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Register a new userSynchronized Planarity with Applications to Constrained Planarity Problems
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Added by
Simon D. Fink
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchronized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of $O(n^{8})$.
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In general, a graph modification problem is defined by a graph modification operation $boxtimes$ and a target graph property ${cal P}$. Typically, the modification operation $boxtimes$ may be vertex removal}, edge removal}, edge contraction}, or edge addition and the question is, given a graph $G$ and an integer $k$, whether it is possible to transform $G$ to a graph in ${cal P}$ after applying $k$ times the operation $boxtimes$ on $G$. This problem has been extensively studied for particilar instantiations of $boxtimes$ and ${cal P}$. In this paper we consider the general property ${cal P}_{{phi}}$ of being planar and, moreover, being a model of some First-Order Logic sentence ${phi}$ (an FOL-sentence). We call the corresponding meta-problem Graph $boxtimes$-Modification to Planarity and ${phi}$ and prove the following algorithmic meta-theorem: there exists a function $f:Bbb{N}^{2}toBbb{N}$ such that, for every $boxtimes$ and every FOL sentence ${phi}$, the Graph $boxtimes$-Modification to Planarity and ${phi}$ is solvable in $f(k,|{phi}|)cdot n^2$ time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifmans Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.
For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. ESA95], has only been recently settled [Radoslav Fulek and Csaba D. Toth. Atomic Embeddability, Clustered Planarity, and Thickenability. To appear at SODA20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Toth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.
We show algorithms for computing representative families for matroid intersections and use them in fixed-parameter algorithms for set packing, set covering, and facility location problems with multiple matroid constraints. We complement our tractability results by hardness results.
This paper formalizes connections between stability of polynomials and convergence rates of Markov Chain Monte Carlo (MCMC) algorithms. We prove that if a (multivariate) partition function is nonzero in a region around a real point $lambda$ then spectral independence holds at $lambda$. As a consequence, for Holant-type problems (e.g., spin systems) on bounded-degree graphs, we obtain optimal $O(nlog n)$ mixing time bounds for the single-site update Markov chain known as the Glauber dynamics. Our result significantly improves the running time guarantees obtained via the polynomial interpolation method of Barvinok (2017), refined by Patel and Regts (2017). There are a variety of applications of our results. In this paper, we focus on Holant-type (i.e., edge-coloring) problems, including weighted edge covers and weighted even subgraphs. For the weighted edge cover problem (and several natural generalizations) we obtain an $O(nlog{n})$ sampling algorithm on bounded-degree graphs. The even subgraphs problem corresponds to the high-temperature expansion of the ferromagnetic Ising model. We obtain an $O(nlog{n})$ sampling algorithm for the ferromagnetic Ising model with a nonzero external field on bounded-degree graphs, which improves upon the classical result of Jerrum and Sinclair (1993) for this class of graphs. We obtain further applications to antiferromagnetic two-spin models on line graphs, weighted graph homomorphisms, tensor networks, and more.
In a emph{fan-planar drawing} of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt, who proved that every $n$-vertex fan-planar drawing has at most $5n-10$ edges, and that this bound is tight for $n geq 20$. We extend their result, both from the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrain
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