國立交通大學
應
應
應用
用
用數
數
數
學
學
學系
系
系
碩
碩
碩 士
士
士 論
論
論 文
文
文
Stein-Lov´
asz
定 理 的 推 廣 及 其 應 用
An Extension of Stein-Lov´
asz Theorem and
Some of its Applications
研 究 生
：李光祥
指導教授：黃大原 教授
An Extension of Stein-Lov´
asz Theorem and Some of
its Applications
研 究 生
：李光祥
Student : Guang-Siang Lee
指導教授：黃大原
Advisor : Tayuan Huang
國 立 交 通 大 學
應
用 數 學 系
碩 士 論 文
A Thesis
Submitted to Department of Applied Mathematics
College of Science
National Chiao Tung University
In Partial Fulfillment of the Requirements
For the Degree of
Master
In
Applied mathematics
June 2009
Hsinchu, Taiwan, Republic of China
誌
誌
誌
謝
謝
謝
這
篇論文的完成，首先要感謝指導教授黃大原教授，從去年暑
假開始就讓我進行論文的相關研究，也常常提供一些學業上規
劃的建議，每當在課業或研究上有疑問的時候，老師總是不厭
其煩的替我解惑，讓我得以順利的完成論文。
同時要感謝傅恆霖教授、陳秋媛教授和翁志文教授的關心與幫
助，使我在大學到研究所的這段時間學到了不少事情。
感謝葉
鴻國教授和傅東山教授擔任我的論文口試委員，葉鴻國
教授提供我一些未來可以繼續研究的方向，傅東山教授則是提
供我一些論文方面編寫的建議，真的是非常感謝。
還有
要感謝132的夥伴：簡文昱、陳哲皓、林奕伸、黃瑞毅、
李柏瑩、黃義閔以及陳鼎三，謝謝他們陪伴我度過碩班這兩
年
，以及對我的照顧。
最後要感謝我的父母，因為有他們的支持，讓我可以無憂無慮
的完成碩士學位。
Stein-Lov´
asz
定 理 的 推 廣 及 其 應 用
研究生
：李光祥
指導教授：黃大原
國 立 交 通 大 學
應
用 數 學 系
摘
摘
摘
要
要
要
Stein-Lov´
asz
定理提供一個演算法的方式來找出好的覆
蓋(covering)
，並且可以用來處理一些組合問題，找出它們的上
界。為了可以用來處理更多的組合問題，在這篇論文裡，我
們將原先的Stein-Lov´asz定理作推廣。此外，我們將利用推廣後
的Stein-Lov´asz定理來處理一些模型，這些模型包括：分離矩
陣(disjunct matrices)、選擇器(selectors)以及系統集(set systems)
，在固定行(column)數的前提之下，分別去找出這些矩陣的最
小
列(row)數的上界。其中分離矩陣和選擇器可以應用在匯集設
計(pooling design)
上。
An Extension of Stein-Lov´
asz Theorem and Some of
its Applications
Student : Guang-Siang Lee
Advisor : Tayuan Huang
Department of Applied Mathematics
Department of Applied Mathematics
National Chiao Tung University
National Chiao Tung University
Abstract
The Stein-Lov´asz theorem provides an algorithmic way to deal with the existence of good coverings and then to derive some upper bounds related to some combinatorial structures. In order to deal with more combinatorial problems, an extension of the classical Stein-Lov´asz theorem, called the extended Stein-Lov´asz theorem, will be given in this thesis. Moreover, we will also discuss applications of the extended Stein-Lov´asz theorem to various models stated as follows:
1. Several disjunct matrices (for group testing purpose) • d-disjunct matrices, (d; z]-disjunct matrices; • (d, r]-disjunct matrices, (d, r; z]-disjunct matrices; • (d, r)-disjunct matrices, (d, r; z)-disjunct matrices;
• (d, s out of r]-disjunct matrices, (d, s out of r; z]-disjunct matrices. 2. Several selectors (for group testing purpose)
• (k, m, n)-selectors, (k, m, n; z)-selectors; • (k, m, c, n)-selectors, (k, m, c, n; z)-selectors. 3. Some set systems (for others)
• uniform (m, t)-splitting systems, uniform (m, t; z)-splitting systems;
• uniform (m, t1, t2)-separating systems, uniform (m, t1, t2; z)-separating systems;
• (v, k, t)-covering designs, (v, k, t; z)-covering designs; • (v, k, t, p)-lotto designs, (v, k, t, p; z)-lotto designs.
Contents
摘要 . . . i
Abstract . . . iii
1 Introduction 1 2 Preliminaries 7 2.1 Several disjunct matrices . . . 7
2.2 Several selectors . . . 9
2.3 Some set systems . . . 11
2.4 Some basic counting results . . . 13
3 The Stein-Lov´asz Theorem and its extension 20 3.1 The Stein-Lov´asz Theorem . . . 20
3.2 Extension of The Stein-Lov´asz Theorem . . . 22
4 Some Applications of the extended Stein-Lov´asz Theorem 28 4.1 Bounds for several disjunct matrices . . . 28
4.2 Bounds for several selectors . . . 42
4.3 Bounds for some set systems . . . 51
5 Conclusion 59 Appendix . . . 60
Chapter 1
Introduction
Let X be a finite set and let Γ be a family of subsets of X. We denote by H = (X, Γ) the hypergraph having X as the set of vertices and Γ as the set of hyperedges. The degree of x ∈ X is the number of hyperedges containing x. Denoted by d(H) the maximum degree in the hypergraph H. For d(H) and other functions to be defined we remove the argument H if no confusion can arise.
A binary matrix M = (mij) of order |Γ| × |X| can be interpreted as a block-point
in-cidence matrix of the hypergraph H, i.e., the rows of M correspond to the hyperedge set {E1, E2, ..., E|Γ|}, and the columns correspond to the vertex set {x1, x2, ..., x|X|}, where
mij =
1 if the hyperedge Ei contains the vertex xj
0 otherwise.
The weight of a binary matrix M is the number of entries with a “1”.
A subset M ⊆ Γ (the same hyperedge may occur more than once) such that each vertex belongs to at most k of its members is called a k-matching of the hypergraph H. The maximum size over all k-matchings of the hypergraph H is denoted by νk(H). A k-matching
is simple if no hyperedge occurs in it more than once. Denoted byν∼k the maximum number
of hyperedges in simple k-matchings, thenν∼k≤ νk.
A subset T ⊆ X (in this thesis, the same vertex does not occur more than once) such that |TT E| ≥ k for any hyperedge E is called a k-cover of the hypergarph H. The minimum size over all k-covers of the hypergraph H is denoted by τk(H). Thus τ (H) = τ1(H) is the
minimum size of a vertex cover of the hypergraph H.
A vector (wE1, wE2, ..., wE|Γ|) with wEi ≥ 0 for each Ei ∈ Γ is called a fractional matching
of the hypergraph H if each entry of the vector (wE1, wE2, ..., wE|Γ|)M is at most 1. A vector
(wx1, wx2, ..., wx|X|) with wxi ≥ 0 for each xi ∈ X is called a fractional cover of the hypergraph
H if each entry of the vector M (wx1, wx2, ..., wx|X|)
t is at least 1. Define ν∗(H) = maxX Ei∈Γ wEi and τ ∗ (H) = min X xi∈X wxi,
where the extrema are taken over all fractional matchings (wE1, wE2, ..., wE|Γ|) and all
frac-tional covers (wx1, wx2, ..., wx|X|), respectively. By the duality theorem of linear programming,
we have ν∗ = τ∗. Then it is easy to see that ν ≤ νk/k ≤ ν∗ = τ∗ ≤ τk/k.
One of the most natural methods to produce a small vertex cover of a given hypergraph H is the so-called “Greedy Cover Algorithm”, which we describe as follows:
1. Let x1 be a vertex with maximum degree.
2. Suppose that x1, x2, ..., xi have been already selected,
if x1, x2, ..., xi cover all hyperedges, then we stop; otherwise,
let xi+1 be a vertex which covers the most number of uncovered hyperedges.
Generally, the greedy cover algorithm is not the best, but we can expect that it gives a rather good estimate. By the greedy cover algorithm, an upper bound for τ (H) was given by Lov´asz [10].
Theorem 1. [10] If H is a hypergraph and any greedy cover algorithm produces t covering vertices, then t ≤ ∼ ν1 1 × 2 + ∼ ν2 2 × 3 + · · · + ∼ νd−1 (d − 1) × d+ ∼ νd d .
Corollary 1. [10] For a hypergraph H, τ (H) ≤ (1 + 1 2 + · · · + 1 d)τ ∗ (H) < (1 + ln d)τ∗(H).
Hence we have the following theorem (for completeness, we also give a proof). Theorem 2. For a hypergraph H = (X, Γ),
τ (H) < |X| minE∈Γ|E|
(1 + ln 4), where 4 = maxx∈X|{E : E ∈ Γ with x ∈ E}|.
