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What Would Edmonds Do? Augmenting Paths and Witnesses for DegreeBounded MSTs
 IN PROCEEDINGS OF APPROX/RANDOM
, 2005
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Degree Bounded Network Design with Metric Costs
"... Given a complete undirected graph, a cost function on edges and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for simple connectivity requirement such as finding a spanning ..."
Abstract

Cited by 7 (3 self)
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Given a complete undirected graph, a cost function on edges and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NPhard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies triangle inequalities, there are constant factor approximation algorithms for various degree bounded network design problems. • Global edgeconnectivity: There is a (2 + 1 k)approximation algorithm for the minimum bounded degree kedgeconnected subgraph problem. • Local edgeconnectivity: There is a 6approximation algorithm for the minimum bounded degree Steiner network problem. • Global vertexconnectivity: There is a (2 + k−1 n + 1 k)approximation algorithm for the minimum bounded degree kvertexconnected subgraph problem. • Spanning tree: There is an (1 + 1 d−1)approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with smallest possible maximum degree, and the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides’ algorithm for metric TSP. The main technical tool is a simplicitypreserving edge splittingoff operation, which is used to “shortcut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.