Some bounds on the coupon collector problem

This work addresses on the coupon collector problem and its generalization introduced by Flajolet, Gardy, and Thimonier. In our main results, we show a ratio limit theorem for the random time of the generalized coupon collector problem, and, further, we give the leading term and the geometric rate for the distribution of this random time, when the number of throws is large. For the classical coupon collector problem, we give a bound on the conditional second moment for the number of visits to the coupons, relying strongly on a result of Holst on extremal distributions. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

Some results on the spectral reconstruction problem

This paper is concerned with the spectral version of the reconstruction conjecture: Whether a graph with n>2 vertices is determined (up to isomorphism) by the collection of its spectrum and the spectrum of its vertex-deleted graphs? Some positive results as well as a method for constructing counterexamples to the problem are provided.     

Some upper bounds for density of function spaces

Let Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X which is closed under finite unions. We proved that the density of the space Cα(X,Y) is at most iw(X)⋅d(Y) where iw(X) denotes the i-weight of the Tychonoff space X, and d(Y) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space Ck([0,1],Y) is κ².     

New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C_r be the unique cycle of G. The graph G-E(C_r) is a forest of r rooted trees T₁,T₂,…,T_r with root vertices v₁,v₂,…,v_r, respectively. Let

     

Some upper and lower bounds on PSD-rank

Positive semidefinite rank (PSD-rank) is a relatively new complexity measure on matrices, with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix M as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.     

Asymptotics in the random assignment problem

Summary We show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation. The method involves construction of an infinite limit structure, in terms of which the limit constant is defined. But we cannot improve on the known numerical bounds for the limit.Research supported by NSF Grant MCS90-01710     

Sharp upper bounds on the distance spectral radius of a graph

Let M=(_mij)$M=(m_{ij})$ be a nonnegative irreducible n×n$n\times n$ matrix with diagonal entries 0. The largest eigenvalue of M is called the spectral radius of the matrix M  , denoted by ρ(M)$\rho (M)$. In this paper, we give two sharp upper bounds of the spectral radius of matrix M. As corollaries, we give two sharp upper bounds of the distance matrix of a graph.     

Computing tight upper bounds on the algebraic connectivity of certain graphs

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of:

the generalized Bethe tree B,

a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B,

a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices,

a graph obtained from the cycle C_r by attaching B, by its root, to each vertex of the cycle, and

a tree obtained from the path P_r by attaching B, by its root, to each vertex of the path,

is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.

     

Asymptotics for the random coupon collector problem

We develop techniques of computing the asymptotics of the expected number of items that one has to check in order to detect all N existing kinds, as N → ∞. The occurring frequencies of the differend kinds are random variables.     

New upper bounds for Estrada index of bipartite graphs

Suppose G is a graph and λ₁,λ₂,…,λ_n are the eigenvalues of G. The Estrada index EE(G) of G is defined as the sum of e^λ_i, 1≤i≤n. In this paper some new upper bounds for the Estrada index of bipartite graphs are presented. We apply our result on a (4,6)-fullerene to improve our bound given in an earlier paper.     

Improved lower bounds on the length of Davenport-Schinzel sequences

We derive lower bounds on the maximal length _s(n) of (n, s) Davenport Schinzel sequences. These bounds have the form _2s=1(n)=(n^s(n)), where(n) is the extremely slowly growing functional inverse of the Ackermann function. These bounds extend the nonlinear lower bound ₃ (n)=(n(n)) due to Hart and Sharir [5], and are obtained by an inductive construction based upon the construction given in [5].Work on this paper has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation.     

On a problem of spencer

LetX ₁, ...,X _n be events in a probability space. Let ϱ_i be the probabilityX _i occurs. Let ϱ be the probability that none of theX _i occur. LetG be a graph on [n] so that for 1 ≦i≦n X _i is independent of ≈X _j ‖(i, j)∉G≈. Letf(d) be the sup of thosex such that if ϱ₁, ..., ϱ _n ≦x andG has maximum degree ≦d then ϱ>0. We showf(1)=1/2,f(d)=(d−1) ^d−1 d ^−d ford≧2. Hence df(d)=1/e. This answers a question posed by Spencer in [2]. We also find a sharp bound for ϱ in terms of the ϱ _i andG.     

An optimal lower bound on the number of variables for graph identification

In this paper we show that (n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-variable language with counting is equivalent to the (k–1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant becausen variables obviously suffice to identify graphs onn vertices.Research supported by NSF grant CCR-8709818.Research supported by NSF grant CCR-8805978 and Pennsylvania State University Research Initiation grant 428-45.Research supported by NSF grants DCR-8603346 and CCR-8806308.     

New upper bounds on the spectral radius of trees with the given number of vertices and maximum degree

This paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree. We obtain the new upper bounds on the spectral radius of the trees, and the results are the best upper bounds expressed by the number of vertices and maximum degree, at present.     

Tight bounds for minimax grid matching with applications to the average case analysis of algorithms

The minimax grid matching problem is a fundamental combinatorial problem associated with the average case analysis of algorithms. The problem has arisen in a number of interesting and seemingly unrelated areas, including wafer-scale integration of systolic arrays, two-dimensional discrepancy problems, and testing pseudorandom number generators. However, the minimax grid matching problem is best known for its application to the maximum up-right matching problem. The maximum up-right matching problem was originally defined by Karp, Luby and Marchetti-Spaccamela in association with algorithms for 2-dimensional bin packing. More recently, the up-right matching problem has arisen in the average case analysis of on-line algorithms for 1-dimen-sional bin packing and dynamic allocation.In this paper, we solve both the minimax grid matching problem and the maximum up-right matching problem. As a direct result, we obtain tight upper bounds on the average case behavior of the best algorithms known for 2-dimensional bin packing, 1-dimensional on-line bin packing and on-line dynamic allocation. The results also solve a long-open question in mathematical statistics.This research was supported by Air Force Contracts AFOSR-82-0326 and AFOSR-86-0078, NSF Grant 8120790, and DARPA contract N00014-80-C-0326. In addition, Tom Leighton was supported by an NSF Presidential Young Investigator Award with matching funds from Xerox and IBM.     

Spectral upper bounds for the order of a k-regular induced subgraph

Let G be a simple graph with least eigenvalue λ and let S be a set of vertices in G which induce a subgraph with mean degree k. We use a quadratic programming technique in conjunction with the main angles of G to establish an upper bound of the form |S|?inf{(k+t)q_G(t):t>-λ} where q_G is a rational function determined by the spectra of G and its complement. In the case k=0 we obtain improved bounds for the independence number of various benchmark graphs.     

On upper bounds for Laplacian graph eigenvalues

In this paper, we obtain the following upper bound for the largest Laplacian graph eigenvalue λ(G):

     

An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree

A Bethe tree B_d,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2?j?k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of B_d,k. Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of B_d,k. Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree T. These upper bounds are given in terms of the largest vertex degree and the radius of T, and they are attained if and only if T is a Bethe tree.     

Some topics in regenerative steady-state simulation

This paper offers a short introduction to the regenerative method of steady-state simulation output analysis. The paper also contains several new results. In particular, it is shown that regenerative methods necessarily apply to steady-state simulations that are well-posed in a certain precise sense. The paper also describes a bias-reduction algorithm that takes advantage of regenerative structure.