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OFaTEHNOGY

0.

by

T. L. Magnanti OR 013-73 January, 1973

Work Performed Under Contract No. DAHC04-70-C-0058 U. S. Army Research Office (Durham) Dept. of the Army Project No. P-9239-M

MIT DSR 72650

Operations Research Center

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Cambridge, Massachusetts 02139

Reproduction in whole or in part is permitted for any purpose of the United States Government

ABS TRACT

Let el, el, e, e'..., e , e' be the elements of matroid M. 1' 1J 2 2' n n

Suppose that {ele _{2,...,en} is a base of M and that every circuit of M} contains at least m+l elements. We prove that there exist at least 2m

bases, called complementary bases, of M with the property that only one of each complementary pair ej,e' is contained in any base.

We also prove an analogous result for the case where E is partitioned into E1,E2...,E and the initial base contains

IEjl

I. Introduction

Let el,el,e,e2,... 2 e n,e ' be the elements of matroid M. A police

force of n men wishes to patrol a base of the matroid. Suppose that policeman k is only allowed to patrol either of ek or e and that

{el,e2,...,e n} is a base of the matroid, called a patrolable (complementary)

base. How many patrolable bases are there?

In [2], Dantzig considered graphic matroids. He showed that if there is one patrolable spanning tree of a graph G (with no loops or parallel edges), then there necessarily must be a second. Later Adler [1] strengthened this result by showing that in this case there must necessarily be at least four.

Here we show that these results extend to general matroids and that the number of complementary bases depends upon the circuit structure of the matroid. If every circuit of M contains more than m elements, we show that there are at least 2m complementary bases. We also generalize this

result to the case when policeman k can patrol dk of the (dk+l) elements

of the matroid assigned to him.

Our proofs are based on Lawler's matroid intersection algorithm [4] and provide an algorithm for determining alternate complementary bases. These proofs could just as well have been based upon Theorem 2c of [3]. In fact, Dantzig's paper was motivated by a discussion held with Fulkerson concerning

the results in [3]. Dantzig and Adler's results were based upon complemen-tary pivot theory of mathematical programming. Consequently, our results strengthen the well known link (via the assignment problem of network theory,

for example) between matroid algorithms and standard results in mathematical programming.

II. Preliminaries

If S E and eE, we denote SU{e} and S-{e} by respectively S+e and S-e. Also ISI denotes the cardinality of S. Beside the matroid inter-section algorithm, we only require very basic properties of matroids, so for convenience we review them here.

For our purposes, we define a matroid M to be a finite set E together with a non-empty family t of subsets of E satisfying

(i) I c I1 E implies I E J

(ii) Given A c E all maximal sets (with respect to inclusion) of

9

Maximal sets in 3 are called bases of M. Subsets of E not contained in are called dependent sets and minimal dependent sets are referred to as circuits.

These definitions easily imply:

(Ml) If S E and (S) = {ICS : I E , then (S,(S)) is a matroid. n

(M2) Suppose that E = U Ei where the sets Ei are pairwise disjoint,

and that d = (dt,...,dn) is a vector with non-negative integer components. Then with g = {I C E : II nEiJ < di}, (E,d) is a matroid called a d-partition matroid. When each d = 1, we call

(E,J) a unitary partition matroid.

(M3) If I is a base of M and e I, then I + e contains a unique circuit. Let M1 = (E,J1) and M2 = (E,,2) be two matroids over the same ground

set E. A maximum cardinality set in J 1/A2 may be found by,using the following

Matroid Intersection Algoritbn 4]: Assume E = {el,...,e }. Let L:E-{O,1,...,n+l} be a labelling function to be defined during the algorithm and let S be a set of "scanned elements." We call an element e of E labelled if L(e)c{O,...,n} and scanned if eS.

Initially, L(ej) = n+l for all e and S=O. Assume that a set I 1nI 2 is given (possibly I = ).

(a) For lj<n and eI, if I+e l, set L(ej) = 0.

(b) If all labelled elements have been scanned, terminate. I is optimal. Otherwise go to (c).

(c) Let ej be labelled and unscanned. Change S to S+ej. If e I, set L(e) = j for each unlabelled eE-I such that I-ej+ec l; then go to (b). If eI, then if there is a circuit C of M2

with C I+e., set L(e) = j for each unlabelled eC and go to (b). If no such C exists, go to (d).

