Matrices in ${\cal A}(R,S)$ with minimum $t$-term ranks
Ros\'ario Fernandes, Henrique F. da Cruz, Susana Palheira
TL;DR
This paper investigates the conditions under which matrices with prescribed row and column sums can achieve their minimum $t$-term ranks, extending understanding of combinatorial matrix properties for various $t$ values.
Contribution
It establishes conditions for the existence of matrices in ${ m extbf{A}}(R,S)$ that realize all minimum $t$-term ranks for any $t \,\geq\, 1$, advancing combinatorial matrix theory.
Findings
Derived necessary and sufficient conditions for such matrices.
Extended previous results to all $t \geq 1$.
Provided a framework for constructing matrices with minimal $t$-term ranks.
Abstract
Let and be two sequences of nonnegative integers in nonincreasing order and with the same sum, and let be the class of all -matrices having row sum and column sum . For a positive integer , the -term rank of a -matrix is defined as the maximum number of 's in with at most one in each column, and at most 's in each row. In this paper, we address conditions for the existence of a matrix in a class that realizes all the minimum -term ranks, for .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsDigital Image Processing Techniques · graph theory and CDMA systems · Medical Image Segmentation Techniques
Matrices in with minimum -term ranks
Rosário Fernandes
CMA and Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa
2829-516 Caparica, Portugal Corresponding author. Email: [email protected]. This work was partially supported by project UID/MAT/00297/2019.
Henrique F. da Cruz
Departamento de Matemática da Universidade da Beira Interior,
Rua Marquês D’Avila e Bolama, 6201-001 Covilhã, Portugal Email: [email protected] work was partially supported by project UID/MAT/00212/2019.
Susana C. Palheira
Faculdade de Ciências e Tecnologia
Universidade Nova de Lisboa
2829-516 Caparica, Portugal Email: [email protected].
Abstract
Let and be two sequences of nonnegative integers in nonincreasing order and with the same sum, and let be the class of all -matrices having row sum and column sum . For a positive integer , the -term rank of a -matrix is defined as the maximum number of ’s in with at most one in each column, and at most ’s in each row. In this paper, we address conditions for the existence of a matrix in a class that realizes all the minimum -term ranks, for .
Keywords: t-term rank, (0,1)-matrix, Gale-Ryser Theorem, Network flows
AMS Subject of Classification: 05A15, 05C50, 05D15
1 Introduction
Consider the following problem: “We are organizing a social dinner for the participants of a mathematical meeting. The participants are from different countries. We intent to seat the participants in tables, each one with a fix number of seats, so that in each table there are no two participants with the same nationality. The number of seats is equal to the number of participants and there is a possible distribution in these conditions. After dinner, there will be a show. Before the show, the host will call some participants for a joke. The host will choose the maximum number of participants selecting, at most, one per table and, at most, one per country. As we do not know when the artists will be ready for the show, the host will repeat the joke with another group, with the maximum number of participants, again choosing, at most, one per table and one per country, but the countries are now different from the countries selected in the first group. The host will repeat this joke until the show begins, or all countries were chosen. How to distribute the participants in the tables so that after rounds of jokes the total number of evolved participants is as small as possible?”
The above problem is a classical problem of distributing elements into sets with some constraints. An important tool for solving these kind of problems are matrices whose entries are just [math]’s and ’s, the -matrices. In fact, Ford and Fulkerson noted that a -matrix can be regarded as distributing elements into sets: the ’s in row designate the elements that occur in the th set, and the ’s in column designate the sets that contain the th element (see [10]). Therefore, the -matrices are essential tools in many combinatorial investigations, and hence these matrices are among those who have been received more attention in the last years (see [1, 2, 6, 7, 8]. Using a -matrices for modeling a certain distribution of the participants into the tables, in our problem, the number of selected participants after rounds is a combinatorial parameter (properties which are invariant under arbitrary permutations of its rows and columns) of a -matrix called -term rank. The -term rank of -matrix , denoted , is the maximum number of s in with at most one in each column and at most s in each row. When , we have the well-known term rank of , denoted . In terms of the incidence matrix of sets vs. elements, the -term rank of a -matrix is the maximum number of distinct elements that we have if we choose at most elements in each set, associated. So a solution for the above problem is a -matrix with prescribed row and column sum vectors, that realizes all the minimum -term ranks, for . In this paper, we focus our attention on this kind of -matrices, that is, -matrices with prescribed row and columns sum vectors with all the minimum -term ranks, for .
