A linear input dependence model for interdependent networks

TL;DR

This paper introduces a linear relaxation model for interdependent networks with binary input dependencies, proposing randomized rounding schemes and a generalized network simplex algorithm for efficient approximation of solutions.

Contribution

It presents a novel linear relaxation model for interdependent networks and develops a generalized network simplex algorithm for solving the associated optimization problem.

Findings

Effective randomized rounding schemes for the model

A generalized network simplex algorithm for efficient computation

Modeling of interrelated systems with input dependencies

Abstract

We consider a linear relaxation of a generalized minimum-cost network flow problem with binary input dependencies. In this model the flows through certain arcs are bounded by linear (or more generally, piecewise linear concave) functions of the flows through other arcs. This formulation can be used to model interrelated systems in which the components of one system require the delivery of material from another system in order to function (for example, components of a subway system may require delivery of electrical power from a separate system). We propose and study randomized rounding schemes for how this model can be used to approximate solutions to a related mixed integer linear program for modeling binary input dependencies. The introduction of side constraints prevents this problem from being solved using the well-known network simplex algorithm, however by characterizing its basis structure we develop a generalization of network simplex algorithm that can be used for its {computationally} efficient solution.