先锋雷达，璀璨皓星

日前,瑞士腕表品牌雷达（RADO）于上海举行了盛大的新品上市发布会,设计与高科技材质并行的先锋腕表Hv—perChrome皓星系列耀世降临。发布会上,英国首席网球选手、目前ATP单打世界排名第三的网坛金童、雷达品牌形象代言人安迪．穆雷（AndyMurray）亲临现场,成为雷达表挑战与突破的精神化身。     

A Unified Approach to Approximating Partial Covering Problems

An instance of the generalized partial cover problem consists of a ground set U and a family of subsets ${\mathcal{S}}\subseteq 2^{U}$ . Each element e??U is associated with a profit p(e), whereas each subset $S\in \mathcal{S}$ has a cost c(S). The objective is to find a minimum cost subcollection $\mathcal{S}'\subseteq \mathcal{S}$ such that the combined profit of the elements covered by $\mathcal{S}'$ is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e??U uncovered, we incur a penalty of ??(e). The goal is to identify a subcollection $\mathcal{S}'\subseteq \mathcal{S}$ that minimizes the cost of $\mathcal{S}'$ plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results.     

On-Line and Off-Line Approximation Algorithms for Vector Covering Problems

This paper deals with vector covering problems in d -dimensional space. The input to a vector covering problem consists of a set X of d -dimensional vectors in [0,1] ^d . The goal is to partition X into a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NP-complete, and we are mainly interested in its on-line and off-line approximability. For the on-line version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d≥ 2 dimensions. This result contradicts a statement of Csirik and Frenk in [5] where it is claimed that, for d≥ 2 , no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d≥ 2 , no on-line algorithm can have a worst case ratio better than 2/(2d+1) . For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(1/ log d) . For d=2 , we present a very fast and very simple off-line approximation algorithm that has worst case ratio 1/2 . Moreover, we show that a method from the area of compact vector summation can be used to construct off-line approximation algorithms with worst case ratio 1/d for every d≥ 2 . Received November 1996; revised March 1997.     

On Lazy Bin Covering and Packing problems

In this paper, we study two interesting variants of the classical bin packing problem, called Lazy Bin Covering (LBC) and Cardinality Constrained Maximum Resource Bin Packing (CCMRBP) problems. For the offline LBC problem, we first prove the approximation ratio of the First-Fit-Decreasing and First-Fit-Increasing algorithms, then present an APTAS. For the online LBC problem, we give a competitive analysis for the algorithms of Next-Fit, Worst-Fit, First-Fit, and a modified HARMONIC_M algorithm. The CCMRBP problem is a generalization of the Maximum Resource Bin Packing (MRBP) problem Boyar et al. (2006) [1]. For this problem, we prove that its offline version is no harder to approximate than the offline MRBP problem.     

Conflict Free Version of Covering Problems on Graphs: Classical and Parameterized

Theory of Computing Systems - Let π be a family of graphs. In the classical π-Vertex Deletion problem, given a graph G and a positive integer k, the objective is to check whether there...     

图的最小顶点覆盖问题的面上DNA解法

1994年，Adleman提出一种解决NP完全问题的新方法-DNA计算．之后又出现了许多关于DNA计算的改进操作并增加了其可靠性，其中面上操作是一种很有效的方法．本文利用DNA计算的固态处理(面上计算)解决了图论中又-NP完全问题一图的最小顶点覆盖问题．构造了含有6个顶点10条边的图的顶点集子集对应的数据池之后，进行了一系列的合成、杂交、清洗、变性等生物操作，得到所有覆盖对应的DNA序列，然后通过编址过程得到所要求的最小覆盖．     

Experimental Analysis of Approximation Algorithms for the Vertex Cover and Set Covering Problems

Several approximation algorithms with proven performance guarantees have been proposed to find approximate solutions to classical combinatorial optimization problems. However, theoretical results may not reflect the experimental performance of the proposed algorithms. As a consequence, a question arises: how “far” from the theoretically proved performance are the experimental results? We conduct a controlled empirical study of approximation algorithms for the Vertex Cover and the Set Covering Problems. Many authors have proposed approximation algorithms for those problems. Our main goal is to better understand their strengths, weaknesses, and operation. Although we implement more than one algorithm to find feasible solutions to either problems, this work does not emphasize competition between them. The quality of the solutions related to the theoretical performance guarantees are analyzed instead. The computational experiments showed that the proven performance guarantees of all tested algorithms did not forecast well the empirical performance.     

变精度覆盖粗糙集模型的推广研究

基于多数包含关系及误差参数β（0≤β＜O．5）,提出了两种基于对象邻域的变精度覆盖粗糙集模型,讨论了模型中β上、下近似算子的性质;从对偶性角度出发推广了β上近似、β下近似算子apcβ（X）与^—apcβ（X）,得到了两对对偶的上、下近似算子apc′β（X）与^—apc′β（X）和apc″β（X）与^—apc″β（X）;研究了这些近似算子的性质及相互关系。     

One for the Price of Two: a Unified Approach for Approximating Covering Problems

We present a simple and unified approach for developing and analyzing approximation algorithms for covering problems. We illustrate this on approximation algorithms for the following problems: Vertex Cover, Set Cover, Feedback Vertex Set, Generalized Steiner Forest, and related problems. The main idea can be phrased as follows: iteratively, pay two dollars (at most) to reduce the total optimum by one dollar (at least), so the rate of payment is no more than twice the rate of the optimum reduction. This implies a total payment (i.e., approximation cost) ≤ twice the optimum cost. Our main contribution is based on a formal definition for covering problems, which includes all the above fundamental problems and others. We further extend the Bafna et al. extension of the Local-Ratio theorem. Our extension eventually yields a short generic r -approximation algorithm which can generate most known approximation algorithms for most covering problems. Another extension of the Local-Ratio theorem to randomized algorithms gives a simple proof of Pitt's randomized approximation for Vertex Cover. Using this approach, we develop a modified greedy algorithm, which for Vertex Cover gives an expected performance ratio ≤ 2 . Received September 17, 1997; revised March 5, 1998.     

