Distributed coloring in sparse graphs with fewer colors

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in \mbox{polylog}(n) rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.Comment: 16 pages, 4 figures - An extended abstract of this work was presented at PODC'18 (ACM Symposium on Principles of Distributed Computing

Mapping Tasks to Interactions for Graph Exploration and Graph Editing on Interactive Surfaces

Graph exploration and editing are still mostly considered independently and systems to work with are not designed for todays interactive surfaces like smartphones, tablets or tabletops. When developing a system for those modern devices that supports both graph exploration and graph editing, it is necessary to 1) identify what basic tasks need to be supported, 2) what interactions can be used, and 3) how to map these tasks and interactions. This technical report provides a list of basic interaction tasks for graph exploration and editing as a result of an extensive system review. Moreover, different interaction modalities of interactive surfaces are reviewed according to their interaction vocabulary and further degrees of freedom that can be used to make interactions distinguishable are discussed. Beyond the scope of graph exploration and editing, we provide an approach for finding and evaluating a mapping from tasks to interactions, that is generally applicable. Thus, this work acts as a guideline for developing a system for graph exploration and editing that is specifically designed for interactive surfaces.Comment: 21 pages, minor corrections (typos etc.

Learning Combinatorial Node Labeling Algorithms

We present a graph neural network to learn graph coloring heuristics using reinforcement learning. Our learned deterministic heuristics give better solutions than classical degree-based greedy heuristics and only take seconds to evaluate on graphs with tens of thousands of vertices. As our approach is based on policy-gradients, it also learns a probabilistic policy as well. These probabilistic policies outperform all greedy coloring baselines and a machine learning baseline. Our approach generalizes several previous machine-learning frameworks, which applied to problems like minimum vertex cover. We also demonstrate that our approach outperforms two greedy heuristics on minimum vertex cover

Visualized Algorithm Engineering on Two Graph Partitioning Problems

Concepts of graph theory are frequently used by computer scientists as abstractions when modeling a problem. Partitioning a graph (or a network) into smaller parts is one of the fundamental algorithmic operations that plays a key role in classifying and clustering. Since the early 1970s, graph partitioning rapidly expanded for applications in wide areas. It applies in both engineering applications, as well as research. Current technology generates massive data (“Big Data”) from business interactions and social exchanges, so high-performance algorithms of partitioning graphs are a critical need. This dissertation presents engineering models for two graph partitioning problems arising from completely different applications, computer networks and arithmetic. The design, analysis, implementation, optimization, and experimental evaluation of these models employ visualization in all aspects. Visualization indicates the performance of the implementation of each Algorithm Engineering work, and also helps to analyze and explore new algorithms to solve the problems. We term this research method as “Visualized Algorithm Engineering (VAE)” to emphasize the contribution of the visualizations in these works. The techniques discussed here apply to a broad area of problems: computer networks, social networks, arithmetic, computer graphics and software engineering. Common terminologies accepted across these disciplines have been used in this dissertation to guarantee practitioners from all fields can understand the concepts we introduce

Coloring and constructing (hyper)graphs with restrictions

We consider questions regarding the existence of graphs and hypergraphs with certain coloring properties and other structural properties. In Chapter 2 we consider color-critical graphs that are nearly bipartite and have few edges. We prove a conjecture of Chen, Erdős, Gyárfás, and Schelp concerning the minimum number of edges in a “nearly bipartite” 4-critical graph. In Chapter 3 we consider coloring and list-coloring graphs and hypergraphs with few edges and no small cycles. We prove two main results. If a bipartite graph has maximum average degree at most 2(k−1), then it is colorable from lists of size k; we prove that this is sharp, even with an additional girth requirement. Using the same approach, we also provide a simple construction of graphs with arbitrarily large girth and chromatic number (first proved to exist by Erdős). In Chapter 4 we consider list-coloring the family of kth power graphs. Kostochka and Woodall conjectured that graph squares are chromatic-choosable, as a strengthening of the Total List Coloring Conjecture. Kim and Park disproved this stronger conjecture, and Zhu asked whether graph kth powers are chromatic-choosable for any k. We show that this is not true: we construct families of graphs based on affine planes whose choice number exceeds their chromatic number by a logarithmic factor. In Chapter 5 we consider the existence of uniform hypergraphs with prescribed degrees and codegrees. In Section 5.2, we show that a generalization of the graphic 2-switch is insufficient to connect realizations of a given degree sequence. In Section 5.3, we consider an operation on 3-graphs related to the octahedron that preserves codegrees; this leads to an inductive definition for 2-colorable triangulations of the sphere. In Section 5.4, we discuss the notion of fractional realizations of degree sequences, in particular noting the equivalence of the existence of a realization and the existence of a fractional realization in the graph and multihypergraph cases. In Chapter 6 we consider a question concerning poset dimension. Dorais asked for the maximum guaranteed size of a subposet with dimension at most d of an n-element poset. A lower bound of sqrt(dn) was observed by Goodwillie. We provide a sublinear upper bound

Japanese irogane alloys and patination – a study of production and application

Japanese metalworkers use a wide range of irogane alloys (shakudo, shibuichi), which are colored with a single patination solution (niiro). This approach allows different alloys to be combined in one piece and patinated, producing a multi-colored piece of metalwork. At present the production of irogane alloys and their patination is an unreliable process. This study aims to develop reliable alloy production and a safe, easy-to use and repeatable patination process using standard ingredients available from chemical suppliers. The study has examined the production of shakudo and shibuichi alloys, characterizing the alloys produced by casting into cloth molds in hot water, into steel molds, and produced using continuous casting. The influence of traditional polishing methods was assessed using surface texture (Sa) measurements. Traditional rokusho, an ingredient of the niiro solution, was analyzed by XRF and XRD. Niiro patinated surfaces on a range of alloys were examined using XRD and L*a*b* color measurements.</p