arXiv subjects

Petr Hlineny

Publications and source records attributed to Petr Hlineny.

8 recordsLinked to original sources

arXiv2019

Toroidal grid minors and stretch in embedded graphs

We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding density parameter, the stretch of an embedded graph G, and use it to bound the toroidal expanse from above and from below within a constant factor depending only on the genus and the maximum degree. We also show that these parameters are tightly related to the planar crossing number of G. As a consequence of our bounds, we derive an efficient constant factor approximation algorithm for the toroidal expanse and for the crossing number of a surface-embedded graph with bounded maximum degree.

math.CO↗

arXiv2016

First order limits of sparse graphs: Plane trees and path-width

Nesetril and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no limit modeling (an analytic representation of the limit). On the positive side, every first order convergent sequence of trees or graphs with no long path (graphs with bounded tree-depth) has a limit modeling. We strengthen these results by showing that every first order convergent sequence of plane trees (trees with embeddings in the plane) and every first order convergent sequence of graphs with bounded path-width has a limit modeling.

math.CO↗

arXiv2015

FO Model Checking of Interval Graphs

We study the computational complexity of the FO model checking problem on interval graphs, i.e., intersection graphs of intervals on the real line. The main positive result is that FO model checking and successor-invariant FO model checking can be solved in time O(n log n) for n-vertex interval graphs with representations containing only intervals with lengths from a prescribed finite set. We complement this result by showing that the same is not true if the lengths are restricted to any set that is dense in an open subset, e.g., in the set $(1, 1 + \varepsilon)$.

cs.DM↗

arXiv2015

A Tighter Insertion-based Approximation of the Crossing Number

Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F. We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee -- depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph G+F, while computing the crossing number of G+F exactly is NP-hard already when |F|=1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.

cs.DM↗

arXiv2015

Kernelizing MSO Properties of Trees of Fixed Height, and Some Consequences

Fix an integer h>=1. In the universe of coloured trees of height at most h, we prove that for any graph decision problem defined by an MSO formula with r quantifiers, there exists a set of kernels, each of size bounded by an elementary function of r and the number of colours. This yields two noteworthy consequences. Consider any graph class G having a one-dimensional MSO interpretation in the universe of coloured trees of height h (equivalently, G is a class of shrub-depth h). First, class G admits an MSO model checking algorithm whose runtime has an elementary dependence on the formula size. Second, on G the expressive powers of FO and MSO coincide (which extends a 2012 result of Elberfeld, Grohe, and Tantau).

cs.DM↗

arXiv2012

Dynamic Scope-Based Dijkstra's Algorithm

We briefly report on the current state of a new dynamic algorithm for the route planning problem based on a concept of scope (the static variant presented at ESA'11, HM2011A). We first motivate dynamization of the concept of scope admissibility, and then we briefly describe a modification of the scope-aware query algorithm of HM2011A to dynamic road networks. Finally, we outline our future work on this concept.

cs.DS↗

arXiv2011

Multi-Stage Improved Route Planning Approach: theoretical foundations

A new approach to the static route planning problem, based on a multi-staging concept and a \emph{scope} notion, is presented. The main goal (besides implied efficiency of planning) of our approach is to address---with a solid theoretical foundation---the following two practically motivated aspects: a \emph{route comfort} and a very \emph{limited storage} space of a small navigation device, which both do not seem to be among the chief objectives of many other studies. We show how our novel idea can tackle both these seemingly unrelated aspects at once, and may also contribute to other established route planning approaches with which ours can be naturally combined. We provide a theoretical proof that our approach efficiently computes exact optimal routes within this concept, as well as we demonstrate with experimental results on publicly available road networks of the US the good practical performance of the solution.

cs.DS↗

arXiv2011

Generalized Maneuvers in Route Planning

We study an important practical aspect of the route planning problem in real-world road networks -- maneuvers. Informally, maneuvers represent various irregularities of the road network graph such as turn-prohibitions, traffic light delays, round-abouts, forbidden passages and so on. We propose a generalized model which can handle arbitrarily complex (and even negative) maneuvers, and outline how to enhance Dijkstra's algorithm in order to solve route planning queries in this model without prior adjustments of the underlying road network graph.

cs.DS↗