A Generalized Approximation Framework for Fractional Network Flow and Packing Problems

TL;DR

This paper introduces a generalized approximation framework for fractional network flow and packing problems, enabling efficient algorithms for complex and diverse network optimization tasks with broad applicability.

Contribution

It extends the fractional packing framework to polyhedral cones with exponential size, providing new approximation algorithms accessible via oracles, simplifying existing solutions, and enabling FPTAS for various network flow problems.

Findings

Developed approximation algorithms for fractional packing over cones.

Enabled FPTAS for budget-constrained, multicommodity, and generalized flows.

Simplified proofs and extended applicability of network flow approximations.

Abstract

We generalize the fractional packing framework of Garg and Koenemann to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form $\max\{c^T x : Ax \leq b, x \in C \}$, where the matrix $A$ contains no negative entries and $C$ is a cone that is generated by a finite set $S$ of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approximability of the packing problem. In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort. In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows. Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof.