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

TL;DR

This paper introduces Flexible Graph Connectivity (FGC), a new network design problem that allows for different reliability levels on edges, and develops approximation algorithms that perform well on this complex, non-uniform problem.

Contribution

It proposes a general algorithmic framework for approximating FGC, unifying and extending techniques for related network design problems.

Findings

Approximation algorithms with ratios close to best known bounds for special cases.

A novel combination of weight-scaling, exchange bijections, and min-max-min optimization techniques.

Effective handling of non-uniform reliability levels in network design.

Abstract

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 (TAP) 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 \emph{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. While this non-uniform nature of the problem makes it appealing from the modeling perspective, it also renders most existing algorithmic tools for dealing with network design problems unfit for approximating FGC. In this paper we develop a general algorithmic approach for approximating FGC that yields approximation algorithms with ratios that are very close to the best known bounds for many special cases, such as 2-ECSS and TAP. 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.