Related papers: Diversities and the Geometry of Hypergraphs

Graphs are a natural representation of data from various contexts, such as social connections, the web, road networks, and many more. In the last decades, many of these networks have become enormous, requiring efficient algorithms to cut…

One of the major results of [N. Robertson and P. D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 63(1):65--110, 1995], also known as the weak structure theorem, revealed the local structure of graphs…

Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach…

The central theorem of topological graph theory states that the graph minor relation is a well-quasi-order on graphs. It has far-reaching consequences, in particular in the study of graph structures and the design of (parameterized)…

Flow Matching has recently gained attention in generative modeling as a simple and flexible alternative to diffusion models. While existing statistical guarantees adapt tools from the analysis of diffusion models, we take a different…

We introduce and study a novel problem of computing a shortest path tree with a minimum number of non-terminals. It can be viewed as an (unweighted) Steiner Shortest Path Tree (SSPT) that spans a given set of terminal vertices by shortest…

This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field.…

Kernel mean embeddings are a popular tool that consists in representing probability measures by their infinite-dimensional mean embeddings in a reproducing kernel Hilbert space. When the kernel is characteristic, mean embeddings can be used…

The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper, we provide the first algorithmic applications of tree-cut…

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several…

We initiate the algorithmic study of retracting a graph into a cycle in the graph, which seeks a mapping of the graph vertices to the cycle vertices, so as to minimize the maximum stretch of any edge, subject to the constraint that the…

We introduce a flow-dependent version of the quadratic Steiner tree problem in the plane. An instance of the problem on a set of embedded sources and a sink asks for a directed tree $T$ spanning these nodes and a bounded number of Steiner…

We connect several notions relating the structural and dynamical properties of a graph. Among them are the topological entropy coming from the vertex shift, which is related to the spectral radius of the graph's adjacency matrix, the…

Fitting distances to tree metrics and ultrametrics are two widely used methods in hierarchical clustering, primarily explored within the context of numerical taxonomy. Given a positive distance function…

Flow matching has emerged as a powerful framework for generative modeling through continuous normalizing flows. We investigate a potential topological constraint: when the prior distribution and target distribution have mismatched topology…

We propose a novel node embedding of directed graphs to statistical manifolds, which is based on a global minimization of pairwise relative entropy and graph geodesics in a non-linear way. Each node is encoded with a probability density…

Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…

Optimal transport is widely used in pure and applied mathematics to find probabilistic solutions to hard combinatorial matching problems. We extend the Wasserstein metric and other elements of optimal transport from the matching of sets to…

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multi-commodity minimum-cost network flow problem can be formulated as a multi-marginal optimal transport…

Hyperbolic geometry has emerged as an effective latent space for representing complex networks, owing to its ability to capture hierarchical organization and heterogeneous connectivity patterns using low-dimensional embeddings. As a result,…