On minimum Venn diagrams
Sofia Brenner, Petr Gregor, Torsten M\"utze, Francesco Verciani

TL;DR
This paper constructs near-minimum crossing $n$-Venn diagrams for all $n eq 7$, especially for powers of two, by leveraging hypercube partitions, thus advancing the understanding of minimal Venn diagram configurations.
Contribution
It provides asymptotic constructions of minimum or near-minimum crossing Venn diagrams for all $n eq 7$, including explicit diagrams for $n=8$ and for $n=2^k$, using hypercube partition techniques.
Findings
Constructed an 8-Venn diagram with 40 crossings, close to the lower bound of 37.
Developed diagrams for $n=2^k$ with crossings at most $(1+o(1))L_n$.
Established a method to generate diagrams with minimal or near-minimal crossings for all $n eq 7$.
Abstract
An -Venn diagram is a diagram in the plane consisting of simple closed curves that intersect only finitely many times such that each of the possible intersections is represented by a single connected region. An -Venn diagram has at most crossings, and if this maximum number of crossings is attained, then only two curves intersect in every crossing. To complement this, Bultena and Ruskey considered -Venn diagrams that minimize the number of crossings, which implies that many curves intersect in every crossing. Specifically, they proved that the total number of crossings in any -Venn diagram is at least , and if this lower bound is attained then essentially all curves intersect in every crossing. Diagrams achieving this bound are called minimum Venn diagrams, and are known only for . Bultena and Ruskey…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Combinatorial Mathematics · Stochastic processes and statistical mechanics
