Configurations with few crossings in topological graphs

AbstractIn this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s–t paths, cycles, matchings, and κ-factors for κ∈{1,2}. We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k1−ε for any ε>0, where k is the number of crossings in G.We then give a simple fixed-parameter algorithm that tests in O⋆(2k) time whether G has a crossing-free configuration for any of the above, where the O⋆-notation neglects polynomial terms. For some configurations we have faster algorithms. The respective running times are O⋆(1.9999992k) for spanning trees and O⋆((3)k) for s-t paths and cycles. For spanning trees we also have an O⋆(1.968k)-time Monte-Carlo algorithm. Each O⋆(βk)-time decision algorithm can be turned into an O⋆((β+1)k)-time optimization algorithm that computes a configuration with the minimum number of crossings

Geometric Crossing-Minimization - A Scalable Randomized Approach

We consider the minimization of edge-crossings in geometric drawings of graphs G=(V, E), i.e., in drawings where each edge is depicted as a line segment. The respective decision problem is NP-hard [Daniel Bienstock, 1991]. Crossing-minimization, in general, is a popular theoretical research topic; see Vrt\u27o [Imrich Vrt\u27o, 2014]. In contrast to theory and the topological setting, the geometric setting did not receive a lot of attention in practice. Prior work [Marcel Radermacher et al., 2018] is limited to the crossing-minimization in geometric graphs with less than 200 edges. The described heuristics base on the primitive operation of moving a single vertex v to its crossing-minimal position, i.e., the position in R^2 that minimizes the number of crossings on edges incident to v. In this paper, we introduce a technique to speed-up the computation by a factor of 20. This is necessary but not sufficient to cope with graphs with a few thousand edges. In order to handle larger graphs, we drop the condition that each vertex v has to be moved to its crossing-minimal position and compute a position that is only optimal with respect to a small random subset of the edges. In our theoretical contribution, we consider drawings that contain for each edge uv in E and each position p in R^2 for v o(|E|) crossings. In this case, we prove that with a random subset of the edges of size Theta(k log k) the co-crossing number of a degree-k vertex v, i.e., the number of edge pairs uv in E, e in E that do not cross, can be approximated by an arbitrary but fixed factor delta with high probability. In our experimental evaluation, we show that the randomized approach reduces the number of crossings in graphs with up to 13 000 edges considerably. The evaluation suggests that depending on the degree-distribution different strategies result in the fewest number of crossings

Phase transitions in a gas of anyons

We continue our numerical Monte Carlo simulation of a gas of closed loops on a 3 dimensional lattice, however now in the presence of a topological term added to the action corresponding to the total linking number between the loops. We compute the linking number using certain notions from knot theory. Adding the topological term converts the particles into anyons. Using the correspondence that the model is an effective theory that describes the 2+1-dimensional Abelian Higgs model in the asymptotic strong coupling regime, the topological linking number simply corresponds to the addition to the action of the Chern-Simons term. We find the following new results. The system continues to exhibit a phase transition as a function of the anyon mass as it becomes small \cite{mnp}, although the phases do not change the manifestation of the symmetry. The Chern-Simons term has no effect on the Wilson loop, but it does affect the {\rm '}t Hooft loop. For a given configuration it adds the linking number of the 't Hooft loop with all of the dynamical vortex loops to the action. We find that both the Wilson loop and the 't Hooft loop exhibit a perimeter law even though there are no massless particles in the theory, which is unexpected.Comment: 6 pages, 5 figure

From singularities to graphs

In this paper I analyze the problems which led to the introduction of graphs as tools for studying surface singularities. I explain how such graphs were initially only described using words, but that several questions made it necessary to draw them, leading to the elaboration of a special calculus with graphs. This is a non-technical paper intended to be readable both by mathematicians and philosophers or historians of mathematics.Comment: 23 pages, 27 figures. Expanded version of the talk given at the conference "Quand la forme devient substance : puissance des gestes, intuition diagrammatique et ph\'enom\'enologie de l'espace", which took place at Lyc\'ee Henri IV in Paris from 25 to 27 January 201