Two constructions of new error-correcting pooling designs from orthogonal spaces over a finite field of characteristic 2

In this paper, we construct two classes of t×n,s ^e -disjunct matrix with subspaces in orthogonal space \mathbbF_q⁽²ⁿ⁺¹⁾\mathbb{F}_{q}^{(2\nu+1)} of characteristic 2 and exhibit their disjunct properties. We also prove that the test efficiency t/n of constructions II is smaller than that of D’yachkov et al. (J. Comput. Biol. 12:1129–1136, 2005).     

Improving an upper bound on the size of k-regular induced subgraphs

This paper considers the NP-hard graph problem of determining a maximum cardinality subset of vertices inducing a k-regular subgraph. For any graph G, this maximum will be denoted by α _k (G). From a well known Motzkin-Straus result, a relationship is deduced between α _k (G) and the independence number α(G). Next, it is proved that the upper bounds υ _k (G) introduced in Cardoso et al. (J. Comb. Optim., 14, 455–463, 2007) can easily be computed from υ ₀(G), for any positive integer k. This relationship also allows one to present an alternative proof of the Hoffman bound extension introduced in the above paper. The paper continues with the introduction of a new upper bound on α _k (G) improving υ _k (G). Due to the difficulty of computing this improved bound, two methods are provided for approximating it. Finally, some computational experiments which were performed to compare all bounds studied are reported.     

Finding longest increasing and common subsequences in streaming data

We present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data-streaming model. To decide if the LIS of a given stream of elements drawn from an alphabet αbet has length at least k, we discuss a one-pass algorithm using O(k log αbetsize) space, with update time either O(log k) or O(log log αbetsize); for αbetsize = O(1), we can achieve O(log k) space and constant-time updates. We also prove a lower bound of Ω(k) on the space requirement for this problem for general alphabets αbet, even when the input stream is a permutation of αbet. For finding the actual LIS, we give a ⌈log (1 + 1/ɛ)-pass algorithm using O(k^1+ɛlog αbetsize) space, for any ɛ > 0. For LCS, there is a trivial Θ(1)-approximate O(log n)-space streaming algorithm when αbetsize = O(1). For general alphabets αbet, the problem is much harder. We prove several lower bounds on the LCS problem, of which the strongest is the following: it is necessary to use Ω(n/ρ²) space to approximate the LCS of two n-element streams to within a factor of ρ, even if the streams are permutations of each other. A preliminary version of this paper appears in the Proceedings of the 11th International Computing and Combinatorics Conference (COCOON'05), August 2005, pp. 263–272.     

On the upper total domination number of Cartesian products of graphs

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63–69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ _t (G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Γ _t (G)Γ _t (H)≤2Γ _t (G □ H). Research of M.A. Henning supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

A quadratic lower bound for Rocchio’s similarity-based relevance feedback algorithm with a fixed query updating factor

Rocchio’s similarity-based relevance feedback algorithm, one of the most important query reformation methods in information retrieval, is essentially an adaptive supervised learning algorithm from examples. In practice, Rocchio’s algorithm often uses a fixed query updating factor. When this is the case, we strengthen the linear Ω(n) lower bound obtained by Chen and Zhu (Inf. Retr. 5:61–86, 2002) and prove that Rocchio’s algorithm makes Ω(k(n−k)) mistakes in searching for a collection of documents represented by a monotone disjunction of k relevant features over the n-dimensional binary vector space {0,1} ⁿ , when the inner product similarity measure is used. A quadratic lower bound is obtained when k is linearly proportional to n. We also prove an O(k(n−k)³) upper bound for Rocchio’s algorithm with the inner product similarity measure in searching for such a collection of documents with a constant query updating factor and a zero classification threshold.     

