w c
n0−w k−c
a−1
a−m
{1 + ln[w
c
n0− w k − c
a − 1 a − m
]}
<(a − m + 1)(z − 1) + 1 a − m + 1 (k
c)c(1 + 1
k − c)k−c{1 + k[1 + ln(n
k + 1)] + ln a − 1 a − m
}
as required.
4.3 Bounds for some set systems
Note that upper bounds for the minimum number of blocks of uniform (m, t)-splitting systems were given in [7] by the Lov´asz Local Lemma. Note further that upper bounds for the minimum number of blocks of uniform (m, t)-splitting systems, (v, k, t)-covering designs, and (v, k, t, p)-lotto designs were given in [8] by the classical Stein-Lov´asz theorem, the proofs are included for completeness.
In this section, upper bounds for the minimum number of blocks of uniform (m, t; z)-splitting systems, uniform (m, t1, t2; z)-separating systems, (v, k, t; z)-covering designs and (v, k, t, p; z)-lotto designs will be derived by using the extended Stein-Lov´asz theorem.
Recall that SP (m, t) is the minimum number of blocks of uniform (m, t)-splitting systems.
Theorem 4.3.1. [8]
SP (m, t) <
m
m 2
t
t 2
m−t
m 2−t
2
{1 + ln[
m 2 t 2
2
]}.
Proof. Let A be the binary matrix of order mt × mm 2
with rows and columns indexed by {T | T ∈ [m]t } and Γ = {B | B ∈ [m]m
2
} respectively. The entry of A at the row indexed by the T and the column indexed by the vector B ∈ Γ is 1 if B splits T ; and 0 otherwise.
Observe that each row of A has weight and each column of A has weight
a =
By the Stein-Lov´asz theorem, there exists a submatrix M of A of order mt × N having no zero rows, where
N <
It is straightforward to show that the columns of M form an uniform (m, t)-splitting system with N blocks, as required.
Recall that SP (m, t; z) is the minimum number of blocks of uniform (m, t; z)-splitting systems.
with rows and columns indexed by {T | T ∈ [m]t } and Γ = {B | B ∈ [m]m
2
} respectively. The entry of A at the row indexed by the T and the column indexed by the vector B ∈ Γ is 1 if B splits T ; and 0 otherwise.
Observe that each row of A has weight
v = t and each column of A has weight
a =
By the extended Stein-Lov´asz theorem, there exists a submatrix M of A of order mt × N with each row weight at least z, where
N <
z mm 2
v {1 + ln a}.
It is straightforward to show that the columns of M form an uniform (m, t; z)-splitting system with N blocks, as required.
For the sizes of uniform (m, t1, t2)-separating systems and uniform (m, t1, t2; z)-separating
with rows and columns in-dexed by {(T1, T2) | T1 ∈ [m]t
1, T2 ∈ [m]t
2 with T1T T2 empty } and Γ = {B | B ∈ [m]m 2
}
respectively. The entry of A at the row indexed by the pair (T1, T2) and the column indexed by the vector B ∈ Γ is 1 if B separates the pair (T1, T2); and 0 otherwise.
Observe that each row of A has weight
v =m − (t1+ t2)
and each column of A has weight
a =
By the Stein-Lov´asz theorem, there exists a submatrix M of A of order [ mt
1
m−t1
t2 ] × N having no zero rows, where
N <
It is straightforward to show that the columns of M form an uniform (m, t1, t2)-splitting system with N blocks, as required.
Recall that SE(m, t1, t2; z) is the minimum number of blocks of uniform (m, t1, t2;
with rows and columns in-dexed by {(T1, T2) | T1 ∈ [m]t
1, T2 ∈ [m]t
2 with T1T T2 empty } and Γ = {B | B ∈ [m]m 2
}
respectively. The entry of A at the row indexed by the pair (T1, T2) and the column indexed by the vector B ∈ Γ is 1 if B separates the pair (T1, T2); and 0 otherwise.
Observe that each row of A has weight
v =m − (t1+ t2)
and each column of A has weight
a =
By the extended Stein-Lov´asz theorem, there exists a submatrix M of A of order [ mt
1
m−t1
t2 ]×
N with each row weight at least z, where
N <
z mm 2
v {1 + ln a}.
It is straightforward to show that the columns of M form an uniform (m, t1, t2; z)-splitting system with N blocks, as required.
