Tilburg University
A short proof of Tutte's characterization of totally unimodular matrices
Gerards, A.M.H.
Publication date:
1987
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Gerards, A. M. H. (1987). A short proof of Tutte's characterization of totally unimodular matrices. (Research
Memorandum FEW). Faculteit der Economische Wetenschappen.
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A SHORT PROOF OF TUTTE'S CHARACTERIZATION OF TOTALLY UNIMODULAR MATRICES
A.M.H. Gerards
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k,:.~.J.'-~á. ~ ;, ::; .
A SHORT PROOF OF TUTTE'S CHARACTERIZATION OF TOTALLY UNIMODULAR MATRICES by
A.M.H. Gerards
ABSTRACT
We give, in terms of totally unimodular matrices, a short and easy proof of Tutte's characterization of regular matroids.
DEPARTMENT OF ECONOMETRICS TILBURG UNIVERSITY
1. INTRODUCTION
We give a short and easy proof of the following well-known result of
Tutte [1958.1965,19~1]:
TUTTE'S THEOREM: Let A be a {0, 1}-matrix. Then the follo~ing are equivaZent:
(i) A has a totally unimodular stgntng, (ii) A cannot be transformed to
1 1 1 0
M(F~):- 1 1 0 1
1 0 1 1
by applyíng (repeatedly) the follomíng operatíons: - Delettng ror~s or columns,
(1) - Permuting roros or columns,
- Taking the transposed matrtx,
- Ptvottng over GF(2).
a
The notions used here are: Signtng a {0, 1}-matrix A means replacing some of the 1's in A by -1's. A matrix is called totally untmodular if each of its subdeterminants is 0, 1, or -1. (In particular, all entries of a
to-tally unimodular matrix are 0, 1, or -1.) Pivoting a matrix A(on entry e-t1 of A) means replacing the matrix
e
x x
We consider pivoting over GF(2) as well as over the field R.
Our proof of Tutte's Theorem is in Section 3. Section 2 contains a few well-known and easy-prove preliminar results on graphs, pivoting and to-tal unimodularity.
REMARKS:
Tutte formulated his result in terms of regular matroids (- regular chain groups): A binary matroid is regular if and only if it has neither the Fano-matroid nor its dual as a minor. It is not hard to esthablish the
2
[1976]). Tutte's original proof is very complicated. Shorter and more transparant proofs are given by Seymour [19~97 and Truemper [19~ó,1982]. Zn fact they prove a generalization of Tutte's Theorem: Reid's characterization of GF(3)-representable matroids. Lovasz and Schrijver observed that Reid's Theorem can be proved also along the lines of the proof we present in Section 3. (Reid never published his result. The first proofs that appeared in print are due to Seymour [1979J and to Bixby [19~97. Bixby's proof goes along the lines of Tutte's original proof of his theorem stated above.)
2. PRF,LIMINARIES
2.1 The Bipartite Graph of a Matrix
The proof of Tutte's Theorem in Section 3 uses the following easy graph-theoretic result.
LEMMA 1: Let G be a connected bipartite graph, ~ith no parallel edges. If deletíng any tr~o nodes in the same color-class yields a dtsconnected graph, then G is a path or a circutt.
PROOF: Siippose G is neither a path nor a circuit. Then G has a spanning tree T that is not a path. Hence T has least three end-nodes. At least two of them are in the same color-class of G. Deleting these two nodes from
G results in a connected graph. O
Let A be matrix. Denote the index-set of the rows (columns) of A by R (C respectively). The btpartite graph, G(A), associated mtth A has color-classes R and C. There is an edge from tER to ,fEC in G(A) if the entry in row t and column j of A is non-zero.
2.2 Total Unimodularity
We recall two well-known facts on totally unimodular matrices.
LEMMA 2: Let A be an nXn-matrix with entries from {0, 1, -1}. If G(A) a
cir-cuit, then A is totally unimodular if and only if the number of - 1's in A fs
congruent to n modulo 2.
3
LEMMA :(Camion[1963]). Let MZ and M2 be taio totally untmodular matrices.
Zf MZ and M2 are congruent modulo 2, then MZ can be obtatned from M2 by mul-tiplying some rows and columns by -1.
PROOF: (Paul Seymour) Call an edge in G(M1) (- G(M2)) even if the correspon-ding entries in M1 and M2 are the same. Call the other edges odd. By Lemma 2, each chordless circuit, and hence each circuit, in G(M1) has an even num-ber of odd edges. Therefore, the nodes of G(M1) can be partitioned into two classes, say V1 and V2 such that any edge e is odd if and only if e
connects VI and VZ. Now multiply sll rows and all columns of MZ
corresponding to the nodes in V1, by -1. The resulting matrix is M2. 0
2.3 Pivoting
The following properties of the pivoting operation (2) are easy to
prove:
(3) (i) Pivoting B on -E yields A.
