Flexible Graph Connectivity Approximating Network Design Problems Between 1- and 2-Connectivity

摘要

Graph connectivity and network design problems are among the most fundamental problems in combinatorial optimization. The minimum spanning tree problem, the two edge-connected spanning subgraph problem (2-ECSS) and the tree augmentation problem (WTAP) are all examples of fundamental well-studied network design tasks that postulate different initial states of the network and different assumptions on the reliability of network components. In this paper we motivate and study Flexible Graph Connectivity (FGC), a problem that mixes together both the modeling power and the complexities of all aforementioned problems and more. In a nutshell, FGC asks to design a connected network, while allowing to specify different reliability levels for individual edges. In this paper we develop a general algorithmic approach for approximating FGC that yields approximation algorithms with ratios that are close to the known best bounds for many special cases, such as 2-ECSS and WTAP. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min-max-min optimization problem.