(d) (Augment): Define ej ,...,em = e by jk L(ejk+l) for k=l,...,m-l and L(j) = O. Replace I with I+e -e +...-e +e.

2 m-1 · Reinitiate L(ej) = n+l for all ej, S=O, and return to (a).

III. Complementary Bases of a Matroid

Suppose that E = elee ,e,..,ee, 2 that = E is a

matroid and that I = {ele2 ... ,e } is a base of M1. Note that I contains

only one element from each of the complementary pairs ej,e. We will call such a base a complementary base of M1. Defining 9₂ = {I-E: In{ej,eJ}I < 1, j=l,...,n}, M2 = (E,J92) is a unitary partition matroid and complementary

Let NM denote the number of complementary bases of M1 and as n

varies let~ denote the collection of matroids M1 with the above proper-ties. In this section we prove certain lower bounds on NM1. The results depend heavily upon the circuit structure of M. Accordingly, we define

N(m) = min {NM:M~ and every circuit of M contains > m elements}. Theorem 1: N(1) 2.

Proof: We consider an arbitrary ME,9 ₁) contained in with the property that every circuit of M1 has > 2 elements and show that NM 2.

Let 42 be defined as above. Starting with I = {el,...,e } suppose that we remove el from the system producing the submatroids (E,J1(E)) and (E, 2(E)) where E = E-el. Apply the matroid intersection algorithm to the set I-e₁

contained in , 1(E) n 4₂(E), beginning by labelling all eE such that

I - e + e 4₁(E). Note that we reach step (d) of the algorithm and produce a new complementary basis of M₁ if and only if e' is labelled. Thus, assume

e' is not labelled and let L₁ be the set of labelled elements. Repeat,

1 1

dropping e2,e3,...,en in turn instead of e from E producing the sets of

labelled elements L₂,...,L . If eL for any j, step (d) is applied pro-viding the result. We show that it must be true that eL. for some j.

Suppose not. Then every labelled element in the n applications of the matroid intersection algorithm above will be scanned. But then, the nature

of the optimal matroid intersection algorithm (i.e., an unlabelled element is labelled based only upon the element being scanned) implies that if ejELi, then Lj . Li. Also, since I is a base of M₁ and no circuits have length 1, I+e! contains a circuit C of M₁ with some e C. Property M3 of matroids states that I-ei+ejel, thus eL _i. Start with eL2, say. When e is

scanned e is labelled thus, L L ecL. ome j#2. If eL 1, then

L =L so elL, a contradiction. Thus, assume by relabelling if necessary 1 2 e L1,

that eL2 3. In general, if eLj+l, j=l,...,k, L1 L ... L Lk thenthen

ef' L for j<k as otherwise eL.. But, once we arrive at L L . LC

k

J

e' must be contained in some Lj, giving a contradiction. Therefore, the n

hypothesis that eL. for some j was invalid.

i O

Lemma: Suppose that every circuit in M1 has > m elements. Then

given any m-l elements, say el,...,em 1, of the base I = {el, ...,en},

there is a complementary base I' I of M1 with eEI', j=l,...,m-l.

Proof: Let j£{m,...,n}. I+e! contains a circuit C of M1 since I is

a base. Our hypothesis implies that this circuit contains at least m elements of I. e is consequently labelled in the algorithm if we drop

3

any of these m elements from the system. Thus, using the notation of the previous proof, eL for some i{m,...,n}. If eLj, we augment and have the result. But, eL. for all j{m,...,n} leads to a contradiction by the

argument used in the previous proof. Theorem 2: N(m) > 2

Proof: Suppose that every circuit of the matroid M1 has > m elements.

We prove a slightly stronger result than the theorem's assertion: there exists a rooted directed tree T whose edges are elements of E and which has the following properties.

(i) A complementary pair of elements of E originate from every node of T

m

(ii) Every maximal (with respect to inclusion) directed path originating from the root of T consists of m elements of E and these elements

are contained in a complement fly ase of M1. (Note that we have not excluded an element of E from appearing in Tm more than once.) For example, for m=2 we have the picture

e.

Note that the 2m complementary bases corresponding to the directed paths of (ii) must be distinct, since in tracing back to the root from the endpoints of T any two paths must meet for the first time at some node. The arcs following this node are complementary so the bases are distinct. Thus the stronger result proves the theorem.

