Generalized max-flows and min-cuts in simplicial complexes

Abstract

We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and show that computing such a cut is NP-hard. However, we provide conditions on the simplicial complex for when the cut found by the linear program is a combinatorial cut. For d-dimensional simplicial complexes embedded into R d+1 we provide algorithms operating on the dual graph: computing a maximum flow is dual to computing a shortest path and computing a minimum cut is dual to computing a minimum cost circulation. Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes, prove its correctness, and provide a heuristic which guarantees it to halt.

Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets

“…In addition to the many applications discussed in Section 1.1, the problem of solving linear equations in ∂ 2 ∂ 2 also arises when using Interior Point Methods to solve a generalized max-flow problem in higher-dimensional simplicial complexes as defined in [MN21]. We sketch how this inverse problem arises when using an Interior Point Method in Appendix C. By a similar argument as Lemma 1.3, we can show that if we can solve linear equations in ∂ 2 ∂ 2 to high accuracy in nearly-linear time, then we can solve linear equations in ∂ 2 to high accuracy in nearly-linear time.…”

“…Solving ∂ 2 f = d can be interpreted as computing a flow f in the triangle space of a 2-complex subject to pre-specified edge demands f . Our reduction is inspired by a reduction in [MN21] that proves NP-hardness of computing maximum integral flows in 2-complexes via a reduction from graph 3-coloring problem. However, the correctness of their reduction heavily relies on that the flow values in the 2-complex are 0-1 integers, which does not apply in our setting.…”

“…We employ some basic building blocks used in [MN21] including punctured spheres and tubes. However, we need to carefully arrange and orient the triangles in the 2-complex to encode both the positive and negative coefficients in difference-average equations, and we need to express the averaging relations not covered by the previous work.…”

“…Generalized Flows. One can generalize the notions of flows, demands vectors, circulations, and cuts to higher-dimensional simplicial complexes [DKM15;MN21]. Recall that in a graph, flows and circulations are defined on a vector space over edges, while demands and cuts are defined on a vector space over vertices.…”

The complexity of high-dimensional cuts

“…During the preparation of this article, we became aware of a recent paper by Maxwell and Nayyeri [32] that studies problems similar to the ones we define but from a completely different point of view. While our focus was on surfaces and parameterized complexity, the main focus of their work was to find out the extent to which the conceptual and the algorithmic framework of max-flow min-cut duality generalizes to the case of simplicial complexes.…”