(ii) If A is square, then det(A)-tdet(D-ExyT).
(iii) If A is totally unimodular, then B is totally unimodular. (iv) If G(A) is connected, then G(B) is connected. (Since if G(B) is
disconnected, then (i) implies that G(A) is disconnected too.)
3. PROOF OF TU'I"fE' S THEOREM
The existence of a totally unimodular signing is invariant under the operations (1) (by (3)(iii)). Moreover M(F,~) has no totally unimodular signing. Hence (i) implies (ii). So it remains to prove the reverse implication.
Suppose A is a{0, 1}-matrix, satisfying (ii), with no totally
unimodular signing. We may assume that each proper submatrix of A has a to-tally unimodular signing. So the bipartite graph G(A) is connected. (If not,
A
-r B
Ill 0
0 1
C J
for certain matrices B and C(up to permutation of rows and columns), imply-ing that at least one of B and C has no totally unimodular signimply-ing.) G(A) is not a path or circuit (as otherwise A has trivially a totally unimodular
signing). Hence, by Lemma 1, A or AT is equal to [x~y~ N](up to
4
signing. Moreover, by Lemma 3, these two signings can be chosen such that N is signed in the same way in both cases. Hence A has a
signing A' - [x'~y'I N' ] satisfying:
(4) (i) G(N') is connected,
(ii) Both [x'I N' ] and [y'~ N' ] are totally unimodular. Claim: We may assume that matrix [x'~y'] has a submatrix of the form
( 1 -1, '
Proof: By (iii) and (iv) of Section 2.3, pivoting A' on an entry in N' does
not influence property (4}. Now, pivot A' on an entry in N' such that the smallest submatrix M with determinant not equal to 0, 1, or -1, is as small
as possible. Then M is a 2x2-matrix. (If not, pivot on an entry lying both
in M and N', cf. (3)(ii)).) So M is of the form as in the claim (if
neces-sary multiply x', y', or a row by -1). Moreover, by (4) M has to be a
submatrix of [x'~y']. ~
Denote the row-indices of the two rows of A' in which the submatrix of the claim occurs by a en p. Since G(N'} is connected there exists a path in G(N') from a to p. This path cannot have lenght 2(as such path would correspond with a column of N' with two tl's in the rows a and p, con-tradicting the fact that both [x'~ N' ] and [y'~ N' ] are totally unimodular). From this it follows that A' has a submatrix of the form depicted in the figure below. (If necessary permute rows of A and columns of N', multiply them by -1, or exchange x' and y'.)
1 1 M 1 0... 0 0 0 0... 0 1 1 1 O 1 1 1 -1 M
By pivoting on the underlined entries, deleting the rows and columns con-taining these pivot elements, and multiplying some rows and columns by -1 (and if necessary exchanging x' and y'), we get a submatrix of the form:
~~l~o~
1 -1 0 1
5
It is still the case that deleting any of the first two columns yields a to-tally unimodular matrix. This implies that a- 1 and b- 0. Hence A can be
transformed to M(F7), contradicting our assumption. 0
ACKNOWLEDGEMENT
I thank Alexander Schrijver. Our discusions on the first version of the proof presented here, stimulated to find some shortcuts.
REFERENCES:
[19637 P.Camion, Caratérisation des matrices unimodulaires, Cahiers du
Centre d'Études de Recherche Opérationelle 5(1963)181-190.
[1979] R.E. Bixby, On Reid's characterization of the ternary
matroids, JournaZ of Combtnatortal Theory Series B 26(1979)174-204.
[1979] P.D. Seymour, Matroid representation over GF(3), Journal of
Combinatorial Theory Series B 26(1979)159-173.
[1978] K. Truemper, "On balanced matrices and Tutte's characterization of regular matroids", Working paper, University of Texas,Dallas, 1978.
[1982] K. Truemper, Alpha-balanced graphs and matrices and GF(3)-representability of matroids, Journal of Combinatorial Theory
Series B 32(1982)112-139.
[1958] W.T. Tutte, A homotopy theorem for matroids I, II, Transacttons of
the American Mathemattcal Socíety 88(1958)144-174.
[1965] W.T. Tutte, Lectures on matroids, Journal of Research of the
National Bureau of Standards Sectton B 69(1965)1-47.
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