The cone of flow matrices: Approximation hierarchies and applications

Abstract

Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone scriptK of flow matrices, which is a polyhedral cone generated by the matrices bold1Pboldnormal1PT∈ℝn×n, where boldnormal1P∈ℝn is the incidence vector of the (s,t)− path P. We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc‐representation of a flow, reduce to a linear optimization problem over scriptK. This cone is intractable: we prove that the membership problem associated to scriptK is NP‐complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial‐time the decomposition of a matrix X∈span K as a linear combination of some boldnormal1Pboldnormal1PT's. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of scriptK. We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc‐capacities.

New advances and applications in deterministic and stochastic network optimization

“…Sagnol et al introduce the concept of the cone K of flow matrices. The authors show that several hard flow (or path) optimization problems that cannot be solved by using the standard arc‐representation of a flow, reduce to a linear optimization problem over the cone K .…”

“…Sagnol et al [6] introduce the concept of the cone K of flow matrices. The authors show that several hard flow (or path) optimization problems that cannot be solved by using the standard arc-representation of a flow, reduce to a linear optimization problem over the cone K. The authors prove that the membership problem associated to K is NP-complete and provide two convergent approximation hierarchies, one of them based on a completely positive representation of K. This approach is illustrated by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc-capacities.…”