Consider a rooted directed graph G with a subset of vertices called terminals, where each arc has a positive integer capacity and a non-negative cost. For a given positive integer k, we say that G is k-survivable if every of its subgraphs obtained by removing at most k arcs admits a feasible flow that routes one unit of flow from the root to every terminal. We aim at determining a k-survivable subgraph of G of minimum total cost. We focus on the case where the input graph G is planar and propose a tabu search algorithm whose main procedure takes advantage of planar graph duality properties. In particular, we prove that it is possible to test the k-survivability of a planar graph by solving a series of shortest path problems. Experiments indicate that the proposed tabu search algorithm produces optimal solutions in a very short computing time, when these are known.

考虑一个有根的有向图G，它有一个称为端子的顶点子集，其中每个弧具有正整数容量和非负成本。对于给定的正整数k，如果说通过去除最多k个弧获得的每个子图都接受了一个可行的流动，该流动将单位流量从根路由到每个终端，则G是k-可存活的。我们旨在确定最小总成本的G的k个可生存子图。我们关注输入图G的情况提出一种禁忌搜索算法，该算法的主要过程利用了平面图对偶性。特别地，我们证明可以通过解决一系列最短路径问题来测试平面图的k-生存能力。实验表明，已知的禁忌搜索算法可以在很短的计算时间内产生最优解。