Recall that C(v, k, t) is the minimum number of blocks of uniform (v, k, t)-covering de-signs.
Theorem 4.3.5. [8]
C(v, k, t) <
v t
k t
{1 + ln
k t
}.
Proof. Let A be the binary matrix of order vt × vk with rows and columns indexed by {T | T ∈ [v]t} and Γ = {B | B ∈ [v]k} respectively. The entry of A at the row indexed by the T and the column indexed by the vector B ∈ Γ is 1 if T ⊆ B; and 0 otherwise.
Observe that each row of A has weight v−tk−t, and each column of A has weight kt. By the Stein-Lov´asz theorem, there exists a submatrix M of A of order vt × N having no zero rows, where
N <
v k
v−t k−t
{1 + ln
k t
} =
v t
k t
{1 + ln
k t
}.
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 an (v, k, t)-covering design with N blocks, as required.
Recall that C(v, k, t; z) is the minimum number of blocks of uniform (v, k, t; z)-covering designs.
Theorem 4.3.6.
C(v, k, t; z) < z vt
k t
{1 + ln
k t
}.
Proof. Let A be the binary matrix of order vt × vk with rows and columns indexed by {T | T ∈ [v]t} and Γ = {B | B ∈ [v]k} respectively. The entry of A at the row indexed by the T and the column indexed by the vector B ∈ Γ is 1 if T ⊆ B; and 0 otherwise.
Observe that each row of A has weight v−tk−t, and each column of A has weight kt. By the extended Stein-Lov´asz theorem, there exists a submatrix M of A of order vt × N with each row weight at least z, where
N < z vk
v−t k−t
{1 + ln
k t
} = z vt
k t
{1 + ln
k t
}.
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 an (v, k, t; z)-covering design with N blocks, as required.
Recall that L(v, k, t, p) is the minimum number of blocks of uniform (v, k, t, p)-lotto de-signs.
Theorem 4.3.7. [8]
L(v, k, t, p) <
v k
min(t,k)
X
i=p
t i
v − t k − i
{1 + ln[
min(t,k)
X
i=p
k i
v − k t − i
]}.
Proof. Let A be the binary matrix of order vt × vk with rows and columns indexed by {T | T ∈ [v]t} and Γ = {B | B ∈ [v]k} respectively. The entry of A at the row indexed by the T and the column indexed by the vector B ∈ Γ is 1 if |TT B| ≥ p; and 0 otherwise.
Observe that each row of A has weight
min(t,k)
X
i=p
t i
v − t k − i
,
and each column of A has weight
By the Stein-Lov´asz theorem, there exists a submatrix M of A of order vt × N having no zero rows, where
N <
It is straightforward to show that the columns of M form an (v, k, t, p)-lotto design with N blocks, as required.
Recall that L(v, k, t, p; z) is the minimum number of blocks of uniform (v, k, t, p; z)-lotto designs.
Observe that each row of A has weight
min(t,k) and each column of A has weight
min(t,k)
By the extended Stein-Lov´asz theorem, there exists a submatrix M of A of order vt × N with each row weight at least z, where
N <
zv k
min(t,k)
X
i=p
t i
v − t k − i
{1 + ln[
min(t,k)
X
i=p
k i
v − k t − i
]}.
It is straightforward to show that the columns of M form an (v, k, t, p; z)-lotto design with N blocks, as required.
Chapter 5 Conclusion
In this thesis, we derive the extended Stein-Lov´asz theorem to deal with more combina-torial problems. From the strategy of the proof in Theorem 3.2.1, it is easy to see that the extended Stein-Lov´asz theorem also provides an algorithmic way to dealing with the exis-tence of good coverings and then deriving some upper bounds related to some combinatorial structures in Chapter 4. Note that most of these upper bounds obtained in Chapter 4 are roughly the same as those derived by the basic probabilistic method including the Lov´asz Lo-cal Lemma (see Appendix). Thus, due to its constructive nature, the Stein-Lov´asz theorem can be regarded as a de-randomized algorithm for the probabilistic methods. The relationship between the (extended) Stein-Lov´asz theorem and the Lov´asz Local Lemma deserve further study.
