Resilient Broadcasting via Independent Spanning-Trees

衡量通信网络弹性的经典理论指标是其边缘连通性。然而，对于一些网络问题，这一措施并不为人所知。例如，在弹性广播中，一个规定的顶点r通过一组生成树与每一个其他顶点通信，使得对于每个顶点v，这些生成树中从r到v的路径是边不相交的(这种树被称为独立的)。但目前尚不清楚网络中独立生成树的最大数目与其边连通性的确切关系，也没有更多关于顶点故障的类似问题或寻找这种独立生成树的复杂性的已知。事实上，图论中由来已久的边独立生成树猜想表明，每个k边连通网络包含k棵独立生成树。这一猜想的证明不仅将描述具有弹性广播的网络的特征，而且可以说还提供了这种网络的紧凑设计和其中有效的路由方案所必需的结构洞察力。尽管最近关于小k的这个猜想在结构和算法上取得了进展(在这种情况下猜想是真的)，但到目前为止还没有得到推广到更高k的结果。这个项目的目的是利用最近提出的结构来攻击高k的边独立生成树猜想。我们将使用图论方法来搜索正确的推广到更高的k，但也计算机辅助算法来支持这种搜索。

A classic theoretical measure for the resilience of a communication network is its edge-connectivity. However, for some network problems, this measure is not known to be suitable at all. In resilient broadcasting, for example, one prescribed vertex r communicates with every other vertex through a set of spanning trees such that, for every vertex v, the paths from r to v in these spanning trees are edge-disjoint (such trees are called independent). But it is unknown how exactly the maximal number of independent spanning trees in a network relate to its edge-connectivity, and nothing more is known for the analogous question regarding vertex-failures or for the complexity of finding such independent spanning trees.In fact, the long-standing Edge-Independent Spanning Tree Conjecture in graph theory states that every k-edge-connected network contains k independent spanning trees. A proof of this conjecture would not only characterize the networks in which resilient broadcasting is possible, but arguably also deliver the structural insights that are necessary for the compact design of such networks and for efficient routing schemes in these. Although there is recent structural and algorithmic progress on this conjecture for small k (in which case the conjecture is true), a generalization to higher k was not achieved so far.This project aims at attacking the Edge-Independent Spanning Tree Conjecture for higher k, using the recently proposed structures. We will use graph-theoretic methods in order to search for the right generalization to higher k but also computer-assisted algorithms to support this search.