Approximation Schemes for Minimum-Cost k-Connectivity Problems in Geometric Graphs

We survey the recent progress in the design of approximation schemes for geometric variants of the following classical optimization problem: for a given undirected weighted graph, find its minimum-cost subgraph that satisfies a priori given multi-connectivity requirements. We present the approximation schemes for various geometric minimum-cost k-connectivity problems and for geometric survivability problems, giving a detailed tutorial of the novel techniques developed for these algorithms. We also shortly discuss extensions to include planar graphs. A classical multi-connectivity graph problem is as follows: for a given undirected weighted graph, find its minimumcost subgraph that satisfies a priori given connectivity requirements. Multi-connectivity graph problems are central in algorithmic graph theory and have numerous applications in computer science and operation research, see, e.g., [1, 22, 18, 34, 35]. They also play very important role in the design of networks that arise in practical situations, see, e.g., [1, 22, 30]. Typical application areas include telecommunication, computer and road networks. Low degree connectivity problems for geometrical graphs in the plane can often closely approximate such practical connectivity problems (see, e.g., the discussion in [22, 32, 35]). For instance, they can be used to model the design of low-cost telephone networks that can “survive” some types of edge and node failure. In such a model, the cost of the edge corresponds to the cost of lying a fiber-optic cable between the endpoints of the edge plus the planned cost of the service of the cable. Furthermore, the minimum connectivity requirement for a pair Research supported in part by NSF ITR grant CCR-0313219 and by VR grant 621-2002-4049.