Improved Approximation Algorithms for Maximum Resource Bin Packing and Lazy Bin Covering Problems

In this paper, we study two variants of the bin packing and covering problems called Maximum Resource Bin Packing (MRBP) and Lazy Bin Covering (LBC) problems, and present new approximation algorithms for them. For the offline MRBP problem, the previous best known approximation ratio is frac65frac{6}{5} (=1.2) achieved by the classical First-Fit-Increasing (FFI) algorithm (Boyar et al. in Theor. Comput. Sci. 362(1–3):127–139, 2006). In this paper, we give a new FFI-type algorithm with an approximation ratio of frac8071frac{80}{71} (≈1.12676). For the offline LBC problem, it has been shown in Lin et al. (COCOON, pp. 340–349, 2006) that the classical First-Fit-Decreasing (FFD) algorithm achieves an approximation ratio of frac7160frac{71}{60} (≈1.18333). In this paper, we present a new FFD-type algorithm with an approximation ratio of frac1715frac{17}{15} (≈1.13333). Our algorithms are based on a pattern-based technique and a number of other observations. They run in near linear time (i.e., O(nlog n)), and therefore are practical.     

覆盖Value集

覆盖粗糙集和Vague集都是处理不确定性问题的数学工具,它们分别是粗糙集和模糊集的扩展。已有的覆盖粗糙集模型在求上、下近似时,可能将一些实际上并非肯定属于给定集合的元素纳入到下近似中,而一些可能属于给定集合的元素却没有纳入到上近似中,这就会改变一些元素与给定集合的关系。通过深入分析论域中的元素与其相关覆盖元之间的关系,建立了覆盖Vague集。该覆盖Vague集能够从一种新的角度反映出论域中各元素与给定集合之间的从属程度。进一步研究了覆盖Vague集与覆盖粗糙集中一些重要概念之间的关系。最后讨论了当覆盖退化为划分时覆盖Vague集的特性。     

Covering numbers of rotations

For an irrational rotation, we use the symbolic dynamics on the sturmian coding to compute explicitly, according to the continued fraction approximation of the argument, the measure of the largest Rokhlin stack made with intervals, and the measure of the largest Rokhlin stack whose levels have one name for the coding. Each one of these measures is equal to one if and only if the argument has unbounded partial quotients.     

Covering a string

We consider the problem of finding the repetitive structures of a given stringx. The periodu of the stringx grasps the repetitiveness ofx, sincex is a prefix of a string constructed by concatenations ofu. We generalize the concept of repetitiveness as follows: A stringw covers a stringx if there is a superstring ofx which is constructed by concatenations and superpositions ofw. A substringw ofx is called aseed ofx ifw coversx. We present anO(n logn)-time algorithm for finding all the seeds of a given string of lengthn.Partially supported by SERC Grants GR/F 00898 and GR/J 17844, NATO Grant CRG 900293, ESPRIT BRA Grant 7131 for ALCOMII, and MRC Grant G 9115730.Partially supported by MRC Grant G 9115730 and S.N.U. Posco Research Fund 94-15-1112.     

Covering with Ellipses

We address the problem of how to cover a set of required points by a small number of axis-parallel ellipses that avoid a second set of forbidden points. We study geometric properties of such covers and present an efficient randomized approximation algorithm for the cover construction. This question is motivated by a special pattern recognition task where one has to identify ellipse-shaped protein spots in two-dimensional electrophoresis images.     

Covering with Ellipses

We address the problem of how to cover a set of required points by a small number of axis-parallel ellipses that avoid a second set of forbidden points. We study geometric properties of such covers and present an efficient randomized approximation algorithm for the cover construction. This question is motivated by a special pattern recognition task where one has to identify ellipse-shaped protein spots in two-dimensional electrophoresis images.     

On Rooted Node-Connectivity Problems

Let G be a graph which is k -outconnected from a specified root node r , that is, G has k openly disjoint paths between r and v for every node v . We give necessary and sufficient conditions for the existence of a pair rv,rw of edges for which replacing these edges by a new edge vw gives a graph that is k -outconnected from r . This generalizes a theorem of Bienstock et al. on splitting off edges while preserving k -node-connectivity. We also prove that if C is a cycle in G such that each edge in C is critical with respect to k -outconnectivity from r , then C has a node v , distinct from r , which has degree k . This result is the rooted counterpart of a theorem due to Mader. We apply the above results to design approximation algorithms for the following problem: given a graph with nonnegative edge weights and node requirements c _u for each node u , find a minimum-weight subgraph that contains max {c _u ,c _v } openly disjoint paths between every pair of nodes u,v . For metric weights, our approximation guarantee is 3 . For uniform weights, our approximation guarantee is \min{ 2, (k+2q-1)/k} . Here k is the maximum node requirement, and q is the number of positive node requirements. Received September 15, 1998; revised March 10, 2000, and April 17, 2000.     

变精度覆盖近似算子与覆盖近似算子的关系

变精度覆盖粗糙集模型是在放宽了覆盖标准的前提下给出的,因而导致近似算子发生了变化,但其变化有一定的规律。在介绍了覆盖粗糙集模型和变精度覆盖粗糙集模型的概念的基础上,给出并证明了变精度覆盖粗糙近似算子与覆盖粗糙近似算子之间的关系,即定理1、定理2及其推论。