# Network Design Via Iterative Rounding Of Setpair Relaxations

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A typical problem in network design is to find a minimum-cost sub-network *H* of a given network *G* such that *H* satisfies some prespecified connectivity requirements. Our focus is on approximation algorithms for designing networks that satisfy *vertex connectivity* requirements. Our main tool is a linear programming relaxation of the following *setpair formulation* due to Frank and Jordan: a *setpair* consists of two subsets of vertices (of the given network *G*); each setpair has an integer requirement, and the goal is to find a minimum-cost subset of the edges of *G* sucht hat each setpair is covered by at least as many edges as its requirement. We introduce the notion of *skew bisupermodular* functions and use it to prove that the basic solutions of the linear program are characterized by “non-crossing families” of setpairs. This allows us to apply Jain’s *iterative rounding method* to find approximately optimal integer solutions. We give two applications. (1) In the *k-vertex connectivity problem* we are given a (directed or undirected) graph *G*=(*V,E*) with non-negative edge costs, and the task is to find a minimum-cost spanning subgraph *H* such that *H* is *k*-vertex connected. Let *n*=|*V*|, and let ε<1 be a positive number such that *k*≤(1−ε)*n*. We give an \(
O{\left( {{\sqrt {n/ \in } }} \right)}
\)-approximation algorithm for both problems (directed or undirected), improving on the previous best approximation guarantees for *k* in the range \(
\Omega {\left( {{\sqrt n }} \right)} < k < n - \Omega {\left( 1 \right)}
\). (2)We give a 2-approximation algorithm for the *element connectivity* problem, matching the previous best approximation guarantee due to Fleischer, Jain and Williamson.

## *Mathematics Subject Classification (2000):*

68W25 90C35 05C40 68R10 90C27 90B10 ## Preview

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