平面图上网络流阻断的扩展
Extensions to Network Flow Interdiction on Planar Graphs
摘要：
网络流阻断分析研究的是，通过去除网络中受某个预算约束的组件，网络中的最大流的值可以减少多少。虽然这个问题在一般网络上是强NP完全的，但对于具有单个源和单个汇并且不可能移除顶点的平面网络，已经找到了伪多项式算法。在这项工作中，我们引入伪多项式算法，克服了以前的方法的一些限制。我们提出了一个平面保持变换，允许将顶点删除和顶点容量的伪多项式阻断算法的平面图。此外，一个伪多项式算法被引入的问题，确定最小的阻断预算，这是至少需要使它不可能满足所有的sink节点的需求，在平面网络与多个源和sink满足的供应之和的总和在源节点等于sink节点的需求之和。此外，我们证明了平面图上的k-可分配子图问题可以归结为一个平面图上的网络流阻断问题，该平面图具有多个源和汇以及多项式有界的输入数。然而，仍然不知道这些问题中的任何一个是否可以在多项式时间内解决。
英文摘要：
Network flow interdiction analysis studies by how much the value of a maximum flow in a network can be diminished by removing components of the network constrained to some budget. Although this problem is strongly NP-complete on general networks, pseudo-polynomial algorithms were found for planar networks with a single source and a single sink and without the possibility to remove vertices. In this work we introduce pseudo-polynomial algorithms which overcome some of the restrictions of previous methods. We propose a planarity-preserving transformation that allows to incorporate vertex removals and vertex capacities in pseudo-polynomial interdiction algorithms for planar graphs. Additionally, a pseudo-polynomial algorithm is introduced for the problem of determining the minimal interdiction budget which is at least needed to make it impossible to satisfy the demand of all sink nodes, on planar networks with multiple sources and sinks satisfying that the sum of the supplies at the source nodes equals the sum of the demands at the sink nodes. Furthermore we show that the k-densest subgraph problem on planar graphs can be reduced to a network flow interdiction problem on a planar graph with multiple sources and sinks and polynomially bounded input numbers. However it is still not known if either of these problems can be solved in polynomial time.
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