requirement for this algorithm is essentially that of storing
the reduced auxiliary graph but with all arcs corresponding to different
bases present. Since there are 61E12 bases and for each of these bases
we have to form fundamental circuits with respect to elements outside it,
the number of arcs in this graph is O(E3r), which is therefore also the
storage requirement.
Let us now consider the important special case of the column ma-
troid of a matrix over a field F. In this case, for each base, we first
build the row reduced echelon form corresponding to it. The fundamen-
tal circuits corresponding to the base can be directly read off. In the
reduced auxiliary graph, constructing the arcs corresponding to this base
requires only r(lEl- r) steps.
When a base b is modified into the base hue-i;with i: E C(e, b), to build
the corresponding reduced echelon form we require r((E( - r) elementary
steps. Now if we follow the same stages as in the general case, we see that

56 H. NARAYANAN
Stage 1 requires
O(Er(IEI - r) + r2()EI- r)(lE12- r) + E2(IE12
- r)).
elementary steps (where the third term corresponds to shortest path com-
putations). Stage 2 requires O(r2(lEl -r)E2+ lE14) elementary steps, while
Stage 3 requires the same number of steps as stage 2. Thus in this case we
have an O(E3r2 + jE14) lg ‘th
a or-1 m. The space complexity is as follows: We
would store the matrix in the row reduced form for each of the bases. This
requires 0(E2(Er)) units. To store the reduced auxiliary graph requires
0(E3r) units as before. Thus the space complexity is 0(E3r).
For the above problem, Cunningham [l] presents an algorithm of time
complexity 0(E8r2) and space complexity 0(E4r). He also presents a vari-
ation (using real multiplication and division) that requires 0(E5r2+ E6)
time and O(E2) space.
5. CONCLUSION
In this paper we have introduced a rounding technique which allows us
to reduce the membership problem in terms of a vector w and a polyma-
troid rank function p to more convenient problems in terms of approxima-
tions a and p. We have applied this technique to convert real membership
problems to &tegral ones and also to solve the matroid membership prob-
lem in 0(E3r2) calls to the independence oracle using O(E3r) space. The
methods of this paper can be used to solve the membership problem over
the intersection of two polymatroids, using as a subroutine an algorithm
for finding A(w - p), where w is an integral vector and p is an integral
polymatroid.
The author is grateful to Murali Srinivasan, Department of Mathe-
matics, and to Sachin Patkar and Milind Sohoni, CSE Department, IIT,
Bombay, for numerous discussions. He is also grateful to the anonymous
referee for his constructive criticism and for the improved statement and
proof of Lemma 3.2.
REFERENCES
1 W. H. Cunningham, Testing membership in matroid polyhedra, J. Combin.
Theory Ser. B 36:161-188 (1984).
2 W. H. Cunningham, On submodular function minimization, Combinatoricu,
1985, pp. 185-190.

3 M. Grotschel, L. Lovasz, and A. Schrijver, The ellipsoid method and its
consequences in combinatorial optimisation, Combinatorics 1(2):169-197
(1981).
4 E. Tardos, C. A. Tovey, and M. A. Trick, Layered augmenting path algo-
rithms, Math. Oper. Res. 11:362-370 (1986).
5 J. C. Picard, Maximal closure of a graph and application to combinatorial
problems, Management Sci., 22 (11):1268-1277 (1975/76).
6 D. J. A. Welsh, Matroid Theory, Academic, New York, 1976.
Received 29 September 1990; final manuscript accepted 12 August 1993

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