Discrete cooperative covering problems

A family of discrete cooperative covering problems is analysed in this paper. Each facility emits a signal that decays by the distance and each demand point observes the total signal emitted by all facilities. A demand point is covered if its cumulative signal exceeds a given threshold. We wish to maximize coverage by selecting locations for p facilities from a given set of potential sites. Two other problems that can be solved by the max-cover approach are the equivalents to set covering and p-centre problems. The problems are formulated, analysed and solved on networks. Optimal and heuristic algorithms are proposed and extensive computational experiments reported.     

Bin covering with a general profit function: approximability results

In the recent paper (Benk? et al. 2010) we introduced a new problem that we call Bin Packing/Covering with Delivery, or BP/CD for short. Mainly we mean under this expression that we look for not only a good, but a “good and fast” packing or covering. In the present paper we investigate the offline case. For the analysis, a novel view on “offline optimum” is introduced, which appears to be relevant concerning all problems where a final solution is ordering-dependent. We prove that if the item sizes are not allowed to be arbitrarily close to zero, then an optimal offline solution can be found in polynomial time. On the other hand, for unrestricted problem instances, no polynomial-time algorithm can achieve an asymptotic approximation ratio better than 6/7 if $P\ne NP$ .     

The gradual covering decay location problem on a network

In covering problems it is assumed that there is a critical distance within which the demand point is fully covered, while beyond this distance it is not covered at all. In this paper we define two distances. Within the lower distance a demand point is fully covered and beyond the larger distance it is not covered at all. For a distance between these two values we assume a gradual coverage decreasing from full coverage at the lower distance to no coverage at the larger distance.     

Lasalle method and general decay stability of stochastic neural networks with mixed delays

This paper investigates the general decay pathwise stability conditions on a class of stochastic neural networks with mixed delays by applying Lasalle method. The mixed time delays comprise both time-varying delays and infinite distributed delays. The contributions are as follows: (1)?we extend the Lasalle-type theorem to cover stochastic differential equations with mixed delays; (2)?based on the stochastic Lasalle theorem and the M-matrix theory, new criteria of general decay stability, which includes the almost surely exponential stability and the almost surely polynomial stability and the partial stability, for neural networks with mixed delays are established. As an application of our results, this paper also considers a two-dimensional delayed stochastic neural networks model.     

Dynamical networks: Continuous time and general discrete time models

Dynamical networks are characterized by 1) their topology (structure of the graph of interactions among the elements of a network); 2) the interactions between the elements of the network; 3) the intrinsic (local) dynamics of the elements of the network. A general approach to studying the commulative effect of all these three factors on the evolution of networks of a very general type has been developed in [1]. Besides, in this paper there were obtained sufficient conditions for a global stability (generalized strong synchronization) of networks with an arbitrary topology and the dynamics which is a composition (action of one after another) of a local dynamics of the elements of a network and of the interactions between these elements. Here we extend the results of [1] on global stability (generalized strong synchronization) to the case of a general dynamics in discrete time dynamical networks and to general dynamical networks with continuous time.     

Minimizing a monotone concave function with laminar covering constraints

Let V be a finite set with |V|=n. A family F⊆2^V is called laminar if for all two sets X,Y∈F, X∩Y≠∅ implies X⊆Y or X⊇Y. Given a laminar family F, a demand function , and a monotone concave cost function , we consider the problem of finding a minimum-cost such that x(X)?d(X) for all X∈F. Here we do not assume that the cost function F is differentiable or even continuous. We show that the problem can be solved in O(n²q) time if F can be decomposed into monotone concave functions by the partition of V that is induced by the laminar family F, where q is the time required for the computation of F(x) for any . We also prove that if F is given by an oracle, then it takes Ω(n²q) time to solve the problem, which implies that our O(n²q) time algorithm is optimal in this case. Furthermore, we propose an algorithm if F is the sum of linear cost functions with fixed setup costs. These also make improvements in complexity results for source location and edge-connectivity augmentation problems in undirected networks. Finally, we show that in general our problem requires Ω(2^n/2q) time when F is given implicitly by an oracle, and that it is NP-hard if F is given explicitly in a functional form.     

Elliptic problems on networks with constrictions

We investigate the asymptotic behavior of minimizers to sequences of elliptic variational problems posed on thin three-dimensional domains. These domains arise as thin neighborhoods of artibrary graphs that contain severe constrictions near the graph nodes. We characterize an appropriate limit of minimizers as a function of one variable defined on the graph that necessarily minimizes a one-dimensional variational problem. The most salient feature of these limits of minimizers is the emergence of jump discontinuities across the graph nodes. While the approach can handle quite general elliptic problems, we pay particular attention to an application to generalized Josephson junctions within the Ginzburg-Landau theory of superconductivity. Mathematics Subject Classification (2000) 35Q60, 78M30, 78M35     

Hyperbolic set covering problems with competing ground-set elements

Motivated by a challenging problem arising in wireless network design, we investigate a new nonlinear variant of the set covering problem with hyperbolic objective function. Each ground-set element (user) competes with all its neighbors (interfering users) for a shared resource (the network access time), and the goal is to maximize the sum of the resource share assigned to each ground-set element (the network efficiency) while covering all of them. The hyperbolic objective function privileges covers with limited overlaps among selected subsets. In a sense, this variant lies in between the set partitioning problem, where overlaps are forbidden, and the standard set covering problem, where overlaps are not an issue at all. We study the complexity and approximability of generic and Euclidean versions of the problem, present an efficient Lagrangean relaxation approach to tackle medium-to-large-scale instances, and compare the computational results with those obtained by linearizations.     

