Faster Approximate Lossy Generalized Flow via Interior Point Algorithms

TL;DR

This paper introduces faster approximation algorithms for lossy generalized network flow problems using interior-point methods and nearly linear time linear system solvers, significantly improving efficiency over previous algorithms.

Contribution

It develops a nearly linear time algorithm for solving symmetric M-matrix systems within interior-point methods for generalized flows, enhancing computational efficiency.

Findings

Achieves an pproximate solution in equation O(m^{3/2} log(1/)) time.

Improves previous algorithms by a factor of about m^{1/2} in many cases.

Extends the approach to standard min-cost flow problems with similar improvements.

Abstract

We present faster approximation algorithms for generalized network flow problems. A generalized flow is one in which the flow out of an edge differs from the flow into the edge by a constant factor. We limit ourselves to the lossy case, when these factors are at most 1. Our algorithm uses a standard interior-point algorithm to solve a linear program formulation of the network flow problem. The system of linear equations that arises at each step of the interior-point algorithm takes the form of a symmetric M-matrix. We present an algorithm for solving such systems in nearly linear time. The algorithm relies on the Spielman-Teng nearly linear time algorithm for solving linear systems in diagonally-dominant matrices. For a graph with m edges, our algorithm obtains an additive epsilon approximation of the maximum generalized flow and minimum cost generalized flow in time tildeO(m^(3/2) * log(1/epsilon)). In many parameter ranges, this improves over previous algorithms by a factor of approximately m^(1/2). We also obtain a similar improvement for exactly solving the standard min-cost flow problem.