Appendix
Some upper bounds for the sizes of several disjunct matrices and selectors obtained by the Lov´asz Local Lemma and the classical Stein-Lov´asz theorem are survey in the following.
d-disjunct matrices
t(d, n) ≤ (d + 1)(1 +1
d)d{1 + ln[(d + 1)( d+1n − n−d−1d+1 )]} [13]
(by the Lov´asz Local Lemma)
(d, r]-disjunct matrices
t(n, d, r] ≤ (1 +d
r)r(1 + r
d)d{1 + ln[ nd n−d
r − n−(d+r)d n−(d+r)−d
r ]} [14]
(by the Lov´asz Local Lemma)
(d, r; z]-disjunct matrices
t(n, d, r; z] < z(1 + d
r)r(1 + r
d)d{1 + k[1 + ln(nk + 1)]}, k = d + r [5]
(by the classical Stein-Lov´asz theorem)
(d, r)-disjunct matrices
t(n, d, r) ≤ (1 +d
r)(1 + r
d)dr{1 + ln[ nd n−d
r − n−(d+r)d n−(d+r)−d
r ]} [14]
(by the Lov´asz Local Lemma)
(d, s out of r]-disjunct matrices
t(n, d, r, s] ≤ 1 + ln[ nd n−d
r − n−(d+r)d n−(d+r)−d
r ]
fd,r,s(p) for all 0 < p < 1, where fd,r,s(p) = (1 − p)d[1 −
s−1
X
i=0
r i
pi(1 − p)r−i] [14]
(by the Lov´asz Local Lemma)
(k, m, n)-selectors
ts(k, m, n) < ek2
k − m + 1lnn
k +ek(2k − 1) k − m + 1 [2]
(by the classical Stein-Lov´asz theorem) ts(k, m, n) ≤ m
k
mm![k(1 + 1
k − 1)k−1]{1 + ln[ nk − n−kk ]} [14]
(by the Lov´asz Local Lemma)
(k, m, c, n)-selectors
ts(k, m, c, n) < ekc+1 z lndn
ke − ewkc
z ln c +ekc
z (c + m + k + z − 1), where z = kc − m + 1 [1]
(by the classical Stein-Lov´asz theorem)
Bibliography
[1] A. De Bonis, New combinatorial structures with applications to efficient group testing with inhibitors, J. Combin. Opt., (2008) 15: 77-94.
[2] A. De Bonis, L. Gasieniec, U. Vaccaro, Optimal Two-State Algorithms for Group Testing Problems, SIAM J. Comput. Vol. 34, No. 5, (2005), 1253-1270.
[3] A. De Bonis, U. Vaccaro, Improved algorithms for group testing with inhibitors, Inform.
Process Lett. 67 (1998), 57-64.
[4] H. B. Chen, D. Z. Du and F. K. Hwang, An unexpected meeting of four seemingly unrelated problems: graph testing, DNA complex screeing, superimposed codes and secure key distribution, J. Combin. Opt., (2007) 14:121-129.
[5] H. B. Chen, H. L. Fu and F. K. Hwang, An upper bound of the number of tests in pooling designs for the error-tolerant complex model, Opt. Letters, (2008) 2:425-431.
[6] G. Cohen, S. Litsyn and G. Zemor, On greedy algorithms in coding theory, IEEE Trans-actions on Information Theory, Vol.42, No. 6, (1996), 2053-2057.
[7] D. Deng, D.R. Stinson and R. Wei, The Lov´asz Local Lemma and Its Application to Some Combinatorial Arrays, Designs, Codes and Cryptography 32 (2004) 121-134.
[8] D. Deng, Y. Zhang, P.C. Li and G.H.J. van Rees, The Stein-Lov´asz Theorem and Its Application to Some Combinatorial Arrays, preprint 2008.
[9] D.R. Fulkerson, H. J. Ryser, Width sequences for special classes of (0, 1)-matrices, Canad. J. Math. 15 (1963), 371-396.
[10] L. Lov´asz, On the ratio of optimal integral and fractional covers, Discrete Mathematics, 13 (1975) 383-390.
[11] S. K. Stein, Two combinatorial covering problems, Journal of Combinatorial Theory, Ser. A, 16 (1974), 391-397.
[12] D. R. Stinson and R. Wei, Generalized cover-free families, Discrete Mathematics, 279 (2004), 463-477.
[13] Hong-Gwa Yeh, d-Disjunct matrices: bound and Lov´asz Local Lemma, Discrete Math-ematics, 253 (2002), 97-107.
[14] Kuo-An Yu, Applications of the Lov´asz Local Lemma to pooling designs, Master thesis, NCTU 2007.