70 research outputs found
CoBe -- Coded Beacons for Localization, Object Tracking, and SLAM Augmentation
This paper presents a novel beacon light coding protocol, which enables fast
and accurate identification of the beacons in an image. The protocol is
provably robust to a predefined set of detection and decoding errors, and does
not require any synchronization between the beacons themselves and the optical
sensor. A detailed guide is then given for developing an optical tracking and
localization system, which is based on the suggested protocol and readily
available hardware. Such a system operates either as a standalone system for
recovering the six degrees of freedom of fast moving objects, or integrated
with existing SLAM pipelines providing them with error-free and easily
identifiable landmarks. Based on this guide, we implemented a low-cost
positional tracking system which can run in real-time on an IoT board. We
evaluate our system's accuracy and compare it to other popular methods which
utilize the same optical hardware, in experiments where the ground truth is
known. A companion video containing multiple real-world experiments
demonstrates the accuracy, speed, and applicability of the proposed system in a
wide range of environments and real-world tasks. Open source code is provided
to encourage further development of low-cost localization systems integrating
the suggested technology at its navigation core
The Generalised Colouring Numbers on Classes of Bounded Expansion
The generalised colouring numbers , ,
and were introduced by Kierstead and Yang as
generalisations of the usual colouring number, also known as the degeneracy of
a graph, and have since then found important applications in the theory of
bounded expansion and nowhere dense classes of graphs, introduced by
Ne\v{s}et\v{r}il and Ossona de Mendez. In this paper, we study the relation of
the colouring numbers with two other measures that characterise nowhere dense
classes of graphs, namely with uniform quasi-wideness, studied first by Dawar
et al. in the context of preservation theorems for first-order logic, and with
the splitter game, introduced by Grohe et al. We show that every graph
excluding a fixed topological minor admits a universal order, that is, one
order witnessing that the colouring numbers are small for every value of .
Finally, we use our construction of such orders to give a new proof of a result
of Eickmeyer and Kawarabayashi, showing that the model-checking problem for
successor-invariant first-order formulas is fixed-parameter tractable on
classes of graphs with excluded topological minors
Graph Searching, Parity Games and Imperfect Information
We investigate the interrelation between graph searching games and games with
imperfect information. As key consequence we obtain that parity games with
bounded imperfect information can be solved in PTIME on graphs of bounded
DAG-width which generalizes several results for parity games on graphs of
bounded complexity. We use a new concept of graph searching where several cops
try to catch multiple robbers instead of just a single robber. The main
technical result is that the number of cops needed to catch r robbers
monotonously is at most r times the DAG-width of the graph. We also explore
aspects of this new concept as a refinement of directed path-width which
accentuates its connection to the concept of imperfect information
Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs
Bounded expansion and nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of uniformly sparse graphs which includes the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs. Since their initial definition it was shown that these graph classes can be defined in many equivalent ways: by generalised colouring numbers, neighbourhood complexity, sparse neighbourhood covers, a game known as the splitter game, and many more.
We study the corresponding concepts for directed graphs. We show that the densities of bounded depth directed minors and bounded depth topological minors relate in a similar way as in the undirected case. We provide a characterisation of bounded expansion classes by a directed version of the generalised colouring numbers. As an application we show how to construct sparse directed neighbourhood covers and how to approximate directed distance-r dominating sets on classes of bounded expansion. On the other hand, we show that linear neighbourhood complexity does not characterise directed classes of bounded expansion
Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs
A connected graph G is called matching covered if every edge of G is
contained in a perfect matching. Perfect matching width is a width parameter
for matching covered graphs based on a branch decomposition. It was introduced
by Norine and intended as a tool for the structural study of matching covered
graphs, especially in the context of Pfaffian orientations. Norine conjectured
that graphs of high perfect matching width would contain a large grid as a
matching minor, similar to the result on treewidth by Robertson and Seymour. In
this paper we obtain the first results on perfect matching width since its
introduction. For the restricted case of bipartite graphs, we show that perfect
matching width is equivalent to directed treewidth and thus the Directed Grid
Theorem by Kawarabayashi and Kreutzer for directed \treewidth implies Norine's
conjecture.Comment: Manuscrip
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