Computing Bounds on the Expected Maximum of Correlated Normal Variables

We compute upper and lower bounds on the expected maximum of correlated normal variables (up to a few hundred in number) with arbitrary means, variances, and correlations. Two types of bounding processes are used: perfectly dependent normal variables, and independent normal variables, both with arbitrary mean values. The expected maximum for the perfectly dependent variables can be evaluated in closed form; for the independent variables, a single numerical integration is required. Higher moments are also available. We use mathematical programming to find parameters for the processes, so they will give bounds on the expected maximum, rather than approximations of unknown accuracy. Our original application is to the maximum number of people on-line simultaneously during the day in an infinite-server queue with a time-varying arrival rate. The upper and lower bounds are tighter than previous bounds, and in many of our examples are within 5% or 10% of each other. We also demonstrate the bounds’ performance on some PERT models, AR/MA time series, Brownian motion, and product-form correlation matrices.     

A posteriori error bounds for the empirical interpolation method

We present rigorous a posteriori error bounds for the Empirical Interpolation Method (EIM). The essential ingredients are (i) analytical upper bounds for the parametric derivatives of the function to be approximated, (ii) the EIM “Lebesgue constant,” and (iii) information concerning the EIM approximation error at a finite set of points in parameter space. The bound is computed “off-line” and is valid over the entire parameter domain; it is thus readily employed in (say) the “on-line” reduced basis context. We present numerical results that confirm the validity of our approach.     

Scheduling on parallel identical machines to minimize total tardiness

This paper focuses on the problem of scheduling n independent jobs on m identical parallel machines for the objective of minimizing total tardiness of the jobs. We develop dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as upper bounds obtained from a heuristic algorithm. Computational experiments are performed on randomly generated test problems and results show that the algorithm solves problems with moderate sizes in a reasonable amount of computation time.     

Proof of covering minimality by generalizing the notion of independence

A method is proposed for obtaining lower bounds for the length of the shortest cover and complexity of the minimal cover based on the notion of independent family of sets. For the problem of minimization of Boolean functions, we provide the functions and construct coverings by faces of the set of unit vertices for which the suggested lower bounds can be achieved in the case of five or more variables. The lower bounds, based on independent sets, are unreachable and cannot be used as sufficient conditions for minimality of such functions.     

Bounds for randomly shared risk of heavy-tailed loss factors

For a risk vector V, whose components are shared among agents by some random mechanism, we obtain asymptotic lower and upper bounds for the individual agents’ exposure risk and the aggregated risk in the market. Risk is measured by Value-at-Risk or Conditional Tail Expectation. We assume Pareto tails for the components of V and arbitrary dependence structure in a multivariate regular variation setting. Upper and lower bounds are given by asymptotically independent and fully dependent components of V with respect to the tail index α being smaller or larger than 1. Counterexamples, where for non-linear aggregation functions no bounds are available, complete the picture.     

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

Bounds on degrees of p-adic separating polynomials

We study a discrete optimization problem introduced by Babai, Frankl, Kutin, and Štefankovi? (2001), which provides bounds on degrees of polynomials with p-adically controlled behavior. Such polynomials are of particular interest because they furnish bounds on the size of set systems satisfying Frankl-Wilson-type conditions modulo prime powers, with lower degree polynomials providing better bounds. We elucidate the asymptotic structure of solutions to the optimization problem, and we also provide an improved method for finding solutions in certain circumstances.     

On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

The packing chromatic number χ_ρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced.     

Upper bounds for the height of S-integral points on elliptic curves

We establish new upper bounds for the height of the S-integral points of an elliptic curve. This bound is explicitly given in terms of the set S of places of the number field K involved, but also in terms of the degree of K, as well as the rank, the regulator and the height of a basis of the Mordell–Weil group of the curve. The proof uses the elliptic analogue of Baker’s method, based on lower bounds for linear forms in elliptic logarithms.     

Positive solutions of semilinear biharmonic equations with critical Sobolev exponents

In this paper we study the critical growth biharmonic problem with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the biharmonic problem.     

On Multi-threaded Metrical Task Systems

Traditionally, on-line problems have been studied under the assumption that there is a unique sequence of requests that must be served. This approach is common to most general models of on-line computation, such as Metrical Task Systems. However, there exist on-line problems in which the requests are organized in more than one independent thread. In this more general framework, at every moment the first unserved request of each thread is available. Therefore, apart from deciding how to serve a request, at each stage it is necessary to decide which request to serve among several possibilities.In this paper we introduce Multi-threaded Metrical Task Systems, that is, the generalization of Metrical Task Systems to the case in which there are many threads of tasks. We study the problem from a competitive analysis point of view, proving lower and upper bounds on the competitiveness of on-line algorithms. We consider finite and infinite sequences of tasks, as well as deterministic and randomized algorithms. In this work we present the first steps towards a more general framework for on-line problems which is not restricted to a sequential flow of information.     