Theorem 1 shows that N(1) - 2. Since there are two distinct complementary bases, one contains some e. and the other its complement e . Thus, letting e and e be two arcs originating from the root, the stronger result holds for m=l.

By induction the stronger result holds for m-l, giving the tree T ₁ Given any path of Tm_₁ with elements S = {ele ₂ ... em_l}, say, S is

contained in a complementary base I of M1. By the previous lemma, S is also

contained in at least one other complementary base I' of M_1. Since II', there is an ej. I, e' I'. Extend T 1 by adding e and e to the endpoint

of Tm_₁ incident to e_l Doing this foi all 2 maximal paths of Tl we extend it to T and complete the proof.

The bound given in Theorem 2 is the best possible for let E = {el,... em,e{,...,e'} and let consist of all subsets of E with m or fewer elements. Then every m element set of E with one element from each complementary pair is a complementary base. Thus, in M1 = (E,J1), =

IV. A Generalization

n 1 d.+l

Let E = , E E ={ej,...,ej3 } be a partition of E into pairwise j=l

disjoint sets with d 2 1, and let d = (dl,... ,d ). Suppose that M₁= (E,J1)

is a matroid. A base I of Mt satisfying IInEj = d. is called a

d-complemen-tary base. If each d = i, a d-complementary base is just a complementary base as considered in the last section. In this section, we outline

extensions of previous results applicable to general d.

Let I = A U ... A, A = e,...e1 nAu e j}, be a d-complementary base

Let I = A1U A2 U . . . A n, Aj = { ej,.. ,

of M. Apply the algorithm of Theorem 1 with 2 replaced by the d-partition '1 2 dn matroid 2 = {IgE: IlgEj I dj}, i.e., in turn drop e, el,...,e ,..,e from E and I and apply the matroid intersection algorithm to the resulting

sub-k dj+l

matroid beginning with the set I-ej. If e is labelled when any of 1 d.

ej,... ,e have been dropped, a new d-complementary base has been produced. Otherwise, arguing as in the proof of Theorem 1, let L be the elements

r

labelled when dropping e and applying the matroid intersection algorithm. d.+l

s

Note that if e is scanned in the algorithm each of e s = 1,...,d is d+l +1

labelled, thus e L implies that Lt Ls for all tE{l,...,d.}.

Conse-quently, the argument pr gives:r1

Theorem 1A: If there exists a d-complementary base to the matroid M1 = (E,j1) and each circuit of M1 contains at least

two elements, then M1 contains a second d-complementary

base.

Similarly, arguments analogous to those of section III provide the following results.

Lemma: Suppose that every circuit of M1 contains elements from at

least m+l of the partitions E. Let I = A_1U ...U_{A n}, Aj Ej, be a d-complementary base of M1. Then given any (m-l) of the sets Aj, say

A1,... ,Am_1 there is a d-complementary base I' I of M₁ with A. g I' j = 1,...,m-1.

n

Theorem 2A: Let E = U E. be a partition of E into pairwise disjoint j=l J

sets E,

IEjl

Suppose that every circuit of M1 contains elements from

at least m+l of the partitions E. Then if there is one d-complementary base of M1, there are at least 2m .

Note: To extend the proof of Theorem 2, let the edges of T be subsets A. c E. for some 1 < j < n with Aj = dj and change proper-ties (i) and (ii) of T to:

(i') For any node v of Tm, two edges A # B originate at v and both A and B are contained in E for some 1 j < n.

(ii') Every maximal directed path originating from the root of T consists of m subsets A. of E and the union of these subsets is contained in a complementary base of M1.

Finally, observe that tht hy'ypot' i-, o2 lheorem 2A is much stronger than the statement that every circuit of M contains at least m+l elements. We conjecture that the result is not true with this weaker hypothesis.

University, December 1969.

2. Dantzig, G.B., "Complementary Spair.ing Trees," in J. Abadie (Ed.), Integer and Nonlinear PrPrfmlin.Fg North Holland Publishing Company, (Amsterdam, 1970), pp, 499-505.

3. Edmonds, J. and Fulkerson, DR., "Transversals and Matroid Partition," NBS, Journal of Res., 69B, No. 3 June 1965, pp. 147-153.

4. Lawler, E., "Optimal Matroid Intersections," in Combinatorial Structures and Their Applications Pr ceedings of the Calgary International Conference, Gordon and Breach, 1970, abstract only, p. 233.