Approximation Algorithms For Minimum-Cost Low-Degree Subgraphs

In this thesis we address problems in minimum-cost low-degree network design. In the design of communication networks we often face the problem of building a network connecting a large number of end-hosts. Available hardware or software often imposes additional restrictions on the topology of the network. One example for such a requirement arises in the area of computer networks: We are given a number of client hosts in a network that communicate with each other using a certain communication protocol that limits the number of simultaneously open connections for each host. In graph theoretic terms, where hosts correspond to nodes and network connections are represented by edges, this restriction naturally translates to a bound on the maximum node-degree of the constructed network. In this thesis we first address the problem of finding a spanning tree T for a given undirected graph G = (V, E) with maximum node-degree at most a given parameter B > 1. Among all such trees, we aim to find one that minimizes the total edge-cost for a given cost function c on the edges of G. We develop an algorithm based on Lagrangean relaxation. We show how to compute a spanning tree with maximum node-degree O(B + log(n)) and total cost at most a constant factor worse than the cost of an optimum degree-B-bounded spanning tree in an n-node network. We present a second algorithm that is also based on ideas from Lagrangean relaxation but does not rely on computing a solution to a linear program. The new method can handle non-uniform degree-bounds, i.e. we are given integers B v > 1 for all v ∈ V and the degree of each node v ∈ V is constrained to be at most B v in any feasible solution. The algorithm uses a repeated application of Kruskal's MST algorithm interleaved with a combinatorial update of approximate Lagrangean node-multipliers maintained by the algorithm. These updates cause subsequent repetitions of the spanning tree algorithm to run for longer and longer times, leading to overall progress and a proof of the performance guarantee. Finally, we show how to extend the second algorithm to the case of Steiner trees where we use a primal-dual approximation algorithm due to Agrawal, Klein, and Ravi in place of Kruskal's minimum-cost spanning tree algorithm. The algorithm computes a Steiner tree of maximum degree O(B + log n) and total cost that is within a …