Proof. Let M be the block-point incidence matrix of H. Define wxi =
1 minE∈Γ|E|
for each xi ∈ X. Then each Ei-entry of the vector M (wx1, wx2, ..., wx|X|)
t is
|Ei|
minE∈Γ|E|
≥ 1, i.e., (wx1, wx2, ..., wx|X|) is a fractional cover of H. Hence
τ∗(H) ≤ X
xi∈X
wxi =
|X| minE∈Γ|E|
. By Corollary 1,
τ (H) < (1 + ln d)τ∗(H) ≤ |X| minE∈Γ|E|
(1 + ln 4), as required.
Similarly, by the greedy cover algorithm, an equivalent statement in terms of the point-block incidence matrices of the corresponding hypergraphs was given by Stein [11] indepen-dently.
Theorem 3. [11] Let X be a finite set of cardinality n, and let Γ = {A1, A2, ..., At} be a
family of subsets of X, where |Ai| ≤ a for all 1 ≤ i ≤ t. Assume that each element of X is
in at least q members of the set Γ. Then there is a subfamily of Γ that covers X and has at most n a + t q( 1 2 + 1 3 + · · · + 1 a) members.
Note that Theorem 3 is closely related to work of Fulkerson and Ryser [9] in the 1-width of a (0, 1)-matrix. They define the 1-width of such a matrix, A, as the minimum number of columns that can be selected from A in such a way that each row of the resulting submatrix has at least one 1. In this terminology, Theorem 3 can be restated as follows:
Theorem 4. [11] Let A be a (0, 1)-matrix with n rows and t columns. Assume that each row contains at least q 1’s and each column at most a 1’s. Then the 1-width of A is at most
n a + t q( 1 2+ 1 3+ · · · + 1 a).
Theorem 4 was called the Stein-Lov´asz Theorem in [6] while dealing with the covering problems in coding theory. The Stein-Lov´asz theorem was first used in dealing with the upper bounds for the sizes of (k, m, n)-selectors [2]. Inspired by this work, it was also used in dealing with the upper bounds for the sizes of (d, r; z]-disjunct matrices [5]. Some more applications can also be found in [8]. The notion of (k, m, n)-selectors was first introduced by De Bonis, Gasieniec and Vaccaro in [2], followed by a generalization to the notion of (k, m, c, n)-selectors [1]. A further generalization of (k, m, c, n)-selectors will be given in Chapter 2.
In this thesis, definitions of several properties over binary matrices are considered in Chapter 2 including several disjunct matrices, several selectors and some set systems stated as follows:
1. Several disjunct matrices
• d-disjunct matrices, (d; z]-disjunct matrices; • (d, r]-disjunct matrices, (d, r; z]-disjunct matrices; • (d, r)-disjunct matrices, (d, r; z)-disjunct matrices;
• (d, s out of r]-disjunct matrices, (d, s out of r; z]-disjunct matrices. 2. Several selectors
• (k, m, n)-selectors, (k, m, n; z)-selectors; • (k, m, c, n)-selectors, (k, m, c, n; z)-selectors. 3. Some set systems
• uniform (m, t)-splitting systems, uniform (m, t; z)-splitting systems;
• uniform (m, t1, t2)-separating systems, uniform (m, t1, t2; z)-separating systems;
• (v, k, t)-covering designs, (v, k, t; z)-covering designs; • (v, k, t, p)-lotto designs, (v, k, t, p; z)-lotto designs.
Note that the upper bounds of the sizes of several disjunct matrices and selectors are obtained for group testing purpose, and the upper bounds of the sizes of some set systems are obtained for others. Some formulas are given in Section 2.4 for later simplification purpose used in Chapter 4.
In order to deal with the upper bounds for these binary matrices defined in Chapter 2, an extended Stein-Lov´asz theorem is derived in Chapter 3. Some applications of the determination of some upper bounds of the sizes of various models are considered in Chapter 4. In Section 4.1 and Section 4.2, the extended Stein-Lov´asz theorem will be used in dealing the upper bounds for the sizes of several disjunct matrices and selectors, respectively. Those
upper bounds for the sizes of uniform splitting systems, uniform separating systems, covering designs and lotto designs are given in Section 4.3 respectively.
Chapter 2
Preliminaries
2.1
Several disjunct matrices
A few types of binary matrices, called disjunct matrices, will be introduced in this section, followed by corresponding associated parameters. These families of disjunct matrices will be used as models for pooling designs.
Definition 2.1.1. A binary matrix M of order t × n is called d-disjunct if the union of any d columns does not contain any other column of M , i.e., for any d+1 columns C1, C2, · · · , Cd+1,
|Cd+1\ d
[
i=1
Ci| ≥ 1. The integer t is called the size of the d-disjunct matrix. The minimum
size over all d-disjunct matrices with n columns is denoted by t(n, d).
Definition 2.1.2. A binary matrix M of order t × n is called (d; z]-disjunct if for any d + 1 columns C1, C2, · · · , Cd+1, |Cd+1\
d
[
i=1
Ci| ≥ z. The integer t is called the size of the (d;
z]-disjunct matrix. The minimum size over all (d; z]-z]-disjunct matrices with n columns is denoted by t(n, d; z].
Definition 2.1.3. A binary matrix M of order t × n is called (d, r]-disjunct if the union of any d columns does not contain the intersection of any other r columns of M , i.e., for any d + r columns C1, C2, · · · , Cd+r, | r \ i=1 Ci\ d+r [ i=r+1
Ci| ≥ 1. The integer t is called the size of the
(d, r]-disjunct matrix. The minimum size over all (d, r]-disjunct matrices with n columns is denoted by t(n, d, r].
Definition 2.1.4. A binary matrix M of order t × n is called (d, r; z]-disjunct if for any d + r columns C1, C2, · · · , Cd+r, | r \ i=1 Ci\ d+r [ i=r+1
Ci| ≥ z. The integer t is called the size of the
(d, r; z]-disjunct matrix. The minimum size over all (d, r; z]-disjunct matrices with n columns is denoted by t(n, d, r; z].
Definition 2.1.5. A binary matrix M of order t × n is called (d, r)-disjunct if the union of any d columns does not contain the union of any other r columns of M , i.e., for any d + r columns C1, C2, · · · , Cd+r, | r [ i=1 Ci\ d+r [ i=r+1
Ci| ≥ 1. The integer t is called the size of the
(d, r)-disjunct matrix. The minimum size over all (d, r)-disjunct matrices with n columns is denoted by t(n, d, r).
Definition 2.1.6. A binary matrix M of order t × n is called (d, r; z)-disjunct if for any d + r columns C1, C2, · · · , Cd+r, | r [ i=1 Ci \ d+r [ i=r+1
Ci| ≥ z. The integer t is called the size of
the (d, r; z)-disjunct matrix. The minimum size over all (d, r; z)-disjunct matrices with n columns is denoted by t(n, d, r; z).
Definition 2.1.7. A binary matrix M of order t×n is called (d, s out of r]-disjunct, 1 ≤ s ≤ r, if for any d columns and any other r columns of M , there exists a row index in which none of the d columns appear and at least s of the r columns do. The integer t is called the size of the (d, s out of r]-disjunct matrix. The minimum size over all (d, s out of r]-disjunct matrices with n columns is denoted by t(n, d, r, s].
Definition 2.1.8. A binary matrix M of order t × n is called (d, s out of r; z]-disjunct, 1 ≤ s ≤ r, if for any d columns and any other r columns of M , there exist z row indices in which none of the d columns appear and at least s of the r columns do. The integer t is called the size of the (d, s out of r; z]-disjunct matrix. The minimum size over all (d, s out of r; z]-disjunct matrices with n columns is denoted by t(n, d, r, s; z].
Some subclasses of (d, s out of r; z]-disjunct matrices are listed in the following table. parameters types bounds references
s = r = 1, z = 1 d-disjunct t(d, n) [13] s = r = 1 (d; z]-disjunct s = r, z = 1 (d, r]-disjunct t(n, d, r] [14] s = r (d, r; z]-disjunct t(n, d, r; z] [5] s = 1, z = 1 (d, r)-disjunct t(n, d, r) [14] s = 1 (d, r; z)-disjunct t(n, d, r; z) z = 1 (d, s out of r]-disjunct t(n, d, r, s] [14] (d, s out of r; z]-disjunct t(n, d, r, s; z]
2.2
Several selectors
A few types of binary matrices, called selectors, will be introduced in this section, followed by corresponding associated parameters. These families of selectors will be used as models for pooling designs.
Definition 2.2.1. For integers k, m and n with 1 ≤ m ≤ k ≤ n, a binary matrix M of order t × n is called a (k, m, n)-selector if any t × k submatrix of M contains a submatrix with each row weight exactly one, with at least m distinct rows. The integer t is called the size of the (k, m, n)-selector. The minimum size over all (k, m, n)-selectors is denoted by ts(k, m, n).
Definition 2.2.2. For integers k, m and n with 1 ≤ m ≤ k ≤ n, a binary matrix M of order t × n is called a (k, m, n; z)-selector if any t × k submatrix of M contains z disjoint submatrices with each row weight exactly one, with at least m distinct rows each. The integer t is called the size of the (k, m, n; z)-selector. The minimum size over all (k, m, n; z)-selectors is denoted by ts(k, m, n; z).
