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Register a new userGraph Drawing via Gradient Descent, $(GD)^2$
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Abu Reyan Ahmed
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Graph Drawing via Gradient Descent, $(GD)^2$, that can handle multiple readability criteria. $(GD)^2$ can optimize any criterion that can be described by a smooth function. If the criterion cannot be captured by a smooth function, a non-smooth function for the criterion is combined with another smooth function, or auto-differentiation tools are used for the optimization. Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., vertex resolution, angular resolution, aspect ratio). We provide quantitative and qualitative evidence of the effectiveness of $(GD)^2$ with experimental data and a functional prototype: url{http://hdc.cs.arizona.edu/~mwli/graph-drawing/}.
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We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $Gamma$. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$, and a planar straight-line drawing with spanning ratio $1$ are NP-complete, $exists mathbb R$-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio $1$ to spanning ratio $1+epsilon$ allows us to draw every graph. Namely, we prove that, for every $epsilon>0$, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than $1+epsilon$. Third, our drawings with spanning ratio smaller than $1+epsilon$ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graphs crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with $O(log^2 n)(n + OPT)$ crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Edge Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Edge Planarization problem on graph G, then we can efficiently find a drawing of G with at most $poly(d)cdot kcdot (k+OPT)$ crossings, where $d$ is the maximum degree in G. This result implies an $O(ncdot poly(d)cdot log^{3/2}n)$-approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.
Graph Crossing Number is a fundamental problem with various applications. In this problem, the goal is to draw an input graph $G$ in the plane so as to minimize the number of crossings between the images of its edges. Despite extensive work, non-trivial approximation algorithms are only known for bounded-degree graphs. Even for this special case, the best current algorithm achieves a $tilde O(sqrt n)$-approximation, while the best current negative result is APX-hardness. All current approximation algorithms for the problem build on the same paradigm: compute a set $E$ of edges (called a emph{planarizing set}) such that $Gsetminus E$ is planar; compute a planar drawing of $Gsetminus E$; then add the drawings of the edges of $E$ to the resulting drawing. Unfortunately, there are examples of graphs, in which any implementation of this method must incur $Omega (text{OPT}^2)$ crossings, where $text{OPT}$ is the value of the optimal solution. This barrier seems to doom the only known approach to designing approximation algorithms for the problem, and to prevent it from yielding a better than $O(sqrt n)$-approximation. In this paper we propose a new paradigm that allows us to overcome this barrier. We show an algorithm that, given a bounded-degree graph $G$ and a planarizing set $E$ of its edges, computes another set $E$ with $Esubseteq E$, such that $|E|$ is relatively small, and there exists a near-optimal drawing of $G$ in which only edges of $E$ participate in crossings. This allows us to reduce the Crossing Number problem to emph{Crossing Number with Rotation System} -- a variant in which the ordering of the edges incident to every vertex is fixed as part of input. We show a randomized algorithm for this new problem, that allows us to obtain an $O(n^{1/2-epsilon})$-approximation for Crossing Number on bounded-degree graphs, for some constant $epsilon>0$.
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