New lower bounds for bin packing problems with conflicts

The bin packing problem with conflicts (BPC) consists of minimizing the number of bins used to pack a set of items, where some items cannot be packed together in the same bin due to compatibility restrictions. The concepts of dual-feasible functions (DFF) and data-dependent dual-feasible functions (DDFF) have been used in the literature to improve the resolution of several cutting and packing problems. In this paper, we propose a general framework for deriving new DDFF as well as a new concept of generalized data-dependent dual-feasible functions (GDDFF), a conflict generalization of DDFF. The GDDFF take into account the structure of the conflict graph using the techniques of graph triangulation and tree-decomposition. Then we show how these techniques can be used in order to improve the existing lower bounds.     

Multidimensional dual-feasible functions and fast lower bounds for the vector packing problem

In this paper, we address the 2-dimensional vector packing problem where an optimal layout for a set of items with two independent dimensions has to be found within the boundaries of a rectangle. Many practical applications in areas such as the telecommunications, transportation and production planning lead to this combinatorial problem. Here, we focus on the computation of fast lower bounds using original approaches based on the concept of dual-feasible functions.     

Upper and Lower Bounds on Maximum Nonlinearity of n-input m-output Boolean Function

The paper presents lower and upper bounds on the maximumnonlinearity for an n-input m-output Booleanfunction. We show a systematic construction method for a highlynonlinear Boolean function based on binary linear codes whichcontain the first order Reed-Muller code as a subcode. We alsopresent a method to prove the nonexistence of some nonlinearBoolean functions by using nonexistence results on binary linearcodes. Such construction and nonexistence results can be regardedas lower and upper bounds on the maximum nonlinearity. For somen and m, these bounds are tighter than theconventional bounds. The techniques employed here indicate astrong connection between binary linear codes and nonlinear n-input m-output Boolean functions.     

Decomposition and Dynamic Cut Generation in Integer Linear Programming

Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to generate bounds for mixed-integer linear programming problems. Traditionally, these methods have been viewed as distinct from polyhedral methods, in which bounds are obtained by dynamically generating valid inequalities to strengthen an initial linear programming relaxation. Recently, a number of authors have proposed methods for integrating dynamic cut generation with various decomposition methods to yield further improvement in computed bounds. In this paper, we describe a framework within which most of these methods can be viewed from a common theoretical perspective. We then discuss how the framework can be extended to obtain a decomposition-based separation technique we call decompose and cut. As a by-product, we describe how these methods can take advantage of the fact that solutions with known structure, such as those to a given relaxation, can frequently be separated much more easily than arbitrary real vectors.     

A tighter variant of Jensen’s lower bound for stochastic programs and separable approximations to recourse functions

In this paper, we propose a new method to compute lower bounds on the optimal objective value of a stochastic program and show how this method can be used to construct separable approximations to the recourse functions. We show that our method yields tighter lower bounds than Jensen’s lower bound and it requires a reasonable amount of computational effort even for large problems. The fundamental idea behind our method is to relax certain constraints by associating dual multipliers with them. This yields a smaller stochastic program that is easier to solve. We particularly focus on the special case where we relax all but one of the constraints. In this case, the recourse functions of the smaller stochastic program are one dimensional functions. We use these one dimensional recourse functions to construct separable approximations to the original recourse functions. Computational experiments indicate that our lower bounds can significantly improve Jensen’s lower bound and our recourse function approximations can provide good solutions.     

Continuous location problems and Big Triangle Small Triangle: constructing better bounds

The Big Triangle Small Triangle method has shown to be a powerful global optimization procedure to address continuous location problems. In the paper published in J. Global Optim. (37:305–319, 2007), Drezner proposes a rather general and effective approach for constructing the bounds needed. Such bounds are obtained by using the fact that the objective functions in continuous location models can usually be expressed as a difference of convex functions. In this note we show that, exploiting further the rich structure of such objective functions, alternative bounds can be derived, yielding a significant improvement in computing times, as reported in our numerical experience.     

Further results on correlation functions of de Bruijn sequences

In this paper we develop some bounds for the correlation functions of sequences with period 2 ⁿ and specific linear complexities. These bounds are also applicable to the correlation functions of de Bruijn sequences with spann and specific linear complexities. It is interesting that a conjecture of Chan, Games and Key's for the case ofn=2 ^m can be proved easily by using the results developed here. Their conjecture asserts that there are no de Bruijn sequences with spann and linear complexity 2 ^n–1+n+1. The comjecture was proved by Games by a different method.Institute of Systems Science, Academic Sinica     

The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed. We study the effect of the use of transformations on the approximation of univariate (convex) functions. In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions. Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations. We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.     

Bounds for Analytical Functions of Matrices

If f is an analytic function bounded on a convex domain of the complex plane and A a square matrix whose spectrum is included in this domain, the function f(A) is well defined. In this paper we study bounds for ||f(A)|| uniform with respect to the functions f bounded by 1, and uniform with respect to the matrices A whose the numerical ranges are included in the domain. We show that these bounds are attained and give explicit formulae in some 2-dimensional cases.     

New bounds for expected delay in FIFO M/G/c queues

Most bounds for expected delay, E[D], in GI/GI/c queues are modifications of bounds for the GI/GI/1 case. In this paper we exploit a new delay recursion for the GI/GI/c queue to produce bounds of a different sort when the traffic intensity p = λ/μ = E[S]/E[T] is less than the integer portion of the number of servers divided by two. (S AND T denote generic service and interarrival times, respectively.) We derive two different families of new bounds for expected delay, both in terms of moments of S AND T. Our first bound is applicable when E[S₂] < ∞. Our second bound for the first time does not require finite variance of S; it only involves terms of the form E[S^β], where 1 < β < 2. We conclude by comparing our bounds to the best known bound of this type, as well as values obtained from simulation. This revised version was published online in June 2006 with corrections to the Cover Date.     