Definition 2.2.3. For integers k, m, c and n with 1 ≤ c ≤ k ≤ n and 1 ≤ m ≤ kc, a t × n binary matrix M is called a (k, m, c, n)-selector if any t × k submatrix of M contains a submatrix with each row weight exactly c, with at least m distinct rows. The integer t is
called the size of the (k, m, c, n)-selector. The minimum size over all (k, m, c, n)-selectors is denoted by ts(k, m, c, n).
Definition 2.2.4. For integers k, m, c and n with 1 ≤ c ≤ k ≤ n and 1 ≤ m ≤ kc, a t × n binary matrix M is called a (k, m, c, n; z)-selector if any t × k submatrix of M contains z disjoint submatrices with each row weight exactly c, with at least m distinct rows each. The integer t is called the size of the (k, m, c, n; z)-selector. The minimum size over all (k, m, c, n; z)-selectors is denoted by ts(k, m, c, n; z).
It is interesting to remark that the notion of (k, m, n)-selectors was first introduced by De Bonis, Gasieniec and Vaccaro [2], and it was then generalized to the notion of (k, m, c, n)-selectors [1], which are equivalent to (k, m, 1, n; 1)-n)-selectors and (k, m, c, n; 1)-n)-selectors rsepec-tively. The upper bounds for the sizes of (k, m, n)-selectors and (k, m, c, n)-selectors were studied in [2] and in [1] respectively by the Stein-Lov´asz theorem. The bounds for the sizes of (k, m, c, n; z)-selectors will be derived by the extended Stein-Lov´asz theorem (Theorem 3.2.1) in Chapter 4 (Theorem 4.1.12).
Some subclasses of (k, m, c, n; z)-selectors are listed in the following table. parameters types bounds references c = 1, z = 1 (k, m, n)-selectors ts(k, m, n) [2, 14]
c = 1 (k, m, n; z)-selectors ts(k, m, n; z)
z = 1 (k, m, c, n)-selectors ts(k, m, c, n) [1]
(k, m, c, n; z)-selectors ts(k, m, c, n; z)
The relationship between various models (disjunct matrices, selectors) and nonadaptive group testing are listed below.
1. A (d, r]-disjunct matrix can be used to identify the up-to-d positives on the complex model [4].
2. The property of (h, d)-disjunctness is a necessary condition for identifying the positive set on the (d, h)-inhibitor model [3].
3. There exists a two-state group testing algorithm for finding up-to-d positives out of n items and that uses a number of tests equal to t + k − 1, where t is the size of a (k, d + 1, n)-selector [2].
2.3
Some set systems
Most of the combinatorial structures can be viewed as set systems. We present some relevant definitions. A set system is a pair (X, Γ), where X is a set of points and Γ is a set of subsets of X, called blocks. A set sysyem (X, Γ) is called k-uniform if |B| = k for each B ∈ Γ.
Definition 2.3.1. Let m and t be even integers with 2 ≤ t ≤ m. An uniform (m, t)-splitting system is a pair (X, Γ) where |X| = m, Γ is a family of m2–subsets of X, called blocks such that for every T ⊆ X with |T | = t, there exists a block B ∈ Γ such that |TT B| = t
2, i.e.,
B splits T . The system (X, Γ) is also called a t-splitting system. The minimum number of blocks over all t-splitting systems is denoted by SP (m, t).
Definition 2.3.2. Let m and t be even integers with 2 ≤ t ≤ m, and let z be a positive integer. An uniform (m, t; z)-splitting system is a pair (X, Γ) where |X| = m, Γ is a family of m2–subsets of X, called blocks such that for every T ⊆ X with |T | = t, there exist z blocks B ∈ Γ such that |TT B| = t
2, i.e., B splits T . The system (X, Γ) is also called a (t;
z)-splitting system. The minimum number of blocks over all (t; z)-z)-splitting systems is denoted by SP (m, t; z).
Definition 2.3.3. Let m be an even integer, and let t1, t2be positive integers with t1+t2 ≤ m.
An uniform (m, t1, t2)-separating system is a pair (X, Γ) where |X| = m, Γ is a family of m
2–subsets of X, called blocks such that for every T1, T2 ⊆ X , where |Ti| = ti for i = 1, 2
and |T1T T2| = ∅, there exists a block B ∈ Γ for which either T1 ⊆ B, T2T B = ∅ or
(t1, t2)-separating system. The minimum number of blocks over all (t1, t2)-separating systems
is denoted by SE(m, t1, t2).
Definition 2.3.4. Let m be an even integer, and let t1, t2, z be positive integers with t1+t2 ≤
m. An uniform (m, t1, t2; z)-separating system is a pair (X, Γ) where |X| = m, Γ is a family
of m2–subsets of X, called blocks such that for every T1, T2 ⊆ X , where |Ti| = ti for i = 1, 2
and |T1T T2| = ∅, there exist z blocks B ∈ Γ for which either T1 ⊆ B, T2T B = ∅ or T2 ⊆ B,
T1T B = ∅, i.e., T1, T2 are separated by B. The system (X, Γ) is also called a (t1, t2;
z)-separating system. The minimum number of blocks over all (t1, t2; z)-separating systems is
denoted by SE(m, t1, t2; z).
Definition 2.3.5. Let v, k, and t be positive integers with t ≤ k ≤ v. A (v, k, t)-covering design is a pair (X, Γ) where |X| = v, Γ is a family of k-subsets of X, called blocks such that for every T ⊆ X with |T | = t, there exists a block B ∈ Γ containing T . The minimum number of blocks over all (v, k, t)-covering designs is denoted by C(v, k, t).
Definition 2.3.6. Let v, k, t and z be positive integers with t ≤ k ≤ v. A (v, k, t; z)-covering design is a pair (X, Γ) where |X| = v, Γ is a family of k-subsets of X, called blocks such that for every T ⊆ X with |T | = t, there exist z blocks B ∈ Γ containing T . The minimum number of blocks over all (v, k, t; z)-covering designs is denoted by C(v, k, t; z).
Definition 2.3.7. Let v, k, t, and p be positive integers with p ≤ t, k ≤ v. A (v, k, t, p)-lotto design is a pair (X, Γ) where |X| = v, Γ is a family of k-subsets of X, called blocks such that for every T ⊆ X with |T | = t, there exists a block B ∈ Γ such that |TT B| ≥ p. The minimum number of blocks over all (v, k, t, p)-lotto designs is denoted by L(v, k, t, p).
Definition 2.3.8. Let v, k, t, p and z be positive integers with p ≤ t, k ≤ v. A (v, k, t, p; z)-lotto design is a pair (X, Γ) where |X| = v, Γ is a family of k-subsets of X, called blocks such that for every T ⊆ X with |T | = t, there exist z blocks B ∈ Γ such that |TT B| ≥ p. The minimum number of blocks over all (v, k, t, p; z)-lotto designs is denoted by L(v, k, t, p; z).
Note that when p = t, a (v, k, t, t; z)-lotto design will be reduced to a (v, k, t; z)-covering design. The related bounds are summarized in the following table.
types bounds references uniform (m, t)-splitting systems SP (m, t) [8]
uniform (m, t; z)-splitting systems SP (m, t; z) uniform (m, t1, t2)-separating systems SE(m, t1, t2)
uniform (m, t1, t2; z)-separating systems SE(m, t1, t2; z)
(v, k, t)-covering designs C(v, k, t) [8] (v, k, t; z)-covering designs C(v, k, t; z)
(v, k, t, p)-lotto designs L(v, k, t, p) [8] (v, k, t, p; z)-lotto designs L(v, k, t, p; z)
2.4
Some basic counting results
Stein-Lov´asz theorem and its extension will be used to estimate the upper bounds of the sizes for pooling designs of various models. In order to give upper bounds for the above mentioned parameters, the following results involving binomial coefficients will be involved. Lemma 2.4.3 will be used in showing appropriate values of w for pooling designs of various models. We need information regarding the maximum of the function
f (w) =n − w d n − w − d r − s w s
with various r and s when dealing with possible upper bounds for t of various models. Lemmas 2.4.4 ∼ 2.4.6 will be used in the simplifications of the bounds Mv , and ln a respectively in the expression Mv (1 + ln a) found in the Stein-Lov´asz theorem (Theorem 3.1.1).
Lemma 2.4.1. The function
f (x) = 1 + ln x x is strictly decreasing on (1, ∞).
Proof. f0(x) = 1 x · x − (1 + ln x) · 1 x2 = − ln x x2 < 0
for all x ∈ (1, ∞), as required. Lemma 2.4.2. a b ≤ a b b! ≤ ( ea b ) b Proof. Since ex = X n≥0 xn n!, we have e x ≥ x b
b! for each x, thus e
b ≥ b b b! and hence 1 ≤ b!eb bb . Therefore, a b = a! (a − b)!b! = a(a − 1) · · · (a − b + 1) b! ≤ ab b! ≤ ab b! · b!eb bb = ( ea b ) b, as required.
