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The solution to a matroid problem of Knuth
In thigncrtewe answera question
posed by Knuth in his recent paper “Random
matroids”.
E”rtrutb des&.xs in [2] an algorithm which, subject to certain arbitrary choices, is ~&D&C of constructing any matroid. In effect, this algorithm shows how to ob~ai$d ‘“erectians? (as defined by Crapo [l]) of a given matroid. This has also -been &laneby Roberts [3], althoqgh Knuth’s work shows that the procedure in [3] can be @mpMied. Her&e will be interested only in a derivative of Knuth’s algorithm, namely the s~-@td "free completion (erection) of a clutter”. In [2j it is asked, roughly speaking, whether or not this “free completion” can be used to generate nonmatioidd objec& We will show,, using very simple methods, that the answer is no. Some familiarity with basic matroid theory is assumed in this note. However, we a@@ here a few definitions in order to fix terminology. \ k&~#-:b +f&ite seti A matraid M on E is a collection of so-called independent I . h ##Be& of:E s&sQing the following axioms: (i) 4 E M; (ii) I E JE M 3 I E M; (iii) _ I& ## .a@ IJ[> ill+ 3a E J\ I s.t. I U u E M. Now fix a matroi(d M on E and take , ,&J$-%&4$d?me the ra& of X by r(X) = max (III: 1~ X, I E w. The set X is .s&#,fa,w closed if X = E or r(X U Q) > r(X) for all QE Z \ X; furthermore, X is a #~~pe##unc if it is closed and r(X) = r(E) - 1. Our final definitions are not as sta*Td as the previous ones: We define a clmrttetR on E to be a collection SC q$&ts of E s.t. X,Y E R and X E Y 3 X = Y. The clutter R IS wtatroi&zl if ii 1s a . .col@tion of rank k closed sets of some m;dt.roid on E. *h
IAt R be a clutter on E. Define Ii:= {XU~xXER,aEE\X}.
If X.YEK, %Se?! and ~ZER s.t. ZI>X~Y, set &k=(&\{X,Y})U{XUY}. Repeat this operation until .@ stabilizes and denote the rem?: by F(R). F(R) is call& the free 87
R.E. Bixby
88
completion of R. (Knuth has shown that F(R) is independent of the order in which the operations on R are performed.)
Problem [2]. If R is not matroidal can we have F(R) ~ {E}? The significance of this problem can be understood by considering a result from [2]: If R ~ {E} and R is the collection of rank k closed sets of some matroid, then F(R) is the (generally nontrivial, i.e., not equal to {E}) collection of rank k + 1 closed sets of some matroid. Thus when R is matroidal, F(R) is matroidal. The idea of Knuth's question is to decide whether F can be used in another way to generate clutters which are not matroidal. The answer to this question is provided by Theorem 2.2. To facilitate the proof of Theorem 2.2 we first state a lemma which is equivalent to the fact that the complements of the hyperplanes in a matroid are the circuits of the dual matroid.
Lemma 2.1. A clutter R is matroidal if and only if
(.)
x, Y E R, X ~
Y, a ~ X U Y ~ 3Z e R S.t. Z ;2 (X n Y) U a.
Theorem 2.2. If a clutter R is not matroidal, then F(R) = {E}. Proof. Because F(R) is clearly a clutter it suffices to show E E F(R). Since by Lemma 2.1, (*) fails, 3X,YeR and a~XU Y S.t. X~ Y and iIZeR, with Z ;2 (X n Y) U a. Let WE F(R) be S.t. W;2 X U a. We prove W = E. Let W' E F(R) be S.t. W ' ;2 Y U a. If W' ~ W, then W' n W;2 (X n Y) U a ~ iI Z E R S.t. Z;2 W' n w, a contradiction. Hence, \V = W';2 X U Y U a. Now suppose 3b~ W. There are two cases to consider. Case 1. Assume iIZ e R S.t. Z;2 (X n Y) U b. Then, as above, 3 W' e F(R) S.t. W' ;2 X U Y U b. But then W' ~ Wand W' n W;2 XU Y. This is contrary to the definition of F(R) since R is a clutter. Case 2. Assume 3Z e R, S.t. Z ;2 (X n Y) U b. Let \V' E F(R) be S.t. W' ;2 Z U a. Then W' #:- Wand W' n W;2 (X n Y) U a is contrary to the choice of X, Y and a.
References [1] H.H. Crapo, Erecting geometries. International Conference on Combinatorial Mathematics, Ann. N.Y. Acad. Sci. 175 (1970) 89-92. [2] D.E. Knuth. Random matroids, Discrete Math. 12 (1975) 341-358. [3] l. Roberts, All Erections of a Combinatorial Geometry and Their Automorphism Groups. in: Lecture Notes in Mathematics 452 (Springer. Berlin. 1975).
posed by Knuth in his recent paper “Random
matroids”.
