1,720,964 research outputs found
Computing Measures of Non-Planarity
Full text linkPlanar graphs have a rich history that dates back to the 18th Century. They form one of the core concepts of graph theory. In computational graph theory, they offer broad advantages to algorithm design and many groundbreaking results are based on them. Formally, a given graph is either planar or non-planar. However, there exists a diverse set of established measures to estimate how far away from being planar any given graph is. In this thesis, we aim at evaluating and improving algorithms to compute these measures of non-planarity. Particularly, we study (1) the problem of finding a maximum planar subgraph, i.e., a planar subgraph with the least number of edges removed; (2) the problem of embedding a graph on a lowest possible genus surface; and finally (3) the problem of drawing a graph such that there are as few edge crossings as possible. These problems constitute classical questions studied in graph drawing and each of them is NP-hard. Still, exact (exponential time) algorithms for them are of high interest and have been subject to study for decades. We propose novel mathematical programming models, based on different planarity criteria, to compute maximum planar subgraphs and low-genus embeddings. The key aspect of our most successful new models is that they carefully describe also the relation between embedded (sub-)graphs and their duals. Based on these models, we design algorithms that beat the respective state-of-the-art by orders of magnitude. We back these claims by extensive computational studies and rigorously show the theoretical advantages of our new models. Besides exact algorithms, we consider heuristic and approximate approaches to the maximum planar subgraph problem. Furthermore, in the realm of crossing numbers, we present an automated proof extraction to
easily verify the crossing number of any given graph; a new hardness result for a subproblem that arises, e.g., when enumerating simple drawings; and resolve a conjecture regarding high node degree in minimal obstructions for low crossing number
Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast
Full text linkThe NP-hard Maximum Planar Subgraph problem asks for a planar subgraph H of a given graph G such that H has maximum edge cardinality. For more than two decades, the only known non-trivial exact algorithm was based on integer linear programming and Kuratowski's famous planarity criterion. We build upon this approach and present new constraint classes - together with a lifting of the polyhedron - to obtain provably stronger LP-relaxations, and in turn faster algorithms in practice. The new constraints take Euler's polyhedron formula as a starting point and combine it with considering cycles in G. This paper discusses both the theoretical as well as the practical sides of this strengthening
An ILP-based Proof System for the Crossing Number Problem
Full text linkFormally, approaches based on mathematical programming are able to find provably optimal solutions.
However, the demands on a verifiable formal proof are typically much higher than the guarantees
we can sensibly attribute to implementations of mathematical programs. We consider this in the context of the crossing number problem, one of the most prominent problems in topological graph theory. The problem asks for the minimum number of edge crossings in any drawing of a given graph. Graph-theoretic proofs for this problem are known to be notoriously hard to obtain. At the same time, proofs even for very specific graphs are often of interest in crossing number research, as they can, e.g., form the basis for inductive proofs.
We propose a system to automatically generate a formal proof based on an ILP computation. Such a proof is (relatively) easily verifiable, and does not require the understanding of any complex ILP codes. As such, we hope our proof system may serve as a showcase for the necessary steps and central design goals of how to establish formal proof systems based on mathematical programming formulations
Stronger ILPs for the Graph Genus Problem
Full text linkThe minimum genus of a graph is an important question in graph theory and a key ingredient in several graph algorithms. However, its computation is NP-hard and turns out to be hard even in practice. Only recently, the first non-trivial approach - based on SAT and ILP (integer linear programming) models - has been presented, but it is unable to successfully tackle graphs of genus larger than 1 in practice.
Herein, we show how to improve the ILP formulation. The crucial ingredients are two-fold. First, we show that instead of modeling rotation schemes explicitly, it suffices to optimize over partitions of the (bidirected) arc set A of the graph. Second, we exploit the cycle structure of the graph, explicitly mapping short closed walks on A to faces in the embedding.
Besides the theoretical advantages of our models, we show their practical strength by a thorough experimental evaluation. Contrary to the previous approach, we are able to quickly solve many instances of genus > 1
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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