Approximate Parametric Computation of Minimum-Cost Flows with Convex Costs

TL;DR

This paper introduces two algorithms for approximately solving a parametric minimum-cost flow problem with convex costs, providing explicit error bounds and testing on real-world traffic and gas data.

Contribution

It develops two novel algorithmic approaches for approximate parametric minimum-cost flows with convex costs, including explicit error bounds and practical testing.

Findings

Both methods achieve controlled approximation errors.

Algorithms perform well on real-world traffic and gas network data.

Explicit bounds on step sizes ensure solution accuracy.

Abstract

This paper studies a variant of the minimum-cost flow problem in a graph with convex cost function where the demands at the vertices are functions depending on a one-dimensional parameter $\lambda$. We devise two algorithmic approaches for the approximate computation of parametric solutions for this problem. The first approach transforms an instance of the parametric problem into an instance with piecewise quadratic cost functions by interpolating the marginal cost functions. The new instance can be solved exactly with an algorithm we developed in prior work. In the second approach, we compute a fixed number of non-parametric solutions and interpolate the resulting flows yielding an approximate solution for the original, parametric problem. For both methods we formulate explicit bounds on the step sizes used in the respective interpolations that guarantee relative and absolute error margins. Finally, we test our approaches on real-world traffic and gas instances in an empirical study.