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 $\vec{m}_\text{src}$ cannot reach a marking $\vec{m}_\text{tgt}$, then there is a formula $\varphi$ of Presburger arithmetic such that: $\varphi(\vec{m}_\text{src})$ holds; $\varphi$ is forward invariant, i.e., $\varphi(\vec{m})$ and $\vec{m} \rightarrow \vec{m}'$ imply $\varphi(\vec{m}'$); and $\neg \varphi(\vec{m}_\text{tgt})$ 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 $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the set of their center points. A hyperplane \Hy is an \emph{$m$-separator} for \D if each closed halfspace bounded by \Hy contains at least $m$ points from $P$. This generalizes the notion of halving hyperplanes, which correspond to $n/2$-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 $\alpha n$-separator for balls in $\R^d$, for any $0<\alpha<1/2$, that intersects at most $cn^{(d-1)/d}$ balls, for some constant $c$ that depends on $d$ and $\alpha$. The number of intersected balls is best possible up to the constant $c$. Secondly, we present a near-linear time algorithm to construct an $(n/2-o(n))$-separator in $\R^d$ that intersects $o(n)$ balls. Finally, we give a linear-time algorithm to construct a halving line in $\R^2$ that intersects $O(n^{(5/6)+\epsilon})$ 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 $\text{Mod}(S_g)$ asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to \text{Mod}(S_g)$ 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 $\text{Mod}(S_g)$. 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 $k$-planar graphs, where each edge is crossed by at most $k$ other edges; and, outer $k$-quasi-planar graphs where no $k$ edges can mutually cross. We show that the outer $k$-planar graphs are $(\lfloor\sqrt{4k+1}\rfloor+1)$-degenerate, and consequently that every outer $k$-planar graph can be $(\lfloor\sqrt{4k+1}\rfloor+2)$-colored, and this bound is tight. We further show that every outer $k$-planar graph has a balanced separator of size $O(k)$. This implies that every outer $k$-planar graph has treewidth $O(k)$. For fixed $k$, these small balanced separators allow us to obtain a simple quasi-polynomial time algorithm to test whether a given graph is outer $k$-planar, i.e., none of these recognition problems are NP-complete unless ETH fails. For the outer $k$-quasi-planar graphs we prove that, unlike other beyond-planar graph classes, every edge-maximal $n$-vertex outer $k$-quasi planar graph has the same number of edges, namely $2(k-1)n - \binom{2k-1}{2}$. We also construct planar 3-trees that are not outer $3$-quasi-planar. Finally, we restrict outer $k$-planar and outer $k$-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence on the boundary is a cycle in the graph. For each $k$, we express closed outer $k$-planarity and \emph{closed outer $k$-quasi-planarity} in extended monadic second-order logic. Thus, closed outer $k$-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 $M$ and $N$ is maximal with respect to the weak order among matroids having $M$ as a submatroid, with complementary contraction equal to $N$. Any minor of the free product of $M$ and $N$ is a free product of a repeated truncation of the corresponding minor of $M$ with a repeated Higgs lift of the corresponding minor of $N$. 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 $\cal C$ of a family of matroids $\cal M$ that is closed under formation of minors and free products: namely, $\cal C$ is cofree, cogenerated by the set of irreducible matroids belonging to $\cal M$.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 $\mathbb{R}^3$. 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