Development of Optimal Algorithms for Partitioning Geometric Data

在过去的两年里，我们一直在尝试开发一种有效的算法，以优化分区几何数据，将其作为二维网格上的平面和像素数据的点集。我们研究过的问题和结果表明，我们发现的问题是最大的，正如我们所遵循的那样。（1）我们可以在一个带有单一仓库的树形网络上交易车辆。客户位于树木的各个部分，每个客户都有积极的需求。客户的需求是由一辆有有限容量的固定车辆提供的。这就意味着一个客户的需求是不可分割的。我们考虑了一些问题，以找到一套最低总长度的旅行。在第一年，我们展示了这个问题是NP--完整的，并提出了一个1.5-近似算法。我们还做了一些计算实验。在第二年，近似值提高到了1.35，并进一步完善了第一个算法。（2）我们考虑在多个最优标准下找到一个最优间隔的数组和一个区域中的两个最优间隔的问题。在具体情况下，我们将考虑找出一个间隔I [1，n]的问题，以最大化interclass variance。我们将为这个问题提供O（n log n）时间算法。我们将此算法扩展到两个维度的情况。例如，给了一个N×N二维数组，寻找一个具有最大互类变量的矩形子数组R的问题。We Management ed an O（NイイD13イエD1）algorithm。（3）我们考虑了找到两个维度关联规则的问题，并根据半定义编程开发了一个算法。

Over the last two years, we have tried to develop efficient algorithms for optimally partitioning geometric data such as point set in the plane and pixel data on two-dimensional grid. The problems we studied and results we obtained are summarized as follows.(1) we deal with a vehicle routing on a tree-shaped network with a single depot. Customers are located on vertices of the tree, and each customer has a positive demand. Demands of customers are served by a fleet of identical vehicles with limited capacity. It is assumed that the demand of a customer is splittable. We considered the problem for finding a set of tours with minimum total lengths. In the first year, we showed that the problem is NP-complete and proposed a 1.5-approximation algorithm for the problem. We also performed some computational experiments. In the second year, the approximation ration is improved to 1.35 by further refining the first algorithm.(2) We considered the problem of finding an optimal interval in one-dimensional array and a region in two-dimensional array under several optimality criteria. In particular, we shall consider the problem of finding an interval I∈[1,n] that maximizes the interclass variance. We shall present an O(n log n)time algorithm for this problem. We then extend this algorithm to two-dimensional case. Namely, given a N×N two-dimensional array, the problem seeks to find a rectangular subarray R with maximum interclass variance. We developed an O(NィイD13ィエD1)algorithm.(3) We considered the problem of finding two-dimensional association rules for categorical attributes and developed an algorithm based on semidefinite programming.