Recursive Linear Bounds for the Vertex Chromatic Number of the Pancake Graph

The pancake graph has been the subject of research. While studies on the various aspects of the graph are abundant, results on the chromatic properties may be further enhanced. Revolving around such context, the paper advances an alternative method to produce novel linear bounds for the vertex chromatic number of the pancake graph. The accompanying demonstration takes advantage of symmetries inherent to the graph, capturing the prefix reversal of subsequences through a homomorphism. Contained within the argument is the incorporation of known vertex chromatic numbers for certain orders of pancake graphs, rendering tighter bounds possible upon the release of new findings. In closing, a comparison with existing bounds is done to establish the relative advantage of the proposed technique

Origami constraints on the initial-conditions arrangement of dark-matter caustics and streams

In a cold-dark-matter universe, cosmological structure formation proceeds in rough analogy to origami folding. Dark matter occupies a three-dimensional 'sheet' of free- fall observers, non-intersecting in six-dimensional velocity-position phase space. At early times, the sheet was flat like an origami sheet, i.e. velocities were essentially zero, but as time passes, the sheet folds up to form cosmic structure. The present paper further illustrates this analogy, and clarifies a Lagrangian definition of caustics and streams: caustics are two-dimensional surfaces in this initial sheet along which it folds, tessellating Lagrangian space into a set of three-dimensional regions, i.e. streams. The main scientific result of the paper is that streams may be colored by only two colors, with no two neighbouring streams (i.e. streams on either side of a caustic surface) colored the same. The two colors correspond to positive and negative parities of local Lagrangian volumes. This is a severe restriction on the connectivity and therefore arrangement of streams in Lagrangian space, since arbitrarily many colors can be necessary to color a general arrangement of three-dimensional regions. This stream two-colorability has consequences from graph theory, which we explain. Then, using N-body simulations, we test how these caustics correspond in Lagrangian space to the boundaries of haloes, filaments and walls. We also test how well outer caustics correspond to a Zel'dovich-approximation prediction.Comment: Clarifications and slight changes to match version accepted to MNRAS. 9 pages, 5 figure

An improved bound on the chromatic number of the Pancake graphs

In this paper an improved bound on the chromatic number of the Pancake graph $P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an equitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficient dominating sets is given and optimal equitable $4$-colorings are considered for small $n$. It is conjectured that the chromatic number of $P_n$ coincides with its equitable chromatic number for any $n\geqslant 2$

Computing a maximum clique in geometric superclasses of disk graphs

In the 90's Clark, Colbourn and Johnson wrote a seminal paper where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of d-dimensional (unit) balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs. However, the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks

Dynamic Algorithms for Graph Coloring

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for $(\Delta+1)$- vertex coloring and $(2\Delta-1)$-edge coloring in a graph with maximum degree $\Delta$. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a $(\Delta+1)$-vertex coloring with $O(\log \Delta)$ expected amortized update time. (2) We present a deterministic algorithm which maintains a $(1+o(1))\Delta$-vertex coloring with $O(\text{poly} \log \Delta)$ amortized update time. (3) We present a simple, deterministic algorithm which maintains a $(2\Delta-1)$-edge coloring with $O(\log \Delta)$ worst-case update time. This improves the recent $O(\Delta)$-edge coloring algorithm with $\tilde{O}(\sqrt{\Delta})$ worst-case update time by Barenboim and Maimon.Comment: To appear in SODA 201

Interpretation of Natural-language Robot Instructions: Probabilistic Knowledge Representation, Learning, and Reasoning

A robot that can be simply told in natural language what to do -- this has been one of the ultimate long-standing goals in both Artificial Intelligence and Robotics research. In near-future applications, robotic assistants and companions will have to understand and perform commands such as set the table for dinner'', make pancakes for breakfast'', or cut the pizza into 8 pieces.'' Although such instructions are only vaguely formulated, complex sequences of sophisticated and accurate manipulation activities need to be carried out in order to accomplish the respective tasks. The acquisition of knowledge about how to perform these activities from huge collections of natural-language instructions from the Internet has garnered a lot of attention within the last decade. However, natural language is typically massively unspecific, incomplete, ambiguous and vague and thus requires powerful means for interpretation. This work presents PRAC -- Probabilistic Action Cores -- an interpreter for natural-language instructions which is able to resolve vagueness and ambiguity in natural language and infer missing information pieces that are required to render an instruction executable by a robot. To this end, PRAC formulates the problem of instruction interpretation as a reasoning problem in first-order probabilistic knowledge bases. In particular, the system uses Markov logic networks as a carrier formalism for encoding uncertain knowledge. A novel framework for reasoning about unmodeled symbolic concepts is introduced, which incorporates ontological knowledge from taxonomies and exploits semantically similar relational structures in a domain of discourse. The resulting reasoning framework thus enables more compact representations of knowledge and exhibits strong generalization performance when being learnt from very sparse data. Furthermore, a novel approach for completing directives is presented, which applies semantic analogical reasoning to transfer knowledge collected from thousands of natural-language instruction sheets to new situations. In addition, a cohesive processing pipeline is described that transforms vague and incomplete task formulations into sequences of formally specified robot plans. The system is connected to a plan executive that is able to execute the computed plans in a simulator. Experiments conducted in a publicly accessible, browser-based web interface showcase that PRAC is capable of closing the loop from natural-language instructions to their execution by a robot

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz