New approximation algorithms and structural results for oblivious multicommodity flow and zero-extension

This dissertation concerns approximation algorithms for oblivious multicommodity flow and zero-extension, both of which problems are fundamental to the theory of algorithms. Oblivious multicommodity flow is the problem of designing an oblivious routing scheme for arbitrary multicommodity flow demands in a fixed, capacitated graph. An oblivious routing scheme for a graph G is a specification, for each pair u, v of vertices, of a probability distribution over simple u − v paths in G; when required to send some amount of flow from u to v, the scheme sends it fractionally over all such paths with the appropriate probability for each one. In addition, the probability distributions for different vertex pairs must be independent of one another; as a consequence, the routing paths for different vertex pairs do not depend upon one another, and hence the scheme is called oblivious. Algorithmically, our goal is to pick the probability distributions so as to minimize the maximum congestion on any edge relative to the best possible. Zero-extension is a graph partitioning problem that is a direct generalization of the multi-way cut problem (Dahlhaus et al. 1994). For a graph G with a cost function on edges, and a set of terminals T among the vertices, a zero-extension of T is a function from vertices to terminals. Given in addition a metric on T, our algorithmic goal is to find such a function that minimizes the sum, over all edges, of the cost of the edge times the distance between the assignments of its endpoints under the metric. Our contributions are as follows: first, we give approximation algorithms that significantly improve the state of the art and our understanding of these problems. Second, we investigate how the techniques developed for and used in these algorithms have applications to better algorithms for a host of related theoretical problems. Finally, we formulate abstract structural results fundamental to our work which, we argue, are the key to understanding these and perhaps future problems.