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Register a new userDrawing Graphs as Spanners
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Added by
Fabrizio Frati
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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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/}.
It was recently shown that a version of the greedy algorithm gives a construction of fault-tolerant spanners that is size-optimal, at least for vertex faults. However, the algorithm to construct this spanner is not polynomial-time, and the best-known polynomial time algorithm is significantly suboptimal. Designing a polynomial-time algorithm to construct (near-)optimal fault-tolerant spanners was given as an explicit open problem in the two most recent papers on fault-tolerant spanners ([Bodwin, Dinitz, Parter, Vassilevka Williams SODA 18] and [Bodwin, Patel PODC 19]). We give a surprisingly simple algorithm which runs in polynomial time and constructs fault-tolerant spanners that are extremely close to optimal (off by only a linear factor in the stretch) by modifying the greedy algorithm to run in polynomial time. To complement this result, we also give simple distributed constructions in both the LOCAL and CONGEST models.
It is well known that any graph admits a crossing-free straight-line drawing in $mathbb{R}^3$ and that any planar graph admits the same even in $mathbb{R}^2$. For a graph $G$ and $d in {2,3}$, let $rho^1_d(G)$ denote the minimum number of lines in $mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. - For $din{2,3}$, we prove that deciding whether $rho^1_d(G)le k$ for a given graph $G$ and integer $k$ is ${existsmathbb{R}}$-complete. - Since $mathrm{NP}subseteq{existsmathbb{R}}$, deciding $rho^1_d(G)le k$ is NP-hard for $din{2,3}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${existsmathbb{R}}subseteqmathrm{PSPACE}$, both $rho^1_2(G)$ and $rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $rho^1_2$ or $rho^1_3$ sometimes require irrational coordinates. - Let $rho^2_3(G)$ be the minimum number of planes in $mathbb{R}^3$ needed to cover a straight-line drawing of a graph $G$. We prove that deciding whether $rho^2_3(G)le k$ is NP-hard for any fixed $k ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $mathrm{P}=mathrm{NP}$.
Consider a unit interval $[0,1]$ in which $n$ points arrive one-by-one independently and uniformly at random. On arrival of a point, the problem is to immediately and irrevocably color it in ${+1,-1}$ while ensuring that every interval $[a,b] subseteq [0,1]$ is nearly-balanced. We define emph{discrepancy} as the largest imbalance of any interval during the entire process. If all the arriving points were known upfront then we can color them alternately to achieve a discrepancy of $1$. What is the minimum possible expected discrepancy when we color the points online? We show that the discrepancy of the above problem is sub-polynomial in $n$ and that no algorithm can achieve a constant discrepancy. This is a substantial improvement over the trivial random coloring that only gets an $widetilde{O}(sqrt n)$ discrepancy. We then obtain similar results for a natural generalization of this problem to $2$-dimensions where the points arrive uniformly at random in a unit square. This generalization allows us to improve recent results of Benade et al.cite{BenadeKPP-EC18} for the online envy minimization problem when the arrivals are stochastic.
In the Metric Capacitated Covering (MCC) problem, given a set of balls $mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $mathcal{B}subseteq mathcal{B}$ and an assignment of the points in $P$ to the balls in $mathcal{B}$ such that each point is assigned to a ball that contains it and each ball is assigned with at most $U$ points. MCC achieves an $O(log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(log |P|)$ even with $beta < 3$ factor expansion of the balls. Bandyapadhyay~{et al.} [SoCG 2018, DCG 2019] showed that one can obtain an $O(1)$-approximation for the problem with $6.47$ factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound $3$ and the upper bound $6.47$. In this current work, we show that it is possible to obtain an $O(1)$-approximation with only $4.24$ factor expansion of the balls. We also show a similar upper bound of $5$ for a more generalized version of MCC for which the best previously known bound was $9$.
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