Approximating the chromatic index of multigraphs

It is well known that if G is a multigraph then χ′(G)≥χ′^*(G):=max {Δ(G),Γ(G)}, where χ′(G) is the chromatic index of G, χ′^*(G) is the fractional chromatic index of G, Δ(G) is the maximum degree of G, and Γ(G)=max {2|E(G[U])|/(|U|−1):U⊆V(G),|U|≥3, |U| is odd}. The conjecture that χ′(G)≤max {Δ(G)+1,⌈Γ(G)⌉} was made independently by Goldberg (Discret. Anal. 23:3–7, 1973), Anderson (Math. Scand. 40:161–175, 1977), and Seymour (Proc. Lond. Math. Soc. 38:423–460, 1979). Using a probabilistic argument Kahn showed that for any c>0 there exists D>0 such that χ′(G)≤χ′^*(G)+c χ′^*(G) when χ′^*(G)>D. Nishizeki and Kashiwagi proved this conjecture for multigraphs G with χ′(G)>⌊(11Δ(G)+8)/10⌋; and Scheide recently improved this bound to χ′(G)>⌊(15Δ(G)+12)/14⌋. We prove this conjecture for multigraphs G with $\chi'(G)>\lfloor\Delta(G)+\sqrt{\Delta(G)/2}\rfloor$\chi'(G)>\lfloor\Delta(G)+\sqrt{\Delta(G)/2}\rfloor , improving the above mentioned results. As a consequence, for multigraphs G with $\chi'(G)>\Delta(G)+\sqrt {\Delta(G)/2}$\chi'(G)>\Delta(G)+\sqrt {\Delta(G)/2} the answer to a 1964 problem of Vizing is in the affirmative.     

The orbit problem is in the GapL hierarchy

The Orbit problem is defined as follows: Given a matrix A∈ℚ ^n×n and vectors x,y∈ℚ ⁿ , does there exist a non-negative integer i such that A ⁱ x=y. This problem was shown to be in deterministic polynomial time by Kannan and Lipton (J. ACM 33(4):808–821, 1986). In this paper we place the problem in the logspace counting hierarchy GapLH. We also show that the problem is hard for C₌L with respect to logspace many-one reductions.     

L(j,k)- and circular L(j,k)-labellings for the products of complete graphs

Let j and k be two positive integers with j≥k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the λ _j,k -number of G, denoted by λ _j,k (G). A σ(j,k)-circular labelling with span m of a graph G is a function f:V(G)→{0,1,…,m−1} such that |f(u)−f(v)| _m ≥j if u and v are adjacent; and |f(u)−f(v)| _m ≥k if u and v are at distance two apart, where |x| _m =min {|x|,m−|x|}. The minimum m such that there exists a σ(j,k)-circular labelling with span m for G is called the σ _j,k -number of G and denoted by σ _j,k (G). The λ _j,k -numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28–35). This paper determines the λ _j,k -numbers of direct products of two complete graphs and the σ _j,k -numbers of direct products and Cartesian products of two complete graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. This work is partially supported by FRG, Hong Kong Baptist University, Hong Kong; NSFC, China, grant 10171013; and Southeast University Science Foundation grant XJ0607230.     

Restricted domination parameters in graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ _m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r _k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r _k (G,γ _m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r _k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r _k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

On the total {k}-domination number of Cartesian products of graphs

Let γ _t ^{k}(G) denote the total {k}-domination number of graph G, and let denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, . As a corollary of this result, we have for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005). The work was supported by NNSF of China (No. 10701068 and No. 10671191).     

Optimal algorithms for online time series search and one-way trading with interrelated prices

The basic models of online time series search and one-way trading are introduced by El-Yaniv et al. in Algorithmica 30(1), 101–139 (2001) where it is assumed that the prices are bounded within interval [m,M] (0<m<M). In this paper, we consider another case where every two consecutive prices are interrelated, that is, the variation range of each price depends on its preceding price. We present optimal deterministic online algorithms for the two problems, respectively. According to one conclusion in Algorithmica 30(1), 101–139 (2001), we further point out that for the case we considered, an optimal deterministic algorithm for the one-way trading problem can be regarded as an optimal randomized one for the time series search problem, and randomization is useless for the one-way trading problem.     

A note on fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges

In the paper “Fault-free Mutually Independent Hamiltonian Cycles in Hypercubes with Faulty Edges” (J. Comb. Optim. 13:153–162, 2007), the authors claimed that an n-dimensional hypercube can be embedded with (n−1−f)-mutually independent Hamiltonian cycles when f≤n−2 faulty edges may occur accidentally. However, there are two mistakes in their proof. In this paper, we give examples to explain why the proof is deficient. Then we present a correct proof. This work was supported in part by the National Science Council of the Republic of China under Contract NSC 95-2221-E-233-002.     

Strongly chordal and chordal bipartite graphs are sandwich monotone

A graph class is sandwich monotone if, for every pair of its graphs G ₁=(V,E ₁) and G ₂=(V,E ₂) with E ₁⊂E ₂, there is an ordering e ₁,…,e _k of the edges in E ₂∖E ₁ such that G=(V,E ₁∪{e ₁,…,e _i }) belongs to the class for every i between 1 and k. In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono (Czechoslov. Math. J. 46:577–583, 1997). So far, very few classes have been proved to be sandwich monotone, and the most famous of these are chordal graphs. Sandwich monotonicity of a graph class implies that minimal completions of arbitrary graphs into that class can be recognized and computed in polynomial time. For minimal completions into strongly chordal or chordal bipartite graphs no polynomial-time algorithm has been known. With our results such algorithms follow for both classes. In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently.     