Lemma 2.4.3. For any positive integers n, d, r, s with k = d + r ≤ n and 1 ≤ s ≤ r, the function f (w) =n − w d n − w − d r − s w s gets its maximum at
w = ns − (k − s) k . Proof. First we note that
f (w) =n − w d n − w − d r − s w s = (n − w)! (n − w − d)!d! · (n − w − d)! (n − w − d − r + s)!(r − s)! · w s · (d + r − s)! (d + r − s)! = n − w d + r − s w s d + r − s d =n − w k − s w s k − s d .
Since f (w + 1) =n − (w + 1) k − s w + 1 s k − s d = ((n − w) − (k − s) n − w n − w k − s ) · ( w + 1 w + 1 − s w s ) ·k − s d = ((n − w) − (k − s) n − w · w + 1 w + 1 − s)f (w), and (n − w) − (k − s) n − w · w + 1 w + 1 − s = (w + 1)(n − w) − (w + 1)(k − s) (w + 1)(n − w) − s(n − w) = 1 if and only if s(n − w) = (w + 1)(k − s), i.e., w = ns − (k − s)
k ; hence (n − w) − (k − s) n − w · w + 1 w + 1 − s ≥ 1 for w ≤ ns − (k − s) k and (n − w) − (k − s) n − w · w + 1 w + 1 − s ≤ 1 for w ≥ ns − (k − s) k . As a consequence, we then have
f (w) is increasing for w ≤ ns − (k − s) k , and f (w) is decreasing for w ≥ ns − (k − s) k as required. By taking s = r = c in f (w) =n − w d n − w − d r − s w s
, we get the quadratic function g(w) =w
c
n − w k − c
(for selectors). Hence we have the following corollary. Corollary 2.4.1. The function
g(w) =w c
n − w k − c
gets its maximum at
w = nc − (k − c) k .
Lemma 2.4.4. For any positive integers n, d, r, s with k = d + r ≤ n and 1 ≤ s ≤ r, n(n − 1) · · · (n − s + 1)(n − s)(n − s − 1) · · · (n − r − d + 1)
n(n −ks) · · · (n − k + ks) · n · (n − k−sk ) · · · (n − k + k−sk ) ≤ 1. Proof. Without loss of generality, let s ≤ k − s and thus 1 ≤ k−sk ≤ 2 ≤ k
s. Moreover, we
note that the left hand side is
n(n − 1) · · · (n − s + 1)(n − s)(n − s − 1) · · · (n − r − d + 1) n(n − ks) · · · (n − k + ks) · n · (n − k−sk ) · · · (n − k + k−sk ) = Y 0≤i≤r+d−1 (n − i) Y 0≤i≤s−1 (n − i · k s) Y 0≤j≤k−s−1 (n − j · k k − s) .
To prove this inequality, we will rearrange the terms in the denominator so that n − t n − f (t) ≤ 1 for each t with 0 ≤ t ≤ r + d − 1, i.e., we will give a bijection
f : {0, 1, ..., r + d − 1} → {i · k
s | 0 ≤ i ≤ s − 1} [
{j · k
k − s | 0 ≤ j ≤ k − s − 1} with the property that f (t) ≤ t for each t. Note that the element 0 will be counted twice as 0·ks and 0·k−sk respectively in the range of the function f . Note also that if i·ks = i+i·k−ss > t, then k−ss < t−ii and hence j ·k−sk = j(1+k−ss ) < j(1+t−ii ) = j(t−it ) = j(jt) = t, where i+j = t.
A such function f is defined recursively as follows. For t = 0, 1, 2, let f (0) = 0k
s, f (1) =
0 k
k−s, f (2) =
k
k−s. For 3 ≤ t ≤ r + d − 1, let i (resp. j) be the smallest positive integers such
that i ·ks (resp. j · k−sk ) /∈ {f (0), f (1), ..., f (t − 1)} if they exist, it follows that t = i + j. 1. Let f (t) = i · k
s if i · k s ≤ t.
2. Otherwise, we define f (t) = j · k−sk .
3. Finally, suppose t is large and there is no such i, note that n − s n ≤ n − s − 1 n − k−sk ≤ · · · ≤ n − r − d + 1 n − k + k−sk and n − r − d + 1 n − k + k−sk < 1, we have n − t
n − (t − s)k−sk ≤ 1 for all s ≤ t ≤ r + d − 1, we define f (t) = (t − s)
k k−s.
Cearly, the function f defined above is 1-1, onto, and f (t) ≤ t for all 0 ≤ t ≤ r + d − 1 as required.
Lemma 2.4.5. For any positive integers n, d, r, s with k = d + r ≤ n and 1 ≤ s ≤ r, let n0 ≥ n be the smallest positive integer such that w = n
0s k is an integer, then n0 d n0− d r n0 − w d n0 − w − d r − s w s ≤ (k s) s( k k − s) k−s r s . Proof. n0 d n0− d r n0− w d n0− w − d r − s w s = (n0)! (n0− d)!d! · (n0− d)! (n0− d − r)!r! (n0− w)! (n0 − w − d)!d!· (n0− w − d)! (n0− w − d − r + s)!(r − s)! · w! (w − s)!s! = 1 r!n 0(n0− 1) · · · (n0− s + 1)(n0− s)(n0− s − 1) · · · (n0− d − r + 1) 1 (r−s)!s!w(w − 1) · · · (w − s + 1)(n 0− w)(n0− w − 1) · · · (n0− w − d − r + s + 1) = 1r s · n0(n0− 1) · · · (n0− s + 1)(n0 − s)(n0− s − 1) · · · (n0− d − r + 1) n0s k ( n0s k − 1) · · · ( n0s k − s + 1)(n 0− n0s k )(n 0− n0s k − 1) · · · (n 0− n0s k − d − r + s + 1) = 1r s · 1 (ks)s(k−s k )k−s · n 0(n0− 1) · · · (n0− s + 1)(n0− s)(n0 − s − 1) · · · (n0− d − r + 1) n0(n0− k s) · · · (n 0− k + k s) · n 0· (n0 − k k−s) · · · (n 0− k + k k−s) ≤ (k s) s( k k − s) k−s r s (by Lemma 2.4.4).
Lemma 2.4.6. For any positive integers n, d, r, s with k = d + r ≤ n and 1 ≤ s ≤ r, let n0 ≥ n be the smallest positive integer such that w = n
0s k is an integer, then ln(n 0− w d n0− w − d r − s w s ) < k[1 + ln(n k + 1)] + ln k − s d .
Proof. First we note that n0 < n + k for such n0. Since a b ≤ (ea b ) b, we have n0− w d n0− w − d r − s w s = (n 0− w)! (n0− w − d)!d! · (n0 − w − d)! (n0− w − d − r + s)!(r − s)! · w s · (d + r − s)! (d + r − s)! = n0 − w d + r − s w s d + r − s d =n 0− w k − s w s k − s d ≤(e(n 0− n0s k ) k − s ) k−s· (e( n0s k ) s ) s·k − s d =(e · n 0 k) k−s· (en0 k) s·k − s d =ek(n 0 k) k·k − s d <ek(n + k k ) k·k − s d =ek(n k + 1) k·k − s d . Therefore, ln(n 0− w d n0− w − d r − s w s ) < ln(ek(n k + 1) k·k − s d ) = k[1 + ln(n k + 1)] + ln k − s d .
The substitutions of w for various subclasses are summarized in the following table: types parameters d-disjunct s = r = 1, z = 1 w = n − d k w = n0 k (d; z]-disjunct s = r = 1 w = n − d k w = n0 k (d, r]-disjunct s = r, z = 1 w = nr − d k w = n0r k (d, r; z]-disjunct s = r w = nr − d k w = n0r k (d, r)-disjunct s = 1, z = 1 w = n − (k − 1) k w = n0 k (d, r; z)-disjunct s = 1 w = n − (k − 1) k w = n0 k (d, s out of r]-disjunct z = 1 w = ns − (k − s) k w = n0s k (d, s out of r; z]-disjunct w = ns − (k − s) k w = n0s k types parameters (k, m, n)-selectors c = 1, z = 1 w = n − (k − 1) k w = n0 k (k, m, n; z)-selectors c = 1 w = n − (k − 1) k w = n0 k (k, m, c, n)-selectors z = 1 w = nc − (k − c) k w = n0c k (k, m, c, n; z)-selectors w = nc − (k − c) k w = n0c k
Chapter 3
The Stein-Lov´
asz Theorem and its
extension
3.1
The Stein-Lov´
asz Theorem
We now introduce the Stein-Lov´asz theorem as follows. The Stein-Lov´asz theorem was first used by Stein [11] and Lov´asz [10] in studying some combinatorial covering problems. In [6], the authors applied this theorem to some problems in coding theory. The Stein-Lov´asz theorem is now stated and the proof is included for completeness [8], with a minor modification.
Theorem 3.1.1. [8] Let A be a (0,1) matrix with N rows and M columns. Assume that each row contains at least v ones, and each column at most a ones. Then there exists an N × K submatrix C with K < (N a) + ( M v ) ln a ≤ ( M v )(1 + ln a), such that C does not contain an all-zero row.
