24,378 research outputs found
Separators in Continuous Petri Nets
Leroux has proved that unreachability in Petri nets can be witnessed by a
Presburger separator, i.e. if a marking cannot reach a
marking , then there is a formula of Presburger
arithmetic such that: holds; is forward
invariant, i.e., and imply
); and holds. While these
separators could be used as explanations and as formal certificates of
unreachability, this has not yet been the case due to their
(super-)Ackermannian worst-case size and the (super-)exponential complexity of
checking that a formula is a separator.
We show that, in continuous Petri nets, these two problems can be overcome.
We introduce locally closed separators, and prove that: (a) unreachability can
be witnessed by a locally closed separator computable in polynomial time; (b)
checking whether a formula is a locally closed separator is in NC (so, simpler
than unreachablity, which is P-complete).
We further consider the more general problem of (existential) set-to-set
reachability, where two sets of markings are given as convex polytopes. We show
that, while our approach does not extend directly, we can still efficiently
certify unreachability via an altered Petri.Comment: Submitted to LMCS as an extension of the FoSSaCS'22 conference
versio
A Linear Separability Criterion for Sets of Euclidean Space
We prove new theorems which describe a necessary and sufficient condition for linear (strong and non-strong) separability and inseparability of the sets in a finite-dimensional Euclidean space. We propose a universal measure for the thickness of the geometric margin (both the strong separation margin (separator) and the margin of unseparated points (pseudo-separator)) formed between the parallel generalized supporting hyperplanes of the two sets which are separated. The introduced measure allows comparing results of linear separation obtained by different techniques for both linearly separable and inseparable sets. An optimization program whose formulation provides a maximum thickness of the separator for the separable sets is considered. When the sets are inseparable, the same solver is guaranteed to construct a pseudo-separator with a minimum thickness. We estimate the distance between the convex and closed sets. We construct a cone of generalized support vectors for hyperplanes, each one of which linearly separates the considered sets. The interconnection of the problem of different types of linear separation of sets with some related problems is studied. © 2012 Springer Science+Business Media, LLC
Halving Balls in Deterministic Linear Time
Let \D be a set of pairwise disjoint unit balls in and the
set of their center points. A hyperplane \Hy is an \emph{-separator} for
\D if each closed halfspace bounded by \Hy contains at least points
from . This generalizes the notion of halving hyperplanes, which correspond
to -separators. The analogous notion for point sets has been well studied.
Separators have various applications, for instance, in divide-and-conquer
schemes. In such a scheme any ball that is intersected by the separating
hyperplane may still interact with both sides of the partition. Therefore it is
desirable that the separating hyperplane intersects a small number of balls
only. We present three deterministic algorithms to bisect or approximately
bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present
a simple linear-time algorithm to construct an -separator for balls
in , for any , that intersects at most
balls, for some constant that depends on and . The number of
intersected balls is best possible up to the constant . Secondly, we present
a near-linear time algorithm to construct an -separator in
that intersects balls. Finally, we give a linear-time algorithm to
construct a halving line in that intersects
disks.
Our results improve the runtime of a disk sliding algorithm by Bereg,
Dumitrescu and Pach. In addition, our results improve and derandomize an
algorithm to construct a space decomposition used by L{\"o}ffler and Mulzer to
construct an onion (convex layer) decomposition for imprecise points (any point
resides at an unknown location within a given disk)
Non-realizability of the Torelli group as area-preserving homeomorphisms
Nielsen realization problem for the mapping class group
asks whether the natural projection has a section. While all the previous results use torsion
elements in an essential way, in this paper, we focus on the much more
difficult problem of realization of torsion-free subgroups of
. The main result of this paper is that the Torelli group has
no realization inside the area-preserving homeomorphisms.Comment: 22 pages, 5 figure
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
A unique factorization theorem for matroids
We study the combinatorial, algebraic and geometric properties of the free
product operation on matroids. After giving cryptomorphic definitions of free
product in terms of independent sets, bases, circuits, closure, flats and rank
function, we show that free product, which is a noncommutative operation, is
associative and respects matroid duality. The free product of matroids and
is maximal with respect to the weak order among matroids having as a
submatroid, with complementary contraction equal to . Any minor of the free
product of and is a free product of a repeated truncation of the
corresponding minor of with a repeated Higgs lift of the corresponding
minor of . We characterize, in terms of their cyclic flats, matroids that
are irreducible with respect to free product, and prove that the factorization
of a matroid into a free product of irreducibles is unique up to isomorphism.
We use these results to determine, for K a field of characteristic zero, the
structure of the minor coalgebra of a family of matroids that
is closed under formation of minors and free products: namely, is
cofree, cogenerated by the set of irreducible matroids belonging to .Comment: Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for
publication in the Journal of Combinatorial Theory (A). See
arXiv:math.CO/0409028 arXiv:math.CO/0409080 for preparatory work on this
subjec
Separating Regular Languages with First-Order Logic
Given two languages, a separator is a third language that contains the first
one and is disjoint from the second one. We investigate the following decision
problem: given two regular input languages of finite words, decide whether
there exists a first-order definable separator. We prove that in order to
answer this question, sufficient information can be extracted from semigroups
recognizing the input languages, using a fixpoint computation. This yields an
EXPTIME algorithm for checking first-order separability. Moreover, the
correctness proof of this algorithm yields a stronger result, namely a
description of a possible separator. Finally, we generalize this technique to
answer the same question for regular languages of infinite words
Self-affine Manifolds
This paper studies closed 3-manifolds which are the attractors of a system of
finitely many affine contractions that tile . Such attractors are
called self-affine tiles. Effective characterization and recognition theorems
for these 3-manifolds as well as theoretical generalizations of these results
to higher dimensions are established. The methods developed build a bridge
linking geometric topology with iterated function systems and their attractors.
A method to model self-affine tiles by simple iterative systems is developed
in order to study their topology. The model is functorial in the sense that
there is an easily computable map that induces isomorphisms between the natural
subdivisions of the attractor of the model and the self-affine tile. It has
many beneficial qualities including ease of computation allowing one to
determine topological properties of the attractor of the model such as
connectedness and whether it is a manifold. The induced map between the
attractor of the model and the self-affine tile is a quotient map and can be
checked in certain cases to be monotone or cell-like. Deep theorems from
geometric topology are applied to characterize and develop algorithms to
recognize when a self-affine tile is a topological or generalized manifold in
all dimensions. These new tools are used to check that several self-affine
tiles in the literature are 3-balls. An example of a wild 3-dimensional
self-affine tile is given whose boundary is a topological 2-sphere but which is
not itself a 3-ball. The paper describes how any 3-dimensional handlebody can
be given the structure of a self-affine 3-manifold. It is conjectured that
every self-affine tile which is a manifold is a handlebody.Comment: 40 pages, 13 figures, 2 table
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