Approximation through local optimality: Designing networks with small degree

Abstract

We give quasipolynomial-time approximation algorithms for designing networks with minimum degree. Using our methods, one can design one-connected networks to meet a variety of connectivity requirements. The degree of the output network is guaranteed to be at most (1 + ε) times optimal, plus an additive error of O(log n/ε) for any ε > 0. We also provide a quasipolynomial-time approximation algorithm for designing a two-edge-connected spanning subgraph of a given two-edgeconnected graph of approximately minimum degree. The performance guarantee is identical to that for one-connected networks. As a consequence of our analysis, we show that the minimum degree in both the problems above is well-estimated by certain polynomially solvable linear programs. This fact suggests that the linear programs we describe could be useful in obtaining optimal solutions via branch-and-bound.