Proof. A constructive approach for producing C is presented. Let Aa = A. We begin by
picking the maximal number Kaof columns from Aa, whose supports are pairwise disjoint and
each column having a ones (perhaps, Ka= 0). Discarding these columns and all rows incident
the columns of Aa−1have at most a−1 ones (indeed, otherwise such a column could be added
to the previously discarded set, contradicting its maximality). Now we remove from Aa−1 a
maximal number Ka−1of columns having a−1 ones and whose supports are pairwise disjoint,
thus getting a ka−1× (M − Ka− Ka−1) matrix Aa−2, where ka−1 = N − aKa− (a − 1)Ka−1.
The process will terminate after at most a steps. The union of the columns of the discarded sets form the desired submatrix C with
K =
a
X
i=1
Ki.
The first step of the algorithm gives
ka = N − aKa,
which we rewrite, setting ka+1 = N , as
Ka = ka+1− ka a . Analogously, Ki = ki+1− ki i , i = 1, ..., a.
Now we derive an upper bound for ki by counting the number of ones in Ai−1 in two ways:
every row of Ai−1 contains at least v ones, and every column at most i − 1 ones, thus
vki ≤ (i − 1)(M − Ka− · · · − Ki) ≤ (i − 1)M. Furthermore, K = a X i=1 Ki = a X i=1 ki+1− ki i = ka+1 a + ka a(a − 1)+ ka−1 (a − 1)(a − 2)+ · · · + k2 2 × 1 − k1 ≤ (N/a) + (M/v)(1/a + 1/(a − 1) + · · · + 1/2),
The greedy procedure as shown in the proof constructs the desired submatrix one column at a time, and hence the algorithm below follows [8].
Algorithm: STEIN-LOV ´ASZ(A)
input: A is an N × M matrix, each column has at most a ones, each row has at least v ones C ← ∅
while A has at least one row
do
find a column c in A having maximum weight delete all rows of A that contain a “1” in column c delete column c from A
output: Returns a submatrix of A with no all-zero row
At each state, a new column is added to the submatrix that maximizes the number of “new” rows that are yet uncovered. When all rows are covered, the algorithm stops. It seems quite interesting that we can use the Stein-Lov´asz theorem to derive bounds for some combinatorial array [8].
3.2
Extension of The Stein-Lov´
asz Theorem
The Stein-Lov´asz theorem can be further extended from rows of the resulting submatrix with weight at least 1 to the case of rows of the resulting submatrix with weight at least z ≥ 1. The bound can be further improved when A is a matrix with constant row weight and column weight as well, i.e., in the language of hypergraphs, uniform and regular.
Theorem 3.2.1. Let A be a (0,1) matrix of order N × M , and let v, a, z be positive integers. Assume that each row contains at least v ones, and each column at most a ones. Then there exists a submatrix C of order N × K with
K < v
v − (z − 1) z( M
v )(1 + ln a), such that each row of C has weight at least z.
More specifically, if the matrix is v-uniform and a-regular, the upper bound can then be reduced to
K < z(M
v )(1 + ln a). The strategy for the proof of Theorem 3.2.1 is as follows:
1. Use the Stein-Lov´asz theorem to obtain a submatrix C1 with each row has weight at
least 1.
2. Choose some columns in the matrix A\C1 to combine with the submatrix C1 to form
a submatrix C2 with each row has weight at least 2.
3. Choose some columns in the matrix A\C2 to combine with the submatrix C2 to form
a submatrix C3 with each row has weight at least 3.
4. Step by step, and finally we obtain the desired submatrix C = Cz with each row has
weight at least z.
Note that this upper bound makes sense only if v v − (z − 1) z( M v )(1 + ln a) < M, i.e., z < v + 1 2 + ln a
in general, or if z(M v )(1 + ln a) < M, i.e., z < v 1 + ln a for the case of uniform and regular.
Proof. A constructive approach for producing C is presented. Let A1 = A. By the
Stein-Lov´asz theorem, there exists an N × M1 submatrix C1(= B
0
1 = B1) of A1 with M1 < M
v (1 + ln a) such that each row of C1 has weight at least 1.
The algorithm used in the proof of the Stein-Lov´asz theorem shows that some rows of C1
have weight exactly 1. Let R1 be the set of indices of those rows and |R1| = r1. Let A2 be the
submatrix of order r1× (M − M1) obtained from A1 by deleting the submatrix C1 and the
i-th row, i /∈ R1 as well. Then each row of A2 contains at least v − 1 ones, and each column
at most a ones. Again, by the Stein-Lov´asz theorem, there exists an r1× M2 submatrix B
0
2
with M2 < M −Mv−11(1 + ln a) such that each row of B
0
2 has weight at least 1. Let B2 be the
matrix of order N × M2 obtained from B
0
2 by adding the i-th row, i /∈ R1. Let C2 be the
matrix of order N × (M1+ M2) obtained by the union of B1 and B2. Then C2 is a submatrix
of A with each row weight at least 2.
Similarly, there exist some rows of C2 that have weight exactly 2. Let R2 be the set of
indices of those rows and |R2| = r2. Continue in this way, we have:
For 2 ≤ i ≤ z,
1. Ai is a matrix of order ri−1× (M − i−1
X
j=1
Mj), and each row contains at least v − (i − 1)
ones, and each column at most a ones.
2. Bi0 is an ri−1× Mi submatrix of Ai with Mi <
M −
i−1
X
j=1
Mj
weight at least 1. For 1 ≤ i ≤ z, 3. Bi is a matrix of order N × Mi. 4. Ci is an N × i X j=1
Mj submatrix of A, and each row has weight at least i.
Hence, C = Cz is the submatrix required, that is,
K = z X j=1 Mj = M1+ M2+ · · · + Mz < M v (1 + ln a) + M − M1 v − 1 (1 + ln a) + · · · + M − z−1 X j=1 Mj v − (z − 1) (1 + ln a) < M v (1 + ln a) + M v − 1(1 + ln a) + · · · + M v − (z − 1)(1 + ln a) = M (1 + ln a)(1 v + 1 v − 1+ · · · + 1 v − (z − 1)) < M (1 + ln a)( 1 v − (z − 1) + 1 v − (z − 1) + · · · + 1 v − (z − 1)) = z v − (z − 1) M (1 + ln a) = v v − (z − 1) z( M v )(1 + ln a), thus gives the result.
More specifically, for the case of uniform and regular, using similar argument as above with a minor modification. First we note that N v = M a by counting the weight of A in two ways. For 2 ≤ i ≤ z , Ai is a matrix of order ri−1× (M −
i−1
X
j=1
Mj), and each row contains
exactly v − (i − 1) ones, and each column at most a ones. Moreover, a lower bound for
i−1
X
j=1
Mj is derived by counting the weight of the submatrix Ci−1in two ways; each row of Ci−1
contains at least i − 1 ones, and each column exactly a ones, thus N (i − 1) ≤ (
i−1 X j=1 Mj)a, and hence M v (i − 1) ≤ i−1 X j=1 Mj for 2 ≤ i ≤ z. Furthermore,
K = z X j=1 Mj = M1+ M2+ · · · + Mz < M v (1 + ln a) + M − M1 v − 1 (1 + ln a) + · · · + M − z−1 X j=1 Mj v − (z − 1) (1 + ln a) ≤ M v (1 + ln a) + M − Mv v − 1 (1 + ln a) + · · · + M − (z − 1) · M v v − (z − 1) (1 + ln a) = M v (1 + ln a) + M v (1 + ln a) + · · · + M v (1 + ln a) = z(M v )(1 + ln a), thus gives the result.
Remark 3.2.1. Since this upper bound makes sense only if z < v + 1 2 + ln a in general, z < v + 1 2 + ln a < v + 1 2 < v + 2 2 = v 2 + 1. Then z − 1 < v 2 and thus v v − (z − 1) = 1 + z − 1 v − (z − 1) < 1 + 1 = 2. Hence K < v v − (z − 1)z M v (1 + ln a) < 2z M v (1 + ln a) for general case.
Similarly, Theorem 3.2.1 can be restated in the language of hypergraphs in the following corollary. Recall that a subset T ⊆ X such that |TT E| ≥ z for any hyperedge E is called a z-cover of the hypergarph H, and the minimum size of a z-cover of the hypergraph H is denoted by τz(H).
Corollary 3.2.1. For a hypergraph H = (X, Γ) and a positive integer z ≥ 2, τz(H) <
2z|X| minE∈Γ|E|
(1 + ln 4), where 4 = maxx∈X|{E : E ∈ Γ with x ∈ E}|.
More specifically, for the case of uniform and regular, we have the following corollary. Corollary 3.2.2. Let H = (X, Γ) be a v-uniform and a-regular hypergraph with vertex set X and edge set Γ, then
τz(H) < z
|X|
v (1 + ln a).
We conjecture that τz(H) ≤ zτ1(H) holds for hypergraphs which are uniform and regular.
However, it is not true in general as shown in the following example. For the hypergraph H with X = {1, 2, 3, ..., 8} and Γ = {{1, 2, 3}, {4, 5, 6}, {1, 7, 8}}. It is easy to see that {1, 4} is a 1-cover with minimum size, hence τ1(H) = 2. Similarly, {1, 2, 4, 5, 7} is a 2-cover with
Chapter 4
Some Applications of the extended
Stein-Lov´
asz Theorem
The Stein-Lov´asz theorem was first used in dealing with the upper bounds for the sizes in the model of (k, m, n)-selecters [2]. Inspired by this work, it was also used in dealing with the upper bounds for the sizes of (d, r; z]-disjunct matrices [5]. In Section 4.1 and Section 4.2, the extended Stein-Lov´asz theorem will be used in dealing the upper bounds for the sizes of several disjunct matrices (Theorem 4.1.1∼ 4.1.8) and selectors (Theorem 4.2.1 ∼ 4.2.4), respectively. Those upper bounds for the sizes of uniform splitting systems, uniform separating systems, covering designs and lotto designs are given in Section 4.3 (Theorem 4.3.1 ∼ 4.3.8) respectively.
4.1
Bounds for several disjunct matrices
Note that upper bounds for the sizes of d-disjunct matrices, (d, r]-disjunct matrices, (d, r)-disjunct matrices and (d, s out of r]-r)-disjunct matrices were given in [13, 14] by the Lov´asz Local Lemma.
Recall that t(n, d) is the minimum size over all d-disjunct matrices with n columns. Theorem 4.1.1. For any positive integers n and d, if k = d + 1 ≤ n, then
t(n, d) < k(k d)
d{1 + k[1 + ln(n
Proof. For 1 ≤ w ≤ n − d, let A be the binary matrix of order [n d n − d 1 ] × n w with rows and columns indexed by {(D, s) | D ∈ [n]
d
, s ∈ [n] \ D} and V = {v | v ∈ {0, 1}n, wt(v) = w} respectively. The entry of A at the row indexed by the pair (D, s) and the column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and the entry of v at s is one; and 0 otherwise.
Observe that each row of A has weight n − (d + 1) w − 1
, and each column of A has weight n − w
d
w 1
. By the Stein-Lov´asz theorem, there exists a submatrix M of A of order [n
d
n − d 1
] × t having no zero rows, where
t < n w n − (d + 1) w − 1 {1 + ln[ n − w d w 1 ]} = n d n − d 1 n − w d w 1 {1 + ln[ n − w d w 1 ]}. Note that the equality is obtained by counting the weight of A in two ways. It is straight-forward to show that the columns of M form a d-disjunct matrix of order t × n. We then have t(n, d) < n d n − d 1 n − w d w 1 {1 + ln[ n − w d w 1 ]}. Let n0 ≥ n be the smallest positive integer such that w = n
0 k is an integer. We have n0 d n0 − d 1 n0− w d w 1 ≤ k( k d) d
by Lemma 2.4.5 (taking s = r = 1), and ln(n 0 − w d w 1 ) < k[1 + ln(n k + 1)]
by Lemma 2.4.6 (taking s = r = 1). Therefore, t(n, d) ≤ t(n0, d) < n0 d n0− d 1 n0 − w d w 1 {1 + ln[ n0− w d w 1 ]} < k(k d) d{1 + k[1 + ln(n k + 1)]} as required.
Recall that t(n, d; z] is the minimum size over all (d; z]-disjunct matrices with n columns. Theorem 4.1.2. For any positive integers n, d and z, if k = d + 1 ≤ n, then
t(n, d; z] < zk(k d)
d{1 + k[1 + ln(n
k + 1)]}. Proof. For 1 ≤ w ≤ n − d, let A be the binary matrix of order [n
d n − d 1 ] × n w with rows and columns indexed by {(D, s) | D ∈ [n]
d
, s ∈ [n] \ D} and V = {v | v ∈ {0, 1}n, wt(v) = w} respectively. The entry of A at the row indexed by the pair (D, s) and the column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and the entry of v at s is one; and 0 otherwise.
Observe that each row of A has weight n − (d + 1) w − 1
, and each column of A has weight n − w
d
w 1
. By the extended Stein-Lov´asz theorem, there exists a submatrix M of A of order [n
d
n − d 1
] × t with each row weight at least z, where
t < z n w n − (d + 1) w − 1 {1 + ln[ n − w d w 1 ]} = zn d n − d 1 n − w d w 1 {1 + ln[ n − w d w 1 ]}. Note that the equality is obtained by counting the weight of A in two ways. It is straightfor-ward to show that the columns of M form a (d; z]-disjunct matrix of order t × n. We then
have t(n, d; z] < zn d n − d 1 n − w d w 1 {1 + ln[ n − w d w 1 ]}. Let n0 ≥ n be the smallest positive integer such that w = n
0 k is an integer. We have n0 d n0 − d 1 n0− w d w 1 ≤ k( k d) d
by Lemma 2.4.5 (taking s = r = 1), and ln(n 0 − w d w 1 ) < k[1 + ln(n k + 1)] by Lemma 2.4.6 (taking s = r = 1). Therefore,
t(n, d; z] ≤ t(n0, d; z] < zn 0 d n0− d 1 n0− w d w 1 {1 + ln[ n0− w d w 1 ]} < zk(k d) d{1 + k[1 + ln(n k + 1)]} as required.
Recall that t(n, d, r] is the minimum size over all (d, r]-disjunct matrices with n columns. Theorem 4.1.3. For any positive integers n, d and r, if k = d + r ≤ n, then
t(n, d, r] < (k r) r(k d) d{1 + k[1 + ln(n k + 1)]}. Proof. For r ≤ w ≤ n − d, let A be the binary matrix of order [n d n − d r ] × n w with rows and columns indexed by {(D, R) | D ∈ [n]
d
, R ∈ [n] r
V = {v | v ∈ {0, 1}n, wt(v) = w} respectively. The entry of A at the row indexed by the pair (D, R) and the column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and the entries of v over R are all one; and 0 otherwise.
Observe that each row of A has weight n − (d + r) w − r
, and each column of A has weight n − w
d
w r
. By the Stein-Lov´asz theorem, there exists a submatrix M of A of order [n
d
n − d r
] × t having no zero rows, where
t < n w n − (d + r) w − r {1 + ln[ n − w d w r ]} = n d n − d r n − w d w r {1 + ln[ n − w d w r ]}. Note that the equality is obtained by counting the weight of A in two ways. It is straight-forward to show that the columns of M form a (d, r]-disjunct matrix of order t × n. We then have t(n, d, r] < n d n − d r n − w d w r {1 + ln[ n − w d w r ]}. Let n0 ≥ n be the smallest positive integer such that w = n
0r k is an integer. We have n0 d n0 − d r n0− w d w r ≤ ( k r) r(k d) d
by Lemma 2.4.5 (taking s = r), and ln(n 0 − w d w r ) < k[1 + ln(n k + 1)]
by Lemma 2.4.6 (taking s = r). Therefore, t(n, d, r] ≤ t(n0, d, r] < n0 d n0− d r n0− w d w r {1 + ln[ n0− w d w r ]} < (k r) r(k d) d{1 + k[1 + ln(n k + 1)]} as required.
Recall that t(n, d, r; z] is the minimum size over all (d, r; z]-disjunct matrices with n columns.
Theorem 4.1.4. [5] For any positive integers n, d, r and z, if k = d + r ≤ n, then t(n, d, r; z] < z(k r) r(k d) d{1 + k[1 + ln(n k + 1)]}. Proof. For r ≤ w ≤ n − d, let A be the binary matrix of order [n
d n − d r ] × n w with rows and columns indexed by {(D, R) | D ∈ [n]
d
, R ∈ [n] r
with DT R empty} and V = {v | v ∈ {0, 1}n, wt(v) = w} respectively. The entry of A at the row indexed by the pair (D, R) and the column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and the entries of v over R are all one; and 0 otherwise.
Observe that each row of A has weight n − (d + r) w − r
, and each column of A has weight n − w
d
w r
. By the extended Stein-Lov´asz theorem, there exists a submatrix M of A of order [n
d
n − d r
] × t with each row weight at least z, where
t < z n w n − (d + r) w − r {1 + ln[ n − w d w r ]} = zn d n − d r n − w d w r {1 + ln[ n − w d w r ]}. Note that the equality is obtained by counting the weight of A in two ways. It is straight-forward to show that the columns of M form a (d, r; z]-disjunct matrix of order t × n. We
then have t(n, d, r] < zn d n − d r n − w d w r {1 + ln[ n − w d w r ]}. Let n0 ≥ n be the smallest positive integer such that w = n
0r k is an integer. We have n0 d n0 − d r n0− w d w r ≤ ( k r) r(k d) d
by Lemma 2.4.5 (taking s = r), and ln(n 0 − w d w r ) < k[1 + ln(n k + 1)] by Lemma 2.4.6 (taking s = r). Therefore,
t(n, d, r; z] ≤ t(n0, d, r; z] < zn 0 d n0− d r n0− w d w r {1 + ln[ n0− w d w r ]} < z(k r) r(k d) d{1 + k[1 + ln(n k + 1)]} as required.
