Related papers: Computing braid groups of graphs with applications…
We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of…
In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators…
We give formulae for the first homology of the $n$-braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the…
The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams…
Most existing motion planning algorithms assume that a map (of some quality) is fully determined prior to generating a motion plan. In many emerging applications of robotics, e.g., fast-moving agile aerial robots with constrained embedded…
Planning for multi-robot teams in complex environments is a challenging problem, especially when these teams must coordinate to accomplish a common objective. In general, optimal solutions to these planning problems are computationally…
Let G be a graph. The (unlabeled) configuration space of n points on G is the space of all n-element subsets of G. The fundamental group of such a configuration space is called a graph braid group. We use a version of discrete Morse theory…
We study the theoretical and practical aspects of computing braids described by approximate descriptions of paths in the plane. Exact algorithms rely on the lexicographic ordering of the points in the plane, which is unstable under…
In the coordinated motion planning problem, we are given a graph together with the starting and destination vertices of $k$ robots. At each time step, any subset of robots may move, each traversing an edge of the graph, provided that no two…
Motion planning is a fundamental problem of robotics with applications in many areas of computer science and beyond. Its restriction to graphs has been investigated in the literature for it allows to concentrate on the combinatorial problem…
Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function.…
We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.
Compact and I/O-efficient data representations play an important role in efficient algorithm design, as memory bandwidth and latency can present a significant performance bottleneck, slowing the computation by orders of magnitude. While…
The aim of this paper is to analyse algorithms for constructing presentations of graph braid groups from the point of view of anyonic quantum statistics on graphs. In the first part of this paper, we provide a comprehensive review of an…
We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological…
The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given…
Automated driving in urban scenarios requires efficient planning algorithms able to handle complex situations in real-time. A popular approach is to use graph-based planning methods in order to obtain a rough trajectory which is…
Path planning for multiple robots is well studied in the AI and robotics communities. For a given discretized environment, robots need to find collision-free paths to a set of specified goal locations. Robots can be fully anonymous,…
This paper describes a framework for multi-robot coordination and motion planning with emphasis on inter-agent interactions. We focus on the characterization of inter-agent interactions with sufficient level of abstraction so as to allow…
In this paper, we address a class of specially structured problems that include speed planning, for mobile robots and robotic manipulators, and dynamic programming. We develop two new numerical procedures, that apply to the general case and…