The Convex Geometry of Network Flows

TL;DR

This paper explores the convex geometry underlying network flow problems, introducing a conic form, analyzing nonconvex fixed-cost flows, and proposing practical rounding heuristics and future research directions.

Contribution

It presents a novel conic formulation of convex flows, analyzes fixed-cost nonconvex flows, and develops practical rounding schemes and heuristics.

Findings

Flow sets have a downward closure property.

Conic form is equivalent and nearly self-dual.

Nonconvex fixed-cost flows have near-integral solutions.

Abstract

In this paper, we derive a number of interesting properties and extensions of the convex flow problem from the perspective of convex geometry. We show that the sets of allowable flows always can be imbued with a downward closure property, which leads to a useful `calculus' of flows, allowing easy combination and splitting of edges. We then derive a conic form for the convex flow problem, which we show is equivalent to the original problem and almost self-dual. Using this conic form, we consider the nonconvex flow problem with fixed costs on the edges, i.e., where there is some fixed cost to send any nonzero flow over an edge. We show that this problem has almost integral solutions by a Shapley--Folkman argument, and we describe a rounding scheme that works well in practice. Additionally, we provide a heuristic for this nonconvex problem which is a simple modification of our original algorithm. We conclude by discussing a number of interesting avenues for future work.