Recall that t(n, d, r) is the minimum size over all (d, r)-disjunct matrices with n columns. Theorem 4.1.5. For any positive integers n, d and r, if k = d + r ≤ n, then
t(n, d, r) < k r(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln k − 1 d }. Proof. For 1 ≤ w ≤ n − d, let A be the binary matrix of order [n
d n − d r ] × n w with rows and columns indexed by {(D, R) | D ∈ [n]
d
, R ∈ [n] r
V = {v | v ∈ {0, 1}n, wt(v) = w} respectively. The entry of A at the row indexed by the pair (D, R) and the column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and at least one entry of v over R is one; and 0 otherwise.
Observe that each row of A has weight
min(r,w) X j=1 r j n − (d + r) w − j
, and each column of A has weight n − w d min(r,w) X j=1 n − w − d r − j w j
. By the Stein-Lov´asz theorem, there exists a submatrix M of A of order [n
d
n − d r
] × t having no zero rows, where
t < n w min(r,w) X j=1 r j n − (d + r) w − j {1 + ln[n − w d min(r,w) X j=1 n − w − d r − j w j ]} = n d n − d r n − w d min(r,w) X j=1 n − w − d r − j w j {1 + ln[n − w d min(r,w) X j=1 n − w − d r − j w j ]}.
Note that the equality is obtained by counting the weight of A in two ways. It is straightfor-ward to show that the columns of M form a (d, r)-disjunct matrix of order t × n. We then have t(n, d, r) < n d n − d r n − w d min(r,w) X j=1 n − w − d r − j w j {1 + ln[n − w d min(r,w) X j=1 n − w − d r − j w j ]} < n d n − d r n − w d n − w − d r − 1 w 1 {1 + ln[ n − w d n − w − d r − 1 w 1 ]}, by Lemma 2.4.1. Let n0 ≥ n be the smallest positive integer such that w = n
0
We have n0 d n0− d r n0− w d n0− w − d r − 1 w 1 ≤ k r(1 + 1 k − 1) k−1
by Lemma 2.4.5 (taking s = 1), and ln(n 0− w d n0− w − d r − 1 w 1 ) < k[1 + ln(n k + 1)] + ln k − 1 d by Lemma 2.4.6 (taking s = 1). Therefore,
t(n, d, r) ≤ t(n0, d, r) < n0 d n0 − d r n0− w d n0− w − d r − 1 w 1 {1 + ln[ n0− w d n0− w − d r − 1 w 1 ]} < k r(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln k − 1 d } as required.
Recall that t(n, d, r; z) is the minimum size over all (d, r; z)-disjunct matrices with n columns.
Theorem 4.1.6. For any positive integers n, d, r and z, if k = d + r ≤ n, then t(n, d, r; z) < zk r(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln k − 1 d }. Proof. For 1 ≤ w ≤ n − d, let A be the binary matrix of order [n
d n − d r ] × n w with rows and columns indexed by {(D, R) | D ∈ [n]
d
, R ∈ [n] r
with DT R empty} and V = {v | v ∈ {0, 1}n, wt(v) = w} respectively. The entry of A at the row indexed by the pair (D, R) and the column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and at least one entry of v over R is one; and 0 otherwise.
Observe that each row of A has weight min(r,w) X j=1 r j n − (d + r) w − j
, and each column of A has weight n − w d min(r,w) X j=1 n − w − d r − j w j
. By the extended Stein-Lov´asz theorem, there exists a submatrix M of A of order [n
d
n − d r
] × t with each row weight at least z, where t < z n w min(r,w) X j=1 r j n − (d + r) w − j {1 + ln[n − w d min(r,w) X j=1 n − w − d r − j w j ]} = zn d n − d r n − w d min(r,w) X j=1 n − w − d r − j w j {1 + ln[n − w d min(r,w) X j=1 n − w − d r − j w j ]}.
Note that the equality is obtained by counting the weight of A in two ways. It is straight-forward to show that the columns of M form a (d, r; z)-disjunct matrix of order t × n. We then have t(n, d, r; z) < zn d n − d r n − w d min(r,w) X j=1 n − w − d r − j w j {1 + ln[n − w d min(r,w) X j=1 n − w − d r − j w j ]} < zn d n − d r n − w d n − w − d r − 1 w 1 {1 + ln[ n − w d n − w − d r − 1 w 1 ]}, by Lemma 2.4.1. Let n0 ≥ n be the smallest positive integer such that w = n
0 k is an integer. We have n0 d n0− d r n0− w d n0− w − d r − 1 w 1 ≤ k r(1 + 1 k − 1) k−1
by Lemma 2.4.5 (taking s = 1), and ln(n 0− w d n0− w − d r − 1 w 1 ) < k[1 + ln(n k + 1)] + ln k − 1 d by Lemma 2.4.6 (taking s = 1). Therefore,
t(n, d, r; z) ≤ t(n0, d, r; z) < zn 0 d n0 − d r n0− w d n0− w − d r − 1 w 1 {1 + ln[ n0 − w d n0 − w − d r − 1 w 1 ]} < zk r(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln k − 1 d } as required.
Recall that t(n, d, r, s] is the minimum size over all (d, s out of r]-disjunct matrices with n columns.
Theorem 4.1.7. For any positive integers n, d, r and s, with 1 ≤ s ≤ r, if k = d + r ≤ n, then t(n, d, r, s] < ( k s) s( k k−s) k−s r s {1 + k[1 + ln( n k + 1)] + ln k − s d }. Proof. For s ≤ w ≤ n − d, let A be the binary matrix of order [n
d n − d r ] × n w with rows and columns indexed by {(D, R) | D ∈ [n]
d
, R ∈ [n] r
with DT R empty} and V = {v | v ∈ {0, 1}n, wt(v) = w} respectively. The entry of A at the row indexed by the pair (D, R) and the column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and at least s entries of v over R are one; and 0 otherwise.
Observe that each row of A has weight
min(r,w) X j=s r j n − (d + r) w − j
has weight n − w d min(r,w) X j=s n − w − d r − j w j
. By the Stein-Lov´asz theorem, there exists a submatrix M of A of order [n
d
n − d r
] × t having no zero rows, where
t < n w min(r,w) X j=s r j n − (d + r) w − j {1 + ln[n − w d min(r,w) X j=s n − w − d r − j w j ]} = n d n − d r n − w d min(r,w) X j=s n − w − d r − j w j {1 + ln[n − w d min(r,w) X j=s n − w − d r − j w j ]}.
Note that the equality is obtained by counting the weight of A in two ways. It is straight-forward to show that the columns of M form a (d, s out of r]-disjunct matrix of order t × n. We then have t(n, d, r, s] < n d n − d r n − w d min(r,w) X j=s n − w − d r − j w j {1 + ln[n − w d min(r,w) X j=s n − w − d r − j w j ]} < n d n − d r n − w d n − w − d r − s w s {1 + ln[ n − w d n − w − d r − s w s ]}, by Lemma 2.4.1. Let n0 ≥ n be the smallest positive integer such that w = n
0s k is an integer. We have n0 d n0− d r n0− w d n0− w − d r − s w s ≤ (k s) s( k k − s) k−s r s by Lemma 2.4.5, and ln(n 0− w d n0− w − d r − s w s ) < k[1 + ln(n k + 1)] + ln k − s d
by Lemma 2.4.6. Therefore, t(n, d, r, s] ≤ t(n0, d, r, s] < n0 d n0− d r n0 − w d n0 − w − d r − s w s {1 + ln[ n0− w d n0− w − d r − s w s ]} < ( k s) s( k k−s) k−s r s {1 + k[1 + ln( n k + 1)] + ln k − s d } as required.
Recall that t(n, d, r, s; z] is the minimum size over all (d, s out of r; z]-disjunct matrices with n columns.
Theorem 4.1.8. For any positive integers n, d, r, s and z, with 1 ≤ s ≤ r, if k = d + r ≤ n, then t(n, d, r, s; z] < z( k s) s( k k−s) k−s r s {1 + k[1 + ln( n k + 1)] + ln k − s d }. Proof. For s ≤ w ≤ n − d, let A be the binary matrix of order [n
d n − d r ] × n w with rows and columns indexed by {(D, R) | D ∈ [n]
d
, R ∈ [n] r
with DT R empty} and V = {v | v ∈ {0, 1}n, wt(v) = w} respectively. The entry of A at the row indexed by the pair (D, R) and the column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and at least s entries of v over R are one; and 0 otherwise.
Observe that each row of A has weight
min(r,w) X j=s r j n − (d + r) w − j
, and each column of A has weight n − w d min(r,w) X j=s n − w − d r − j w j
. By the extended Stein-Lov´asz theorem, there exists a submatrix M of A of order [n
d
n − d r
where t < z n w min(r,w) X j=s r j n − (d + r) w − j {1 + ln[n − w d min(r,w) X j=s n − w − d r − j w j ]} = zn d n − d r n − w d min(r,w) X j=s n − w − d r − j w j {1 + ln[n − w d min(r,w) X j=s n − w − d r − j w j ]}.
