Combinatorics with Applications

结合应用 杰罗尔德河Griggs和Laszlo A. SzekelyThe proposers寻求结果在极值组合，展示结构是关键的应用程序，或提供边界上可以实现的任何算法。 提议者将寻找一个有利可图的交叉的想法和方法，包括代数和分析工具的应用。 具体的问题包括 * 改进当前的子集和的最大集中度的界限， 矢量，特别是数据库安全模型所产生的问题 * 找到更好的算法，图形绘制，发展理论 交叉数的图，并制定出进一步的应用程序， 的理论几何问题 * 开发强大的算法重建非常大的系统发育 树，并分析重建所需的条件， 注意模型中的i.i.d.属性失败 * 完成中的最大非生成集的求解 有限交换群 * 推进大独立集算法的研究 在给定最大度和团大小的图中 * 对允许小阈值函数的图细分问题进行分类 * 领导了图标号理论的发展， 具有多级干扰的最优信道分配问题组合学和图论的研究在广泛的领域中提出，包括那些出现在计算机科学、计算生物学和数论等快速发展的领域中的研究。 问题出现在我们理解离散结构如何工作以及如何最佳使用它们的核心。 他们将研究的问题包括： * 如何优化公众对统计数据库的访问 （包含，比如说，一个部门员工的工资）， 许多查询，每一个都要求一些人的平均工资 员工的集合，可以在不损害 任何个人的工资？* 给定许多人的DNA序列的相应对齐片段， 生物分类群（或物种），如何建立大型系统发育树 反映了它们真正的进化关系* 怎样才能画出一个非常大的网络，使观众能够抓住它呢 从一个清晰的图纸？ * 如何分配给发射机网络的频谱跨度 （无线电台、移动的电话等）最小化，这样通道 分配给附近的发射机必须避免干扰？* 选择大量发射机的有效方法是什么， 没有两个紧密联系在一起，给定一个网络和信息， 网络的复杂性？ 这类基本问题出现在 研究生参与这些问题的研究将使研究人员能够继续他们在工业、政府和学术界成功的职业培训计划。

COMBINATORICS WITH APPLICATIONS Jerrold R. Griggs and Laszlo A. SzekelyThe proposers seek results in extremal combinatorics that exhibit structures which are key to applications, or which provide bounds on what can be achieved by any algorithm. The proposers will search for a profitable crossover of ideas and methods, including applications of algebraic and analytic tools. The particular problems include * improving current bounds on the maximum concentration of subset sums of vectors, especially for problems arising from models of database security * finding better algorithms for graph drawing, developing the theory of crossing numbers of graphs, and working out further applications of of the theory to geometric problems * developing robust algorithms for reconstructing very large phylogenetic trees, and analyzing conditions necessary for reconstruction, with attention to models where the i.i.d. property fails * completing the solution of maximum non-spanning sets in finite abelian groups * advancing the study of algorithms for large independent sets in graphs of given maximum degree and clique size * classifying graph subdivision problems which admit small threshold function * leading the development of the theory of graph labellings spawned by the problem of optimal channel assignments with multiple levels of interferenceResearch in combinatorics and graph theory is proposed in a wide variety of areas, including those that arise in such rapidly developing fields as computer science, computational biology, and number theory. Problems arise at the very core of our understanding of how discrete structures work and how to use them optimally. Among the questions they will study are these: * How can one optimize public access to a statistical database (containing, say, salaries of a department's employees) by maximizing the number of queries, each one asking for the average salary of some collection of the employees, that can be answered without compromising the salary of any individual? * Given corresponding aligned segments of the DNA sequences of many biological taxa (or species), how does one build large phylogenetic trees reflecting their true evolutionary relationship? * How can one draw a very large network such that the viewer can grasp it from a clear drawing? * How can the frequency spectrum span alloted to a network of transmitters (radio stations, mobile phones, etc.) be minimized, such that channels assigned to nearby transmitters must avoid interference? * What is an efficient method to select a large number of transmitters, no two close together, given a network and information bounding the complexity of the network? Fundamental problems of this kind arise in many settings.Involvement of graduate students in research on these problems will enable the investigators to continue their successful training program for careersin industry, government, and academia.