Exact and approximation algorithms for network flow and disjoint-path problems.

Abstract

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 $O(n\sp2$ log n) 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 $t\sb{i}$ and real-valued demands $\rho\sb{i}, 1\leq i \leq k$. We seek to route the demand $\rho\sb{i}$ of each commodity i along a single s-$t\sb{i}$ flow path, so that the total flow routed across any edge e is bounded by the edge capacity $u\sb{e}$. 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 the previously known and achieve significant improvements upon the approximation ratios. In a packing integer program, we seek a vector x of integers, which maximizes $c\sp{T}{\cdot}x$, subject to $Ax\leq b, A, b, c\geq 0.$ The edge and vertex-disjoint path problems together with their multiple-source unsplittable flow generalization are NP-hard problems with a multitude of applications in areas such as routing, scheduling and bin packing. We explore the topic of approximating disjoint-path problems using polynomial-size packing integer programs. Motivated by the disjoint paths applications, we initiate the study of a class of packing integer programs, called column-restricted. We derive approximation algorithms for column-restricted packing integer programs that we believe are of independent interest.