2 Background on
Let , and be two partitions of the same weight (this is, and are integral vectors such that , and ). The class of all -matrices with row sum vector and column sum vector is denoted by . This class has been heavily investigated (see [1, 2, 5] for details) since the fifties, and many notable results have been obtained. The aim of this section is to make a brief sketch of these results.
A question that first arise when we study a class is to know when it is nonempty. This problem was solved independently by Gale (see [11]) and by Ryser (see [14]). In fact, they proved a theorem, now called the Gale-Ryser theorem, which states that the class is nonempty if and only if is majorized by (the conjugate partition of , defined by , for ), i.e.
[TABLE]
where the equality holds for .
In the Ryser’s paper, [14], an algorithm for the construction of a matrix in is also presented. This algorithm, called the Ryser’s algorithm, starts with a -by- -matrix whose row sum vector is and whose column sum vector is the conjugate vector . Thus the 1’s occupy the initial positions in each row. The construction begins by shifting of the last 1’s of certain rows of to column . The 1’s in column are in rows of with largest sum, giving preference to the bottommost positions in case of ties. Reducing by 1 those corresponding to the rows that contains the 1’s placed in column , we obtain a vector which also satisfies the monotonicity assumption. We now proceed inductively to construct the columns . Ryser proved that if is majorized by , then this algorithm can be carried out in order produce a matrix in . When the matrix constructed by Ryser’s algorithm is often called the canonical matrix of .
Another fundamental result due to Ryser is the so called Ryser’s interchange theorem, [2], which states that given two matrices , can be transformed in by a finite sequence of interchanges. An interchange is an operation which replaces a -by- submatrix
[TABLE]
and vice-versa. Therefore, every matrix in can be transformed by interchanges into the canonical matrix of .
Ford and Fulkerson, presented another criterium for the nonemptiness of . Let be the -by- matrix (the rows are indexed in and the columns are indexed in ), such that
[TABLE]
This matrix is called the structure matrix associated with and , and Ford and Fulkerson proved that if and only if each entry of is nonnegative.
Let be a positive integer. Motivated by the study of combinatorial batch codes (see [3, 12, 13]), the authors of [5] defined the -term rank of a -matrix , denoted , as the maximum number of s in with at most one in each column and at most s in each row. When , we have the term rank of , denoted .
By the Knig-Egervry theorem (see [2], p.6) equals the minimum number of lines that cover all the s of :
[TABLE]
In [5] the authors established a generalization of this theorem:
Proposition 1
[5]* Let be an -by-, -matrix and let be a positive integer. Then*
[TABLE]
The term rank and the -term rank are two of several combinatorial parameters of a -matrix. In this paper we turn our attention to the minimal -term rank of a nonempty class , denoted by . So,
[TABLE]
A formula for computing the was first derived by Haber and simplified by Brualdi [2]. A generalization for was stated by Fernandes and da Cruz in [9]. This formula is obtained using a matrix defined as follows: Let be the -by- structure matrix associated with and . Define the matrix , also denoted by by
[TABLE]
for all , and the minimum is taken over all integers and that satisfy
[TABLE]
Proposition 2
[9]* Let and be partitions of the same weight such that the class is nonempty. Let be a positive integer. Then,*
[TABLE]
In [9] the authors proved the following theorem which, in particular, implies the existence of a special matrix in with -term rank .
Theorem 3
[9]* Let and be partitions of the same weight such that the class is nonempty. Then, there is a matrix in all of whose s are contained in the union of its first rows and first columns if and only if *
Our main purpose in this paper is to know if in any nonempty class there is a matrix that realizes all the minimum -term ranks, for . The analogous problem for the maximum -term rank in was solved in [5]. In fact, the authors proved that in any nonempty class there always exists a matrix which realizes all the maximum -term ranks, for , and the authors of [9] conjectured that the same happens for all the minimum -term ranks, with .
Conjecture 4
[9]* If and are partitions of the same weigh such that is nonempty, then there is a matrix such that , for all *
This paper is organized as follows: As we wrote in the Introduction, the -matrices can be regarded as incidence matrices of sets vs. elements, so, in the next section we compute the -term rank of a matrix using the Ford-Fulkerson algorithm. Despite what happens with the maximum -term rank, in the third section we present a class where there is no matrix that realizes all the minimum -term ranks, for . However with some restrictions on and it is possible to prove that there are classes where such matrices exist. This is the subject of the fourth section. The existence of these classes depends of the existence of a special matrix with prescribed zero blocks. In the last section we present an algorithm for construct -matrices with these constraints .