E”rtrutb des&.xs in [2] an algorithm which, subject to certain arbitrary choices, is ~&D&C of constructing any matroid. In effect, this algorithm shows how to ob~ai$d ‘“erectians? (as defined by Crapo [l]) of a given matroid. This has also -been &laneby Roberts [3], althoqgh Knuth’s work shows that the procedure in [3] can be @mpMied. Her&e will be interested only in a derivative of Knuth’s algorithm, namely the s~-@td "free completion (erection) of a clutter”. In [2j it is asked, roughly speaking, whether or not this “free completion” can be used to generate nonmatioidd objec& We will show,, using very simple methods, that the answer is no. Some familiarity with basic matroid theory is assumed in this note. However, we a@@ here a few definitions in order to fix terminology. \ k&~#-:b +f&ite seti A matraid M on E is a collection of so-called independent I . h ##Be& of:E s&sQing the following axioms: (i) 4 E M; (ii) I E JE M 3 I E M; (iii) _ I& ## .a@ IJ[> ill+ 3a E J\ I s.t. I U u E M. Now fix a matroi(d M on E and take , ,&J$-%&4$d?me the ra& of X by r(X) = max (III: 1~ X, I E w. The set X is .s&#,fa,w closed if X = E or r(X U Q) > r(X) for all QE Z \ X; furthermore, X is a #~~pe##unc if it is closed and r(X) = r(E) - 1. Our final definitions are not as sta*Td as the previous ones: We define a clmrttetR on E to be a collection SC q$&ts of E s.t. X,Y E R and X E Y 3 X = Y. The clutter R IS wtatroi&zl if ii 1s a . .col@tion of rank k closed sets of some m;dt.roid on E. *h
IAt R be a clutter on E. Define Ii:= {XU~xXER,aEE\X}.
If X.YEK, %Se?! and ~ZER s.t. ZI>X~Y, set &k=(&\{X,Y})U{XUY}. Repeat this operation until .@ stabilizes and denote the rem?: by F(R). F(R) is call& the free 87
R.E. Bixby
88
completion of R. (Knuth has shown that F(R) is independent of the order in which the operations on R are performed.)
Problem [2]. If R is not matroidal can we have F(R) ~ {E}? The significance of this problem can be understood by considering a result from [2]: If R ~ {E} and R is the collection of rank k closed sets of some matroid, then F(R) is the (generally nontrivial, i.e., not equal to {E}) collection of rank k + 1 closed sets of some matroid. Thus when R is matroidal, F(R) is matroidal. The idea of Knuth's question is to decide whether F can be used in another way to generate clutters which are not matroidal. The answer to this question is provided by Theorem 2.2. To facilitate the proof of Theorem 2.2 we first state a lemma which is equivalent to the fact that the complements of the hyperplanes in a matroid are the circuits of the dual matroid.
Lemma 2.1. A clutter R is matroidal if and only if
(.)
x, Y E R, X ~
Y, a ~ X U Y ~ 3Z e R S.t. Z ;2 (X n Y) U a.
Theorem 2.2. If a clutter R is not matroidal, then F(R) = {E}. Proof. Because F(R) is clearly a clutter it suffices to show E E F(R). Since by Lemma 2.1, (*) fails, 3X,YeR and a~XU Y S.t. X~ Y and iIZeR, with Z ;2 (X n Y) U a. Let WE F(R) be S.t. W;2 X U a. We prove W = E. Let W' E F(R) be S.t. W ' ;2 Y U a. If W' ~ W, then W' n W;2 (X n Y) U a ~ iI Z E R S.t. Z;2 W' n w, a contradiction. Hence, \V = W';2 X U Y U a. Now suppose 3b~ W. There are two cases to consider. Case 1. Assume iIZ e R S.t. Z;2 (X n Y) U b. Then, as above, 3 W' e F(R) S.t. W' ;2 X U Y U b. But then W' ~ Wand W' n W;2 XU Y. This is contrary to the definition of F(R) since R is a clutter. Case 2. Assume 3Z e R, S.t. Z ;2 (X n Y) U b. Let \V' E F(R) be S.t. W' ;2 Z U a. Then W' #:- Wand W' n W;2 (X n Y) U a is contrary to the choice of X, Y and a.
References [1] H.H. Crapo, Erecting geometries. International Conference on Combinatorial Mathematics, Ann. N.Y. Acad. Sci. 175 (1970) 89-92. [2] D.E. Knuth. Random matroids, Discrete Math. 12 (1975) 341-358. [3] l. Roberts, All Erections of a Combinatorial Geometry and Their Automorphism Groups. in: Lecture Notes in Mathematics 452 (Springer. Berlin. 1975).