Related papers: Shortest Paths in a Weighted Simplicial Complex
Among the several topological properties of complex networks, the shortest path represents a particularly important characteristic because of its potential impact not only on other topological properties, but mainly for its influence on…
We present simplicial neural networks (SNNs), a generalization of graph neural networks to data that live on a class of topological spaces called simplicial complexes. These are natural multi-dimensional extensions of graphs that encode not…
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an…
Finding a shortest path in a graph is one of the most classic problems in algorithmic and graph theory. While we dispose of quite efficient algorithms for this ordinary problem (like the Dijkstra or Bellman-Ford algorithms), some slight…
In this article, we present an approximation algorithm for solving the Weighted Region Problem amidst a set of $ n $ non-overlapping weighted disks in the plane. For a given parameter $ \varepsilon \in (0,1]$, the length of the approximate…
We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification…
Continuous 2-dimensional space is often discretized by considering a mesh of weighted cells. In this work we study how well a weighted mesh approximates the space, with respect to shortest paths. We consider a shortest path $…
Computing shortest paths is one of the most fundamental algorithmic graph problems. It is known since decades that this problem can be solved in near-linear time if all weights are nonnegative. A recent break-through by [Bernstein,…
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
One of the most fundamental graph problems is finding a shortest path from a source to a target node. While in its basic forms the problem has been studied extensively and efficient algorithms are known, it becomes significantly harder as…
A {\em parametric weighted graph} is a graph whose edges are labeled with continuous real functions of a single common variable. For any instantiation of the variable, one obtains a standard edge-weighted graph. Parametric weighted graph…
This paper details a new algorithm to solve the shortest path problem in valued graphs. Its complexity is $O(D \log v)$ where $D$ is the graph diameter and $v$ its number of vertices. This complexity has to be compared to the one of the…
Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the…
We introduce and investigate generalizations of interval and proper interval graphs to simplicial complexes, including strong interval, unit interval, and under closed variants. Through equivalent combinatorial and algebraic…
Shortest paths play an important role in mathematical modeling and image processing. Usually, shortest path problems are formulated on planar graphs that consist of vertices and weighted arcs. In this context, one is interested in finding a…
In this paper we introduce the concept of generalized d-graph (admitting cycles) as special dependency-graphs for modelling dynamic programming (DP) problems. We describe the d-graph versions of three famous single-source shortest…
We consider single-source shortest path algorithms that perform a sequence of relaxation steps whose ordering depends only on the input graph structure and not on its weights or the results of prior steps. Each step examines one edge of the…
Simplicial complexes are higher-order combinatorial structures which have been used to represent real-world complex systems. In this paper, we concentrate on the local patterns in simplicial complexes called simplets, a generalization of…
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…
We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension $d$. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance…