Approximating Minimum Communication Cost Spanning Trees and Related Problems

The problem of designing a communication network for a given set of requirements has been studied extensively in the literature, and many di erent variants of it were considered and given either exact solutions or heuristics. We consider the problem of minimum communication cost spanning trees (MCT), introduced in [18] and listed as [ND7] in [15] and [11]. This problem requires selecting a spanning tree of a network that minimizes the total cost of transmitting a given set of communication requirements between n sites over the tree edges. The problem is NP-hard even if all the communication requirements are equal [20]. Besides natural applications to design of communication networks, algorithms for providing approximate solutions to the problem can be used to improve distributed algorithms, and, perhaps surprisingly, the problem is strongly related to a large number of approximation algorithms via probabilistic approximation of metric spaces ([4, 5, 9]) and spreading metrics ([13]). Thus, providing a better approximate solution to the problem directly implies better approximation for other NP-hard problems. In this thesis, we present deterministic algorithms that provide approximate solutions to several cases of the problem: A 3-approximation algorithm for k-source uniformMCT, i.e., when k designated nodes have uniform communication requirements to the nodes in the network. A 2-approximation algorithm for independent-requirements-MCT, where the requirement matrix R is a product of a vector and its transpose, i.e., R = P T P . We demonstrate the applicability of this approximation algorithm in improving the average case behavior of the arrow distributed directory protocol [12]. An O(d logn)-approximation algorithm for Euclidean-MCT, i.e., when the graph is embedded in d-dimensional Euclidean space. In particular, when the dimension is xed, the algorithm provides an approximation ratio of O(logn). The algorithm provides the same approximation ratio when the distance is measured according to any lp norm. An O(log2 n)-approximation algorithm for metric-MCT, i.e., when the graph is complete and metric, namely, the edge weights obey the triangle inequality. An O(2O(plog n log log n))-approximation algorithm for the general MCT problem, by preprocessing the input and applying the techniques of [1, 29]. We further show that the general case of MCT is MAX SNP-hard, and thus does not admit a polynomial time approximation scheme, unless P = NP.