Abstract
Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.
Highlights
One of the most powerful paradigms for drawing a graph is to construct an algebraic formulation for a suitably-defined optimal drawing of the graph and solve this formulation to produce a drawing
It is natural to ask if this preference for numerical solutions over symbolic solutions is inherent in algebraic graph drawing or due to some other phenomena, such as laziness or lack of mathematical sophistication on the part of those who are producing the algebraic formulations
We introduce a framework for deciding whether certain algebraic graph drawing formulations have symbolic solutions, and we show that exact symbolic solutions are, impossible in several algebraic computation models for some simple examples of common algebraic graph drawing formulations, including force-directed graph drawings, spectral graph drawings [40], classical multidimensional scaling [42], and circle packings [39]
Summary
One of the most powerful paradigms for drawing a graph is to construct an algebraic formulation for a suitably-defined optimal drawing of the graph and solve this formulation to produce a drawing. We introduce a framework for deciding whether certain algebraic graph drawing formulations have symbolic solutions, and we show that exact symbolic solutions are, impossible in several algebraic computation models for some simple examples of common algebraic graph drawing formulations, including force-directed graph drawings (in both the Fruchterman– Reingold [24] and Kamada–Kawai [36] approaches), spectral graph drawings [40], classical multidimensional scaling [42], and circle packings [39] (which we review in more detail at the beginning of each section for readers unfamiliar with them). Because of this mathematical foundation, we refer to this topic as the Galois complexity of graph drawing
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