The dominance digraph as a solution to the two-machine flow-shop problem with interval processing times

The flow-shop minimum-length scheduling problem with n jobs processed on two machines is addressed where processing times are uncertain: lower and upper bounds for the random processing time are given before scheduling, but its probability distribution between these bounds is unknown. For such a problem, there often does not exist a dominant schedule that remains optimal for all possible realizations of the job processing times, and we look for a minimal set of schedules that is dominant. Such a minimal dominant set of schedules may be represented by a dominance digraph. We investigate useful properties of such a digraph.     

Continuous-Time Markov Decision Processes with Unbounded Transition and Discounted-Reward Rates

Abstract

In this article, we study continuous-time Markov decision processes in Polish spaces. The optimality criterion to be maximized is the expected discounted criterion. The transition rates may be unbounded, and the reward rates may have neither upper nor lower bounds. We provide conditions on the controlled system's primitive data under which we prove that the transition functions of possibly non-homogeneous continuous-time Markov processes are regular by using Feller's construction approach to such transition functions. Then, under continuity and compactness conditions we prove the existence of optimal stationary policies by using the technique of extended infinitesimal operators associated with the transition functions of possibly non-homogeneous continuous-time Markov processes, and also provide a recursive way to compute (or at least to approximate) the optimal reward values. The conditions provided in this paper are different from those used in the previous literature, and they are illustrated with an example.     

Bounds for the Multicovering Radii of Reed-Muller Codes with Applications to Stream Ciphers

The multicovering radii of a code are recentgeneralizations of the covering radius of a code. For positivem, the m-covering radius of C is the leastradius t such that everym-tuple of vectors is contained in at least one ball of radiust centered at some codeword. In this paper upper bounds arefound for the multicovering radii of first order Reed-Muller codes. These bounds generalize the well-known Norse bounds for the classicalcovering radii of first order Reed-Muller codes. They are exactin some cases. These bounds are then used to prove the existence of secure families of keystreams against a general class of cryptanalytic attacks. This solves the open question that gave rise to the study ofmulticovering radii of codes.     

Upper bounds for parent-identifying set systems

We derive new upper bounds on the size set families having the c-identifiable parent property (c-IPP) and the c-traceability property (c-TA) and compare these bounds to similar results on parent-identifying codes. An earlier version of this paper appeared in [4]. Sandia National Laboratories—This is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.     

Error Bounds for the Large‐Argument Asymptotic Expansions of the Lommel and Allied Functions

In this paper, we reconsider the large‐z asymptotic expansion of the Lommel function and its derivative. New representations for the remainder terms of the asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re‐expansions for these remainder terms and provide their error estimates. Applications to the asymptotic expansions of the Anger–Weber‐type functions, the Scorer functions, the Struve functions, and their derivatives are provided. The sharpness of our error bounds is discussed in detail, and numerical examples are given.     

Designs in Product Association Schemes

A number of important families of association schemes—such as the Hamming and Johnson schemes—enjoy the property that, in each member of the family, Delsarte t-designs can be characterised combinatorially as designs in a certain partially ordered set attached to the scheme. In this paper, we extend this characterisation to designs in a product association scheme each of whose components admits a characterisation of the above type. As a consequence of our main result, we immediately obtain linear programming bounds for a wide variety of combinatorial objects as well as bounds on the size and degree of such designs analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes.     

Dispersing hash functions

We define a family of functions F from a domain U to a range R to be dispersing if for every set S ? U of a certain size and random h ∈ F, the expected value of ∣S∣ – ∣h[S]∣ is not much larger than the expectation if h had been chosen at random from the set of all functions from U to R. We give near‐optimal upper and lower bounds on the size of dispersing families and present several applications where using such a family can reduce the use of random bits compared to previous randomized algorithms. A close relationship between dispersing families and extractors is exhibited. This relationship provides good explicit constructions of dispersing hash functions for some parameters, but in general the explicit construction is left open. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Error Bounds and Stopping Rules for Extrapolation Methods

In this paper, we present two different types of error boundsfor the approximation of functions by extrapolation methods(also called elimination methods). First, we give some a prioritype bounds; by means of these, one can, before starting theextrapolation process, estimate the errors of the extrapolatedvalues. Next, we present the so-called stopping rules; thesecan be used to decide during the process if the desired accuracyhas already been reached. Using the same techniques as for deducingthe error bounds, we then give criteria which help to predictthe form of the resulting error curves. It turns out that theseare in many cases monotone functions. Finally, two numericalexamples illustrate the results of this paper.     

Average distribution of prime ideals in families of number fields

We view an algebraic curve over ℚ as providing a one-parameter family of number fields and obtain bounds for the average value of some standard prime ideal counting functions over these families which are better than averaging the standard estimates for these functions.      

Sharp upper and lower bounds for the Laplacian spectral radius and the spectral radius of graphs

In this paper, sharp upper bounds for the Laplacian spectral radius and the spectral radius of graphs are given, respectively. We show that some known bounds can be obtained from our bounds. For a bipartite graph G, we also present sharp lower bounds for the Laplacian spectral radius and the spectral radius, respectively.