Packing 5-cycles into balanced complete m-partite graphs for odd m

Let be a complete m-partite graph with partite sets of sizes n ₁,n ₂,…,n _m . A complete m-partite graph is balanced if each partite set has n vertices. We denote this complete m-partite graph by K _m(n). In this paper, we completely solve the problem of finding a maximum packing of the balanced complete m-partite graph K _m(n), m odd, with edge-disjoint 5-cycles and we explicitly give the minimum leaves. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. Research of M.-H.W. was supported by NSC 93-2115-M-264-001.     

Pooling designs associated with unitary space and ratio efficiency comparison

Let \mathbbF^(2n+d)_q²\mathbb{F}^{(2\nu+\delta)}_{q^{2}} be a (2ν+δ)-dimensional unitary space of \mathbbF_q²\mathbb{F}_{q^{2}} , where δ=0 or 1. In this paper we construct a family of inclusion matrices associated with subspaces of \mathbbF^(2n+d)_q²\mathbb{F}^{(2\nu+\delta)}_{q^{2}} , and exhibit its disjunct property. Moreover, we compare the ratio efficiency of this construction with others, and find it smaller under some conditions.     

On the distance paired domination of generalized Petersen graphs <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,1) and <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,2)

Let G=(V,E) be a graph without an isolated vertex. A set D⊆V(G) is a k -distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a k-distance paired dominating set for graph G is the k -distance paired domination number, denoted by γ _p ^k (G). In this paper, we determine the exact k-distance paired domination number of generalized Petersen graphs P(n,1) and P(n,2) for all k≥1.     

On domination number of Cartesian product of directed paths

Let γ(G) denote the domination number of a digraph G and let P _m □P _n denote the Cartesian product of P _m and P _n , the directed paths of length m and n. In this paper, we give a lower and upper bound for γ(P _m □P _n ). Furthermore, we obtain a necessary and sufficient condition for P _m □P _n to have efficient dominating set, and determine the exact values: γ(P ₂□P _n )=n, g(P₃\square P_n)=n+é\fracn4ù\gamma(P_{3}\square P_{n})=n+\lceil\frac{n}{4}\rceil, g(P₄\square P_n)=n+é\frac2n3ù\gamma(P_{4}\square P_{n})=n+\lceil\frac{2n}{3}\rceil, γ(P ₅□P _n )=2n+1 and g(P₆\square P_n)=2n+é\fracn+23ù\gamma(P_{6}\square P_{n})=2n+\lceil\frac{n+2}{3}\rceil.     

A polynomial-time perfect sampler for the <Emphasis Type="Italic">Q</Emphasis>-Ising with a vertex-independent noise

We present a polynomial-time perfect sampler for the Q-Ising with a vertex-independent noise. The Q-Ising, one of the generalized models of the Ising, arose in the context of Bayesian image restoration in statistical mechanics. We study the distribution of Q-Ising on a two-dimensional square lattice over n vertices, that is, we deal with a discrete state space {1,…,Q} ⁿ for a positive integer Q. Employing the Q-Ising (having a parameter β) as a prior distribution, and assuming a Gaussian noise (having another parameter α), a posterior is obtained from the Bayes’ formula. Furthermore, we generalize it: the distribution of noise is not necessarily a Gaussian, but any vertex-independent noise. We first present a Gibbs sampler from our posterior, and also present a perfect sampler by defining a coupling via a monotone update function. Then, we show O(nlog n) mixing time of the Gibbs sampler for the generalized model under a condition that β is sufficiently small (whatever the distribution of noise is). In case of a Gaussian, we obtain another more natural condition for rapid mixing that α is sufficiently larger than β. Thereby, we show that the expected running time of our sampler is O(nlog n).     

The Independence Number of Graphs with a Forbidden Cycle and Ramsey Numbers

Let k 5 be a fixed integer and let m = (k – 1)/2. It is shown that the independence number of a C _k-free graph is at least c ₁[ d(v)^{1/(m – 1)}]^{(m – 1)/m} and that, for odd k, the Ramsey number r(C _k, K _n) is at most c ₂(n ^{m + 1}/log n)^1/m , where c ₁ = c ₁(m) > 0 and c ₂ = c ₂(m) > 0.