Efficient algorithms for optimal homology problems and their applications

TL;DR

This paper introduces polynomial-time algorithms for solving optimal homology problems using a multiscale flat norm approach, with applications in geometric network comparison and infrastructure validation.

Contribution

It develops a min-cost flow formulation for optimal homology problems, establishes duality and optimality conditions, and applies the flat norm to compare geometric networks efficiently.

Findings

Polynomial-time algorithms for mSFN in certain complexes.

Flat norm effectively captures network differences.

Application to real power networks demonstrates practical utility.

Abstract

The multiscale simplicial flat norm (MSFN) of a d-cycle is a family of optimal homology problems indexed by a scale parameter {\lambda} >= 0. Each instance (mSFN) optimizes the total weight of a homologous d-cycle and a bounded (d + 1)-chain, with one of the components being scaled by {\lambda}.We propose a min-cost flow formulation for solving instances of mSFN at a given scale {\lambda} in polynomial time in the case of (d + 1)-dimensional simplicial complexes embedded in {R^(d + 1)} and homology over Z. Furthermore, we establish the weak and strong dualities for mSFN, as well as the complementary slackness conditions. Additionally, we prove optimality conditions for directed flow formulations with cohomology over Z+. Next, we propose an approach based on the multiscale flat norm, a notion of distance between objects defined in the field of geometric measure theory, to compute the distance between a pair of planar geometric networks. Using a triangulation of the domain containing the input networks, the flat norm distance between two networks at a given scale can be computed by solving a linear program. In addition, this computation automatically identifies the 2D regions (patches) that capture where the two networks are different. We demonstrate through 2D examples that the flat norm distance can capture the variations of inputs more accurately than the commonly used Hausdorff distance. As a notion of stability, we also derive upper bounds on the flat norm distance between a simple 1D curve and its perturbed version as a function of the radius of perturbation for a restricted class of perturbations. We demonstrate our approach on a set of actual power networks from a county in the USA. Our approach can be extended to validate synthetic networks created for multiple infrastructures such as transportation, communication, water, and gas networks.