Efficient algorithms for optimal homology problems and their applications

TL;DR

This work develops a unified, flow-based framework for solving Optimal Homology Problems, notably the Optimal Homologous Cycle Problem (OHCP) and Multiscale SImplicial Flat Norm (MSFN), by embedding finite complexes in Euclidean space and utilizing dual-flow networks. A key innovation is the hat-augmentation that preserves homology while enabling strong duality results, along with extended analyses for directed (nonnegative) coefficients. The authors derive weak and strong duality theorems, optimality and complementary slackness conditions, and show how these lead to polynomial-time algorithms on embedded complexes via min-cost flow implementations. They further study the stability of the flat norm as a geometry-distance measure, comparing it with Hausdorff distance and establishing perturbation bounds with practical implications for validating synthetic power-distribution networks against real data. The practical impact is demonstrated through 2D planar network comparisons and a detailed structural validation pipeline for synthetic power grids, combining rigorous theory with scalable computational tools.

Abstract

The multiscale simplicial flat norm (MSFN) of a d-cycle is a family of optimal homology problems indexed by a scale parameter λ >= 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 λ.We propose a min-cost flow formulation for solving instances of mSFN at a given scale λ 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.

Figures (29)

• Figure 1: Natural orientation and duality. Left: A facet $\tau$ is on the left of the blue edge $e$, and an inner void $\nu_{{in}}$ is on the right. Middle: The positive direction is clockwise, and hence $e$ is negeative on the boundary of $\tau$ and positive on the boundary of $\nu_{{in}}$. Therefore, the positive dual di-arc $(\tau^{*} \to \nu_{{in}})$ (black in the middle) is oriented from left to right. Right: Negative dual di-arcs are shown in red.
• Figure 2: Left: Example of a $(q + 1)$-complex $\mathcal{K}$ with three inner voids (referred to as blue, green, and red). Middle: Its acyclization $\bar{\mathcal{K}}$ with inner voids plugged by corresponding $(q + 1)$-cells. The cell corresponding to the outer void $\nu_0$ occupies the whole space outside of the complex. Middle: The dual graph $\mathcal{K}^{*}$ with a node for every $(q + 1)$-cell of $\bar{\mathcal{K}}$ (the brown node corresponds to $\nu_0$). Every undirected edge in this figure represents a pair of opposite dual arcs, see Fig. \ref{['fig:ortanddual-clockwise-02']} for details.
• Figure 3: The two kinds of $q$-cycles $\bm{z}$: those that curl in a single direction (Left), and those that curl in both directions -- mixed curling (Right). Double arrows mean that the corresponding $q$-edges appear with coefficient $2$ in the input $q$-cycle.
• Figure 4: The dual cochain $\mathop{\mathrm{dist}}\limits^{*}_{\bm{z}}$ that is constructed for the two examples from Fig. \ref{['fig:types-of-cycles']}. The node's value is defined by its shortest path distance from $\nu_0^{*}$, with the length of dual arcs given by the value of the input $q$-cycle $\bm{z}$. The node's color represents the sign of $\mathop{\mathrm{dist}}\limits^{*}_{\bm{z}}$, and hence the direction of $\bm{z}$ curling around the corresponding $(q + 1)$-cell: green is positive (CW curl), red is negative (CCW curl), and grey is for nodes that are outside of $\bm{z}$, i.e. $\mathop{\mathrm{dist}}\limits_{\bm{z}}$ is zero on such nodes.
• Figure 5: The dual flow networks produced by Algorithm \ref{['algo:ohcp:flow-network']} for the two examples in Fig. \ref{['fig:types-of-cycles']}. Left: In the case of a $q$-cycle curling in a single direction, as a consequence of the partition in Eq. \ref{['eq:recyclization-01']} and \ref{['eq:recyclization-02']}, the void-nodes are connected either to the source node $S$, or to the sink node $T$, but never to both. Right: On the other hand, the homology class of a mixed curling $q$-cycle always contains generators of both CW and CCW directions that, consequently, are connected to the $S$ and $T$ respectively. The voids that are "outside" of $\bm{z}$, i.e. those with $\mathop{\mathrm{dist}}\limits_{\bm{z}} = 0$, become transit nodes -- they are connected to both the source and sink nodes of the flow network. A number of nodes in an OHCP-network is $F+ \beta_q + 1 + 2 = F + \beta_q + 3$, and the number of di-arcs is at least $2E + \beta_q + 1 + 1 = 2E + \beta_q + 2$.

Theorems & Definitions (73)

• Definition 2.B.1
• Definition 2.B.2
• Corollary 2.B.3: Hodge Theorem
• Theorem 2.B.4: Integrality of TU, Theorem 2.1 ohcp2011
• Theorem 2.B.5: Theorem 5.2 ohcp2011
• Theorem 2.B.6: Theorem 4.1 ohcp2011
• Theorem 2.B.7: Theorem 5.7 ohcp2011
• Definition 2.B.8
• Definition 2.B.9
• Definition 2.B.10