Exact and Approximation Algorithms for Network Flow and Disjoint - Path Problems

Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP -hard variants such as disjoint paths and unsplittable flow. Given an n-vertex, m-edge directed network G with real costs on the edges we give new algorithms to compute single-source shortest paths and the minimum mean cycle. Our algorithm is deterministic with O(n logn) expected running time over a large class of input distributions. This is the first strongly polynomial algorithm in over 35 years to improve upon some aspect of the O(nm) running time of the Bellman-Ford shortest-path algorithm. In the single-source unsplittable flow problem, we are given a network G, a source vertex s and k commodities with sinks ti and real-valued demands ρi, 1 ≤ i ≤ k. We seek to route the demand ρi of each commodity i along a single s-ti flow path, so that the total flow routed across any edge e is bounded by the edge capacity ue. This NP-hard problem combines the difficulty of bin-packing with routing through an arbitrary graph and has many interesting and important variations. We give a generic framework, which yields approximation algorithms that are simpler than