3 Network flows and the t-term rank
Another way to obtain the -term rank of a -matrix is using the network flows. Let be the bipartite direct graph where , and there is the edge of , with and , if and only if .
Let be the graph obtained from putting two new vertices, and , and the edges , , for , .
Let be the map from to such that
[TABLE]
The -term rank can also be obtained using the Ford-Fulkerson algorithm [10]:
Let . 2. 2.
If there is a path , in , from to ,
[TABLE]
such that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
then go to . Otherwise, go to . 3. 3.
Let be the map from to such that
[TABLE]
Go to with . 4. 4.
Stop. .
Example 5
Let A=\left[\begin{array}[]{cccc}1&1&1&1\\ 1&0&0&0\\ 1&0&0&0\end{array}\right]. Using the algorithm we can obtain as described:
Let be the set of rows of and be the set of columns of . Let be the direct graph whose set of vertices, , is and set of edges is .
Let be the map from to such that
[TABLE]
We begin the algorithm with and the path
[TABLE]
Let be the map from to such that
[TABLE]
With , the path
[TABLE]
is a path in the conditions of step 2., let be the map from to such that
[TABLE]
With , the path
[TABLE]
is a path in the conditions of step 2., let be the map from to such that
[TABLE]
Since with there is any path in the conditions of step 2., then
[TABLE]
4 A counterexample for conjecture 4
Let , and
The structure matrix associated with and is
[TABLE]
and the matrix is
[TABLE]
The grey entries are the first entries, in each column, that are equal in matrices and .
So, by Proposition 2
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Suppose there is a matrix such that for . Since by Theorem 3, the 1’s of can be covered by one row and nine columns. Using the fact that all columns of has at least a nonzero entry, and has five columns with at least two nonzero entries, then with one row we must cover all 1’s of six columns of . The unique row of with at least nonzero entries is the first. Consequently, we can assume that the first row of is
[TABLE]
Let be the matrix obtained from removing the first row and the last six columns. Since then the 1’s of can be covered with two rows and five columns. Using the above arguments, the 1’s of can be covered with one row and five columns. Note that has nine columns and five of them has at least two nonzero entries. This implies that the row that cover the 1’s of , cover the 1’s in the last four columns of . Only the first and the second rows of has at least four nonzero entries. Let be the matrix obtained from removing the last four columns.
The matrix has five columns and eight rows. Moreover, the row sum vector of is or , and the column sum vector of is .
Since then the 1’s of can be covered with three rows and three columns. We have two cases:
If the row sum vector of is and its column sum is , using the above arguments, all 1’s of will be covered with one row and three columns. This is impossible because five columns of has at least two nonzero entries.
If the row sum vector of is , and its column sum is , using the above arguments, all 1’s of will be covered with two rows (the first row of is one of these two rows) and three columns. This is impossible because four columns of the matrix obtained from removing the first row has at least two nonzero entries.
Therefore, no matrix in realizes all the minimum -term ranks, for .
5 Special partitions and
Let and be partitions of the same weight such that is nonempty. In this section we present conditions for the existence of a matrix in that realizes all the minimum -term ranks, for . So the next result is our main result:
Theorem 6
Let and be be two partitions of the same weight such that , , with , , and is nonempty. Let be a positive integer. If the minimum integers such that and verify , , and for all , , then there exists a matrix such that
[TABLE]
For proving this theorem we start the following lemma:
Lemma 7
[1]* Let and be two partitions of the same weight, and let be an -by- nonnegative integral matrix. There is a nonnegative integral matrix , satisfying*
[TABLE]
[TABLE]
and
[TABLE]
if and only if, for all and ,
[TABLE]
Let be the -by- structure matrix associated with and . For , and , define the nonnegative integer as
[TABLE]
where the minimum is taken over all integers and that satisfy
[TABLE]
and
[TABLE]
To compute a matrix in the conditions of Theorem 3 we can use the modified Ryser’s algorithm (see [4] or Section 6) or the known algorithms from the network flows (see [10]). The following proposition is a generalization of Theorem 3.
Proposition 8
Let and be two partitions of the same weight such that , , and is nonempty. Let be integers such that and . Then there is a matrix in of the form
[TABLE]
if and only if
[TABLE]
Proof. The proof follows the steps of the proof of theorem 3.5.8 in [2]. Let be the -by- matrix such that
[TABLE]
where denote the -by- matrix whose entries are all equal to . By Lemma 7, there is a matrix in of the form (1) if and only if
[TABLE]
for all and .
