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

Bad luck vs. self-inflicted neediness – An experimental investigation of gift giving in a solidarity game

We experimentally examine the impact of self-inflicted neediness on the solidarity behavior of subjects. In one treatment in our solidarity experiment all subjects face the same probability of becoming needy, in the other treatment subjects have the choice between a secure payment and a lottery including a certain probability of becoming needy. Then we ask all subjects how much they will give to losers in their group thus investigating if people are willing to give the same gifts whether or not subjects are responsible for inequality in payoffs. We found evidence for allocative as well as for procedural utility concerns.solidarity game, self-inflicted neediness, responsibility, procedural utility