Exponential lower bounds for some NP-complete problems in a restricted linear decision tree model

LetV be a set inR ⁿ consisting of finitely many hyperplanes. The linear recognition problem given byV is to determine, using ternary comparisons of the form “f(x):0” wheref:R ⁿ →R is a linear function, whether a pointxεR _n is inV. We consider lower bounds on the number of comparisons whenV corresponds to some NP-complete problems. A technique is proposed for proving such bounds. If the tests “f(x):0” are restricted so thatf always defines some hyperplane inV, then some NP-complete problems are shown to have exponential lower bounds inn. Examples of larger classes of linear test functions are found such that the exponential lower bounds are still valid.     

L 2-Decay rate for non-ergodic Jackson network

We establish the additive theorem of L^2-decay rate for multi- dimensional Markov process with independent marginal processes. Using this and the decomposition method, we obtain explicit upper and lower bounds for decay rate of non-ergodic Jackson network. In some cases, we get the exact decay rate.     

Bounds for codes and designs in complex subspaces

We introduce the concepts of complex Grassmannian codes and designs. Let $\mathcal{G}_{m,n}$ denote the set of m-dimensional subspaces of ? ⁿ : then a code is a finite subset of $\mathcal{G}_{m,n}$ in which few distances occur, while a design is a finite subset of $\mathcal{G}_{m,n}$ that polynomially approximates the entire set. Using Delsarte’s linear programming techniques, we find upper bounds for the size of a code and lower bounds for the size of a design, and we show that association schemes can occur when the bounds are tight. These results are motivated by the bounds for real subspaces recently found by Bachoc, Bannai, Coulangeon and Nebe, and the bounds generalize those of Delsarte, Goethals and Seidel for codes and designs on the complex unit sphere.     

Clique partitions of distance multigraphs

We consider the minimum number of cliques needed to partition the edge set of D(G), the distance multigraph of a simple graph G. Equivalently, we seek to minimize the number of elements needed to label the vertices of a simple graph G by sets so that the distance between two vertices equals the cardinality of the intersection of their labels. We use a fractional analogue of this parameter to find lower bounds for the distance multigraphs of various classes of graphs. Some of the bounds are shown to be exact.     

Probabilistic analysis of some euclidean clustering problems

We are given n points distributed randomly in a compact region D of R^m. We consider various optimisation problems associated with partitioning this set of points into k subsets. For each problem we demonstrate lower bounds which are satisfied with high probability. For the case where D is a hypercube we use a partitioning technique to give deterministic upper bounds and to construct algorithms which with high probability can be made arbitrarily accurate in polynomial time for a given required accuracy.     

Integrating heuristic information into exact methods: The case of the vertex p-centre problem

We solve the vertex p-centre problem optimally using an exact method that considers both upper and lower bounds as part of its search engine. Tight upper bounds are generated quickly via an efficient three-level heuristic, which are then used to derive potential ‘lower bounds’ accordingly. These two pieces of information when used together make our chosen exact method more efficient at obtaining optimal solutions relatively quickly. The proposed implementation produced excellent results when tested on the OR Library data set. This integrated approach can be adopted for those exact methods that consider both upper and lower bounds within their search engine and hence provide a wider spectrum of applicability in other hard combinatorial problems.     

Independence number of de Bruijn graphs

In this paper, we give several exact values of the independence number of a de Bruijn graph UB(d,D) and in the other cases, we establish pertinent lower and upper bounds of this parameter. We show that asymptotically, if d is even, the ratio of the number of vertices of a greatest independent set of UB(d,D) is .     

On a method of derivation of lower bounds for the nonlinearity of Boolean functions

The calculation of the exact value of the rth order nonlinearity of a Boolean function (the power of the distance between the function and the set of functions is at most r) or the derivation of a lower bound for it is a complicated problem (especially for r > 1). Lower bounds for nonlinearities of different orders in terms of the value of algebraic immunity were obtained in a number of papers. These estimates turn out to be sufficiently strong if the value of algebraic immunity is maximum or close to maximum. In the present paper, we prove a statement that allows us to obtain fairly strong lower bounds for nonlinearities of different orders and for many functions with low algebraic immunity.     

Hausdorff and upper box dimension estimate of hyperbolic recurrent sets

In this paper, under the open set condition we have determined the lower and upper bounds of the Hausdorff dimensions and the upper box-counting dimensions of the limit sets of hyperbolic recurrent iterated function systems in terms of the unique zeros h and H of the two pressure functions. In addition, we have estimated the bounds of h, H-dimensional Hausdorff and packing measures. The result in this paper generalizes existing results about fractal dimensions of many other iterated function systems.