For and , we write where , and , and we write where , and . We agree to take complements of , , , , , and with respect to , , , , , and , respectively. Then (2) is equivalent to
[TABLE]
[TABLE]
for all , , , , and . Let , , , , and . Since and are nonincreasing, it follows that the last inequality is equivalent to
[TABLE]
[TABLE]
holding for all integers , and , with and all integers , and , with . The last inequality is equivalent to
[TABLE]
where , , , , , , , . Let be any matrix in , and partition according to the diagram
[TABLE]
Then (3) counts and in submatrices of as shown above, where means the negative of the number of in the submatrices of indicated.
On the other hand, the expression counts and as shown below:
[TABLE]
In the matrix, means twice the number of in the corresponding submatrices of , means three times the number of in the corresponding submatrices of , means twice the number of in the corresponding submatrices of , and means three times the number of in the corresponding submatrices of .
It now follows that
[TABLE]
counts and in submatrices of as indicated in the matrix diagram.
[TABLE]
Hence, (4) equals
[TABLE]
Therefore (3) holds if and only if .
Proposition 9
Let and be be two partitions of the same weight such that , , with , , and is nonempty. Let be integers such that and . If and , then
[TABLE]
Proof. Let be an integer such that . Then,
[TABLE]
Since then
[TABLE]
Let be an integer such that . Then,
[TABLE]
Since then
[TABLE]
By definition,
[TABLE]
[TABLE]
where the minimum is taken over all integers and that satisfy
[TABLE]
and
[TABLE]
Therefore, we may conclude that:
- •
If , then ;
- •
If , then ;
- •
If , then ;
- •
If , then .
So, we have four cases:
- •
Case 1 If and , then
[TABLE]
Since , we get
[TABLE]
- •
Case 2 If and , then
[TABLE]
Since , we get
[TABLE]
- •
Case 3 If and , then
[TABLE]
Since , we get
[TABLE]
- •
Case 4 If and , then
[TABLE]
Consequently,
[TABLE]
Lemma 10
Let and be nonnegative integers such that , , and . Then and .
Proof. Since , and , we have
[TABLE]
If then . Using the inequality we get
[TABLE]
Contradiction. So, . Consequently, and
[TABLE]
Therefore, .
Proof of Theorem 6. If , the result follows. Let . Suppose there is no matrix such that
[TABLE]
Let be the greatest integer such that and there is a matrix in such that
[TABLE]
Let a matrix in such that
[TABLE]
So, there is an integer , with , such that
[TABLE]
and
[TABLE]
Consequently,
[TABLE]
By Proposition 2, there are nonnegative integers , such that , , and , .
Then , , and . Using Lemma 10, we get and . By hypothesis, , , and .
Using Proposition 9, we have
[TABLE]
By Proposition 8, we conclude that there exists a matrix in such that all of are contained in the union of the first rows and first columns, and all of are contained in the union of the first row and first columns. Thus, by Proposition 1
[TABLE]
Contradiction. Therefore, there exists a matrix such that
6 Algorithm for construct matrices with fixed zero blocks
Let and be partitions of the same weight. Let and positive integers such that and . Denote by the set of all matrices of whose all 1’s are covered by rows and columns and, by rows and columns. Assume that is nonempty. In this section we present an algorithm for construct a matrix in . This algorithm generalizes the algorithm stated by Brualdi and Dahl in [4] for construct a matrix in (the subset of all matrices of whose all 1’s are covered by rows and columns).
First we describe the modified Ryser algorithm.
Let , .
Let be an integer vector with -coordinates. Let be a permutation of such that the vector verify . We denote by the -by- permutation matrix associated with , and by we denote its inverse.
The modified Ryser algorithm:
Start with an -by- -matrix whose row sum vector is and whose column sum vector is . Thus the 1’s occupy the initial positions in each row. Let be the submatrix obtained from deleting rows . Let be the -by-[math] empty matrix. 2. 2.
For , do:
Shift to column the final ’s in those rows of with the largest sum, with preference given to the lowest rows (those with the largest index) in case of ties. This results in a matrix
[TABLE]
where has columns. 3. 3.
Let be the -by- -matrix whose row sum vector is and whose column sum vector is . Thus the 1’s occupy the initial positions in each row. Let be the submatrix obtained from deleting rows . Let be the -by-[math] empty matrix. 4. 4.
