In this chapter, we study the survivable network design problem. Given an undirected graph G = (V, E) and a connectivity requirement ruv for each pair of vertices u, v, a Steiner network is a subgraph of G in which there are at least ruv edge-disjoint paths between u and v for every pair of vertices u, v. The survivable network design problem is to find a Steiner network with minimum total cost. In the first part of this chapter, we will present the 2-approximation algorithm given by Jain [75] for this problem. We will present his original proof, which introduced the iterative rounding method to the design of approximation algorithms.

Interestingly, we will see a close connection of the survivable network design problem to the traveling salesman problem. Indeed the linear program, the characterization results, and presence of edges with large fractional value are identical for both problems. In the (symmetric) TSP, we are given an undirected graph G = (V, E) and cost function c: E → ℝ+, and the task is to find a minimum-cost Hamiltonian cycle. In the second part of this chapter, we will present an alternate proof of Jain's result, which also proves a structural result about extreme point solutions to the traveling salesman problem.

In the final part of this chapter, we consider the minimum bounded degree Steiner network problem, where we are also given a degree upper bound Bv for each vertex v ∈ V, and the task is to find a minimum-cost Steiner network satisfying all the degree bounds.