Note that the equality is obtained by counting the weight of A in two ways. It is straightfor-ward to show that the columns of M form a (d, s out of r; z]-disjunct matrix of order t × n. We then have t(n, d, r, s; z] < zn d n − d r n − w d min(r,w) X j=s n − w − d r − j w j {1 + ln[n − w d min(r,w) X j=s n − w − d r − j w j ]} < zn d n − d r n − w d n − w − d r − s w s {1 + ln[ n − w d n − w − d r − s w s ]}, by Lemma 2.4.1. Let n0 ≥ n be the smallest positive integer such that w = n
0s k is an integer. We have n0 d n0− d r n0− w d n0− w − d r − s w s ≤ (k s) s( k k − s) k−s r s by Lemma 2.4.5, and ln(n 0− w d n0− w − d r − s w s ) < k[1 + ln(n k + 1)] + ln k − s d
by Lemma 2.4.6. Therefore, t(n, d, r, s; z] ≤ t(n0, d, r, s; z] < zn 0 d n0− d r n0− w d n0− w − d r − s w s {1 + ln[ n0− w d n0− w − d r − s w s ]} < z( k s) s( k k−s) k−s r s {1 + k[1 + ln( n k + 1)] + ln k − s d } as required.
4.2
Bounds for several selectors
The notion of (k, m, n)-selectors was first introduced by De Bonis, Gasieniec and Vaccaro in [2] , and it was then generalized to the notion of (k, m, c, n)-selectors [1]. It is interesting to remark that the notions of (k, m, n)-selecters and (k, m, c, n)-selectors are equivalent to (k, m, 1, n; 1)-selectors and (k, m, c, n; 1)-selectors respectively. Note that upper bounds for the sizes of (k, m, n)-selectors were also given in [14] by the Lov´asz Local Lemma.
Following similar arguments in [2] and [1] with a minor modification, upper bounds for the sizes of several selectors are given below.
Recall that ts(k, m, n) is the minimum size over all (k, m, n)-selectors.
Theorem 4.2.1. ts(k, m, n) < k k − m + 1(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln k − 1 k − m }.
Proof. For 1 ≤ w ≤ n − k + 1, let X = {x ∈ {0, 1}n | wt(x) = w} and U = {u ∈ {0, 1}k | wt(u) = 1}. Moreover, for any A ⊆ U of size r, r = 1, ..., k, and any set S ∈ [n]k, define EA,S = {x ∈ X : x|S ∈ A}.
by Γ = {EA,S ⊆ X | A ⊆ U with |U | = k, |A| = k − m + 1, S ∈ [n]k} and X = {x ∈ {0, 1} n
| wt(x) = w} respectively. The entry of M at the row indexed by the set EA,S and the column
indexed by the vector x ∈ X is 1 if x ∈ EA,S; and 0 otherwise.
Observe that each row of M has weight k−m+11 w−1n−k, and each column of M has weight
w 1
n−w k−1
k−1
(k−m+1)−1. By the Stein-Lov´asz theorem, there exists a submatrix M
0
of M of order [ k−m+1k nk] × t having no zero rows, where
t < n w (k − m + 1) w−1n−k {1 + ln[ w 1 n − w k − 1 k − 1 k − m ]} = k k−m+1 n k w 1 n−w k−1 k−1 k−m {1 + ln[ w 1 n − w k − 1 k − 1 k − m ]}. Note that the equality is obtained by counting the weight of M in two ways.
It suffices to show that the matrix M∗ of order t × n formed by the columns of M0 is a (k, m, n)-selector, that is, any submatrix of k arbitrary columns of M∗ contains a submatrix with each row weight exactly one, with at least m distinct rows.
Let x1, x2, ..., xt be the t rows of M∗ and let T = {x1, x2, ..., xt}. Suppose contradictorily
that there exists a set S ∈ [n]k such that the submatrix M∗|S of M∗ contains a submatrix
with each row weight exactly one, with at most m − 1 distinct rows. Let uj1, uj2, ..., ujq
be such rows, with q ≤ m − 1; let A be any subset of U \{uj1, uj2, ..., ujq} of cardinality
|A| = k − m + 1, then we have TT EA,S = ∅, contradicting the fact that M
0
is a matrix of order [ k−m+1k nk] × t having no zero rows. Hence we have
ts(k, m, n) < k k−m+1 n k w 1 n−w k−1 k−1 k−m {1 + ln[ w 1 n − w k − 1 k − 1 k − m ]}. Let n0 ≥ n be the smallest positive integer such that w = n
0 k is an integer. We have k k−m+1 n0 k w 1 n0−w k−1 k−1 k−m = k! (m−1)!(k−m+1)! (k−1)! (m−1)!(k−m)! · n0 k w 1 n0−w k−1 = k k − m + 1 · n0 k w 1 n0−w k−1 ≤ k k − m + 1(1 + 1 k − 1) k−1
by Lemma 2.4.5 (taking s = r = 1), and ln[w 1 n0− w k − 1 k − 1 k − m ] < k[1 + ln(n k + 1)] + ln k − 1 k − m by Lemma 2.4.6 (taking s = r = 1). Therefore, we have
ts(k, m, n) ≤ts(k, m, n0) < k k−m+1 n0 k w 1 n0−w k−1 k−1 k−m {1 + ln[ w 1 n0− w k − 1 k − 1 k − m ]} < k k − m + 1(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln k − 1 k − m } as required.
Recall that ts(k, m, n; z) is the minimum size over all (k, m, n; z)-selectors.
Theorem 4.2.2. ts(k, m, n; z) < (k − m + 1)(z − 1) + 1 k − m + 1 k(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln k − 1 k − m }. Proof. For 1 ≤ w ≤ n − k + 1, let X = {x ∈ {0, 1}n | wt(x) = w} and U = {u ∈ {0, 1}k | wt(u) = 1}. Moreover, for any A ⊆ U of size r, r = 1, ..., k, and any set S ∈ [n]k, define EA,S = {x ∈ X : x|S ∈ A}.
Let M be the binary matrix of order [ k−m+1k nk] × wn with rows and columns indexed by Γ = {EA,S ⊆ X | A ⊆ U with |U | = k, |A| = k − m + 1, S ∈ [n]k} and X = {x ∈ {0, 1}
n
| wt(x) = w} respectively. The entry of M at the row indexed by the set EA,S and the column
indexed by the vector x ∈ X is 1 if x ∈ EA,S; and 0 otherwise.
Observe that each row of M has weight k−m+11 w−1n−k, and each column of M has weight
w 1
n−w k−1
k−1
(k−m+1)−1. By the extended Stein-Lov´asz theorem, there exists a submatrix M
0
of M of order [ k−m+1k nk] × t with each row weight at least (k − m + 1)(z − 1) + 1, where
t < [(k − m + 1)(z − 1) + 1] n w (k − m + 1) w−1n−k {1 + ln[ w 1 n − w k − 1 k − 1 k − m ]} = [(k − m + 1)(z − 1) + 1] k k−m+1 n k w 1 n−w k−1 k−1 k−m {1 + ln[ w 1 n − w k − 1 k − 1 k − m ]}. Note that the equality is obtained by counting the weight of M in two ways.
It suffices to show that the matrix M∗ of order t × n formed by the columns of M0 is a (k, m, n; z)-selector, that is, any submatrix of k arbitrary columns of M∗ contains z disjiont submatrices with each row weight exactly one, with at least m distinct rows each.
Let x1, x2, ..., xt be the t rows of M∗ and let T = {x1, x2, ..., xt}. Suppose contradictorily
that there exists a set S ∈ [n]k such that the submatrix M∗|S of M∗ contains at most
z − 1 disjoint submatrices with each row weight exactly one, with at least m distinct rows. Moreover, M∗|S contains another disjoint submatrix with at most m − 1 distinct rows with
weight exactly one. Let uj1, uj2, ..., ujq be such rows, with q ≤ m − 1; let A be any subset
of U \{uj1, uj2, ..., ujq} of cardinality |A| = k − m + 1, then we have |T T EA,S| < (k − m +
1)(z − 1) + 1, contradicting the fact that M0 is a matrix of order [ k−m+1k nk] × t with each row weight at least (k − m + 1)(z − 1) + 1. Hence we have
ts(k, m, n; z) < [(k − m + 1)(z − 1) + 1] k−m+1k nk w 1 n−w k−1 k−1 k−m {1 + ln[ w 1 n − w k − 1 k − 1 k − m ]}. Let n0 ≥ n be the smallest positive integer such that w = n
0 k is an integer. We have k k−m+1 n0 k w 1 n0−w k−1 k−1 k−m = k! (m−1)!(k−m+1)! (k−1)! (m−1)!(k−m)! · n0 k w 1 n0−w k−1 = k k − m + 1 · n0 k w 1 n0−w k−1 ≤ k k − m + 1(1 + 1 k − 1) k−1
by Lemma 2.4.5 (taking s = r = 1), and ln[w 1 n0− w k − 1 k − 1 k − m ] < k[1 + ln(n k + 1)] + ln k − 1 k − m