For , do:
Shift to column the final ’s in those rows of with the largest sum, with preference given to the lowest rows (those with the largest index) in case of ties. This results in a matrix
[TABLE]
where has columns. 5. 5.
Let be the row-sum sequence of and be the row-sum sequence of . Let and . 6. 6.
Let be the canonical matrix of .
Output:
[TABLE]
where is the transpose of .
The matrix is the canonical column -submatrix relative to , and . The matrix is the canonical column -submatrix relative to , and .
Example 11
Let and . Then . It is possible to prove that there is a matrix whose all 1’s are covered by rows and columns.
Using last algorithm let
[TABLE]
and
[TABLE]
The following matrices \left[\begin{array}[]{c|c}B_{2,i}&\overline{B}_{9-i}\end{array}\right] are produced using step 2..
[TABLE]
[TABLE]
[TABLE]
In step 3.
[TABLE]
The following matrices \left[\begin{array}[]{c|c}C_{4,i}&\overline{C}_{7-i}\end{array}\right] are produced using step 4..
[TABLE]
[TABLE]
[TABLE]
In this case, in step 5.,
[TABLE]
with row sum sequence ,
[TABLE]
with row sum sequence .
So, and
A canonical matrix of is
[TABLE]
Therefore, the final matrix is
[TABLE]
Theorem 12
Let and be partitions of the same weight. Let and positive integers with , , and . Then there is a matrix
[TABLE]
in the class , where is the canonical column -submatrix relative to , , and , and is the canonical column -submatrix relative to , and , is its transpose, and there are permutation matrices and such that
[TABLE]
is the canonical matrix in the class where it belongs.
Proof. Let
[TABLE]
be a matrix in . From Ryser’s interchange theorem we may apply interchanges to the matrix
[TABLE]
to obtain the matrix
[TABLE]
Applying a similar argument to the matrix
[TABLE]
we obtain the matrix
[TABLE]
Consider the submatrix
[TABLE]
Let and be the row sum vector and the column sum vector of , respectively, and let be the canonical matrix of . Then, there are permutation matrices and such that has row sum vector and column sum vector . Now we replace by and we have the desired matrix.
We can now use Theorem 12 to give an algorithm to construct a matrix in :
Algorithm:
Let and positive integers such that and .
Construct , the canonical column -submatrix, relative to , and . 2. 2.
Construct , the canonical column -submatrix, relative to , and . 3. 3.
Let be the row-sum sequence of and be the row-sum sequence of . Let and . 4. 4.
Let be the canonical matrix of .
Example 13
Let and as in last example. It is possible to prove that there is a matrix whose all 1 s are covered by rows and columns and, by rows and columns.
In this case, , , and .
In last example we constructed the canonical column -submatrix, relative to , and ,
[TABLE]
This matrix has row sum sequence .
The canonical column -submatrix, relative to , and is
[TABLE]
This matrix has row sum sequence .
So, and
A canonical matrix of is
[TABLE]
Therefore, the final matrix is
[TABLE]
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1991.
- 2[2] R.A. Brualdi, Combinatorial Matrix Classes, Cambridge University Press, Cambridge, 2006.
- 3[3] R.A. Brualdi, K.P. Kiernan, S.A. Meyer and M.W. Schroeder, Combinatorial batch codes and transversal matroids, Adv. Math. Commun. , 4 (2010) 419-431 (Erratum 4 (2010) 597).
- 4[4] R.A. Brualdi and G. Dahl, Matrices of zeros and zeros and ones with given line sums and a zero block, Linear Algebra and its Applications , 371 (2003) 191-207.
- 5[5] R.A. Brualdi, Kathleen P. Kiernan, Seth A. Meyer and Michael W. Schroeder, On the t-term rank of a matrix, Linear Algebra and its Applications , 436 (2012) 1632-1643.
- 6[6] R.A. Brualdi, R. Fernandes and S. Furtado, On the Bruhat Order of Labeled Graphs, Discreta Applied Math. , 258 (2019) 49- 64.
- 7[7] H.F. da Cruz, R. Fernandes and S. Furtado, Minimal matrices in the Bruhat order for symmetric (0,1)-matrices, Linear Alg. Appl. , 530 (2017) 160- 184.
- 8[8] R. Fernandes and H.F. da Cruz, An extension of Brualdi ′ s algorithm for the construction of (0,1)-matrices with prescribed row and column sum vectors, Discrete Mathematics , 313 (2013) 2365-2379.