# A 2-Approximation for Minimum Cost {0, 1, 2} Vertex Connectivity

## Abstract

In survivable network design, each pair (*i*, *j*) of vertices is assigned a level of importance *r* _{ij}. The vertex connectivity problem is to design a minimum cost network such that between each pair of vertices with importance level *r*, there are *r vertex* disjoint paths. There is no approximation algorithm known for this general problem. In this paper, we give a 2-approximation for the problem when *r* ∈ {0, 1, 2}^{V xV}, improving on a previous known 3-approximation. This matches the best known approximation for the easier problem that requires that the paths be only edge-disjoint.

Our algorithm extends an iterative rounding algorithm that gives a 2-approximation for the edge-connectivity problem, for arbitrary connectivity requirements *r*. (K. Jain, A factor 2 approximation for the gen- eralized Steiner network problem.) This algorithm relies on well-known uncrossing lemma for tight edge cutsets. Our extension uses a new type of uncrossing lemma for tight cutsets that may include vertices as well as edges.

For *r* ∈ {1, *k*}^{V xV}, *k* ≥ 3, we show that a) uncrossing tight cutsets is not possible, and b) any analysis for iterative rounding that depends directly on the largest fractional value in the linear programming solution cannot provide approximation guarantees better than the maximum connectivity requirement.

## Keywords

Approximation Algorithm Network Design Problem Basic Feasible Solution Vertex Connectivity Connectivity Problem## Preview

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