环形载荷问题的一个改进的上界
An Improved Upper Bound for the Ring Loading Problem
摘要：
环负载问题出现于20世纪90年代，是电信网络（SONET环）的一个重要特例，引起了实践者和理论家的关注。给定$n$节点上的无向循环以及任何一对节点之间的非负需求，环加载问题要求需求的不可拆分路由，使得任何边缘上的最大累积需求最小化。令$L$为该解的值。在这个问题的放松版本中，每个需求可以分为两部分，其中第一部分顺时针路由，而第二部分逆时针路由。用$L^*$表示最小拆分路由解决方案的最大负载。在一篇具有里程碑意义的论文中，Schrijver、Seymour和Winkler [SSW 98]证明了$L \leq L^* + 1.5D$，其中$D$是最大需求值。他们还发现（隐含地）一个使用$L = L^* + 1.01D$的环加载问题的实例。最近，Skutella [Sku16]通过证明$L \leq L^* + \frac{19}{14}D$改进了这些边界，并且存在$L = L^* + 1.1D$的实例。我们通过证明$L \leq L^* + 1.3D$ .我们还采取了第一步，对小的情况下，下限和上限。
英文摘要：
The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on $n$ nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let $L$ be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with $L^*$ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that $L \leq L^* + 1.5D$, where $D$ is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with $L = L^* + 1.01D$. Recently, Skutella [Sku16] improved these bounds by showing that $L \leq L^* + \frac{19}{14}D$, and there exists an instance with $L = L^* + 1.1D$. We contribute to this line of research by showing that $L \leq L^* + 1.3D$. We also take a first step towards lower and upper bounds for small instances.
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