2,796 research outputs found
The Complexity of Simultaneous Geometric Graph Embedding
Given a collection of planar graphs on the same set of
vertices, the simultaneous geometric embedding (with mapping) problem, or
simply -SGE, is to find a set of points in the plane and a bijection
such that the induced straight-line drawings of
under are all plane.
This problem is polynomial-time equivalent to weak rectilinear realizability
of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x)
proved to be complete for , the existential theory of the
reals. Hence the problem -SGE is polynomial-time equivalent to several other
problems in computational geometry, such as recognizing intersection graphs of
line segments or finding the rectilinear crossing number of a graph.
We give an elementary reduction from the pseudoline stretchability problem to
-SGE, with the property that both numbers and are linear in the
number of pseudolines. This implies not only the -hardness
result, but also a lower bound on the minimum size of a
grid on which any such simultaneous embedding can be drawn. This bound is
tight. Hence there exists such collections of graphs that can be simultaneously
embedded, but every simultaneous drawing requires an exponential number of bits
per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is
only
Combinatorial geometry of neural codes, neural data analysis, and neural networks
This dissertation explores applications of discrete geometry in mathematical
neuroscience. We begin with convex neural codes, which model the activity of
hippocampal place cells and other neurons with convex receptive fields. In
Chapter 4, we introduce order-forcing, a tool for constraining convex
realizations of codes, and use it to construct new examples of non-convex codes
with no local obstructions. In Chapter 5, we relate oriented matroids to convex
neural codes, showing that a code has a realization with convex polytopes iff
it is the image of a representable oriented matroid under a neural code
morphism. We also show that determining whether a code is convex is at least as
difficult as determining whether an oriented matroid is representable, implying
that the problem of determining whether a code is convex is NP-hard. Next, we
turn to the problem of the underlying rank of a matrix. This problem is
motivated by the problem of determining the dimensionality of (neural) data
which has been corrupted by an unknown monotone transformation. In Chapter 6,
we introduce two tools for computing underlying rank, the minimal nodes and the
Radon rank. We apply these to analyze calcium imaging data from a larval
zebrafish. In Chapter 7, we explore the underlying rank in more detail,
establish connections to oriented matroid theory, and show that computing
underlying rank is also NP-hard. Finally, we study the dynamics of
threshold-linear networks (TLNs), a simple model of the activity of neural
circuits. In Chapter 9, we describe the nullcline arrangement of a threshold
linear network, and show that a subset of its chambers are an attracting set.
In Chapter 10, we focus on combinatorial threshold linear networks (CTLNs),
which are TLNs defined from a directed graph. We prove that if the graph of a
CTLN is a directed acyclic graph, then all trajectories of the CTLN approach a
fixed point.Comment: 193 pages, 69 figure
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem
Helly meets Garside and Artin
A graph is Helly if every family of pairwise intersecting combinatorial balls
has a nonempty intersection. We show that weak Garside groups of finite type
and FC-type Artin groups are Helly, that is, they act geometrically on Helly
graphs. In particular, such groups act geometrically on spaces with convex
geodesic bicombing, equipping them with a nonpositive-curvature-like structure.
That structure has many properties of a CAT(0) structure and, additionally, it
has a combinatorial flavor implying biautomaticity. As immediate consequences
we obtain new results for FC-type Artin groups (in particular braid groups and
spherical Artin groups) and weak Garside groups, including e.g.\ fundamental
groups of the complements of complexified finite simplicial arrangements of
hyperplanes, braid groups of well-generated complex reflection groups, and
one-relator groups with non-trivial center. Among the results are:
biautomaticity, existence of EZ and Tits boundaries, the Farrell-Jones
conjecture, the coarse Baum-Connes conjecture, and a description of higher
order homological and homotopical Dehn functions. As a mean of proving the
Helly property we introduce and use the notion of a (generalized) cell Helly
complex.Comment: Small modifications according to the referee report, updated
references. Final accepted versio
- …