On the range of a covering function

Let be a finite system of residue classes with the moduli n₁,…,n_k distinct. By means of algebraic integers we show that the range of the covering function is not contained in any residue class with modulus greater one. In particular, the values of w(x) cannot have the same parity.     

Penalty function theory for general convex programming problems

A class of penalty functions for solving convex programming problems with general constraint sets is considered. Convergence theorems for penalty methods are established by utilizing the concept of infimal convergence of a sequence of functions. It is shown that most existing penalty functions are included in our class of penalty functions.     

A general decay result for a memory-type thermoelasticity with second sound

In this paper, we consider an n-dimensional system of thermoelasticity with second sound in the presence of a viscoelastic term acting in the domain. We prove a general decay result from which the usual exponential and polynomial decay results are only special cases.     

覆盖近似空间的连续与同胚映射

利用一般映射研究了覆盖近似空间的一些性质,并证明了一些结论.接着定义了覆盖空间的粗糙连续映射及粗糙同胚映射.最后在覆盖粗糙连续映射和覆盖粗糙同胚映射的条件下,研究了两个覆盖近似空间的有关性质,进而在某种程度上为覆盖近似空间的分类提供了理论依据.     

Packing and covering with integral feasible flows in integral supply-demand networks

Polynomial-time algorithms are presented for solving combinatorial packing and covering problems defined from the integral feasible flows in an integral supply-demand network. These algorithms are also shown to apply to packing and covering problems defined by the minimal integral solutions to general totally unimodular systems. Analogous problems arising from matroid bases are also discussed and it is demonstrated that a means for solving such problems is provided by recent work of Cunningham.     

The different cooperative behaviors on a kind of scale-free networks with identical degree sequence

The power-law degree distribution of scale-free networks plays an important role in the bloom of cooperation in the evolutionary games performed on them. In this paper we apply prisoner’s dilemma and public goods game on a family of scale-free networks with the same degree sequence, and show that power-law behavior alone does not determine the cooperative behavior in scale-free networks. Instead, we present that the direct connections among large-degree nodes have a crucial influence on the evolution of cooperation in the scale-free network family.     

Absolute bounds on optimal cost for a class of set covering problems

We study the class of (m constraint,n variable) set covering problems which have no more thank variables represented in each constraint. Letd denote the maximum column sum in the constraint matrix, letr=[m/d]?1, and letZ _g denote the cost of a greedy heuristic solution. Then we prove $$\begin{gathered} Zg \leqslant 1 + r + m - d - \left[ {mk \cdot MAX\left\{ {\frac{{2r}}{{2n - r - 1}},\ln \frac{n}{{n - r}}} \right\}} \right. \hfill \\ \left. { - kd \cdot MIN\left\{ {\frac{{r(r + 1)}}{{2(n - r)}},n \cdot \ln \frac{{n - 1}}{{n - r - 1}} - 1} \right\}} \right]. \hfill \\ \end{gathered} $$ This provides the firsta priori nontrivial upper bound discovered on heuristic solution cost (and thus on optimal solution cost) for the set covering problem. An example demonstrates that this bound is attainable, both for a greedy heuristic solution and for the optimal solution. Numerical examples show that this bound is substantially better than existing bounds for many problem instances. An important subclass of these problems occurs when the constraint matrix is a circulant, in which casem=n andk=d=[αη] for some 0<α<1. For this subclass we prove $$\mathop {\lim }\limits_{n \to \infty } Zg/n \leqslant \frac{{\alpha ^2 }}{2}[1/\alpha ][1/\alpha ].$$      

Cyclic networks with general blocking and starvation

We introducegeneral starvation and consider cyclic networks withgeneral blocking and starvation (GBS). The mechanism of general blocking allows the server to process a limited number of jobs when the buffer downstream is full, and that of general starvation allows the server to perform a limited number of services in anticipation of jobs that are yet to arrive. The two main goals of this paper are to investigate how the throughput of cyclic GBS networks is affected by varying (1) the total number of jobsJ, and (2) the buffer allocationk=(k₁..., k_m) subject to a fixed total buffer capacityK=k ₁ +... + k_m. In particular, we obtain sufficient conditions for the throughput to be symmetric inJ and to be maximized whenJ=K/2. We also show that the equal buffer allocation is optimal under the two regimes of light or heavy usage. In order to establish these results, we obtain several intermediate structural properties of the throughput, using duality, reversibility, and concavity, which are of independent interest.Research supported in part by NSF Grant No. ECS-8919818.