Showing papers by "David Eppstein published in 2021"

Stack-Number is Not Bounded by Queue-Number

Abstract: We describe a family of graphs with queue-number at most 4 but unbounded stack-number. This resolves open problems of Heath, Leighton and Rosenberg (1992) and Blankenship and Oporowski (1999).

On the Edge Crossings of the Greedy Spanner

Abstract: $t$-spanners are used to approximate the pairwise distances between a set of points in a metric space. They have only a few edges compared to the total number of pairs and they provide a $t$-approximation on the distance of any two arbitrary points. There are many ways to construct such graphs and one of the most efficient ones, in terms of weight and the number of edges of the resulting graph, is the greedy spanner. In this paper, we study the edge crossings of the greedy spanner for points in the Euclidean plane. We prove a constant upper bound for the number of intersections with larger edges that only depends on the stretch factor of the spanner, $t$, and we show there can be more than a bounded number of intersections with smaller edges. Our results imply that greedy spanners for points in the plane have separators of size $\mathcal{O}(\sqrt n)$, that their planarizations have linear size, and that a separator hierarchy for these graphs can be constructed from their planarizations in linear time.

C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Abstract: 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 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Toth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. 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 (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Toth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time $$O(r(n^2) + n^2)$$ for general, possibly non-flat, instances.

Parameterized Complexity of Finding Subgraphs with Hereditary Properties on Hereditary Graph Classes

Abstract: We investigate the parameterized complexity of finding subgraphs with hereditary properties on graphs belonging to a hereditary graph class. Given a graph G, a non-trivial hereditary property \(\varPi \) and an integer parameter k, the general problem \(P(G,\varPi ,k)\) asks whether there exists k vertices of G that induce a subgraph satisfying property \(\varPi \). This problem, \(P(G,\varPi ,k)\) has been proved to be \(\mathsf {NP}\)-complete by Lewis and Yannakakis. The parameterized complexity of this problem is shown to be \(\mathsf {W}[1]\)-complete by Khot and Raman, if \(\varPi \) includes all trivial graphs (graphs with no edges) but not all complete graphs and vice versa; and is fixed-parameter tractable, otherwise. As the problem is \(\mathsf {W}[1]\)-complete on general graphs when \(\varPi \) includes all trivial graphs but not all complete graphs and vice versa, it is natural to further investigate the problem on restricted graph classes.

On Polyhedral Realization with Isosceles Triangles

Abstract: Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer i such that all convex i-iterated Kleetopes have a scalene face. However, we also show that all Kleetopes of triangulated polyhedral graphs have non-convex non-self-crossing realizations in which all faces are isosceles. We answer another question of Malkevitch by observing that a spherical tiling of Dawson (Renaissance Banff, Bridges Conference, pp. 489–496, 2005) leads to a fourth infinite family of convex polyhedra in which all faces are congruent isosceles triangles, adding one to the three families previously known to Malkevitch. We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.

Parameterized Complexity of Finding Subgraphs with Hereditary Properties on Hereditary Graph Classes

Abstract: We investigate the parameterized complexity of finding subgraphs with hereditary properties on graphs belonging to a hereditary graph class. Given a graph $G$, a non-trivial hereditary property $\Pi$ and an integer parameter $k$, the general problem $P(G,\Pi,k)$ asks whether there exists $k$ vertices of $G$ that induce a subgraph satisfying property $\Pi$. This problem, $P(G,\Pi,k)$ has been proved to be NP-complete by Lewis and Yannakakis. The parameterized complexity of this problem is shown to be W[1]-complete by Khot and Raman, if $\Pi$ includes all trivial graphs but not all complete graphs and vice versa; and is fixed-parameter tractable (FPT), otherwise. As the problem is W[1]-complete on general graphs when $\Pi$ includes all trivial graphs but not all complete graphs and vice versa, it is natural to further investigate the problem on restricted graph classes. Motivated by this line of research, we study the problem on graphs which also belong to a hereditary graph class and establish a framework which settles the parameterized complexity of the problem for various hereditary graph classes. In particular, we show that: $P(G,\Pi,k)$ is solvable in polynomial time when the graph $G$ is co-bipartite and $\Pi$ is the property of being planar, bipartite or triangle-free (or vice-versa). $P(G,\Pi,k)$ is FPT when the graph $G$ is planar, bipartite or triangle-free and $\Pi$ is the property of being planar, bipartite or triangle-free, or graph $G$ is co-bipartite and $\Pi$ is the property of being co-bipartite. $P(G,\Pi,k)$ is W[1]-complete when the graph $G$ is $C_4$-free, $K_{1,4}$-free or a unit disk graph and $\Pi$ is the property of being either planar or bipartite.

Limitations on Realistic Hyperbolic Graph Drawing

Abstract: We show that several types of graph drawing in the hyperbolic plane require features of the drawing to be separated from each other by sub-constant distances, distances so small that they can be accurately approximated by Euclidean distance. Therefore, for these types of drawing, hyperbolic geometry provides no benefit over Euclidean graph drawing.

A Stronger Lower Bound on Parametric Minimum Spanning Trees.

Abstract: We prove that, for an undirected graph with n vertices and m edges, each labeled with a linear function of a parameter \(\lambda \), the number of different minimum spanning trees obtained as the parameter varies can be \(\varOmega (m\log n)\).

A Stronger Lower Bound on Parametric Minimum Spanning Trees

Abstract: We prove that, for an undirected graph with $n$ vertices and $m$ edges, each labeled with a linear function of a parameter $\lambda$, the number of different minimum spanning trees obtained as the parameter varies can be $\Omega(m\log n)$.

Optimal Spanners for Unit Ball Graphs in Doubling Metrics.

Abstract: Resolving an open question from 2006, we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple $\mathcal{O}(\log^*n)$-round distributed algorithm that given a unit ball graph $G$ with $n$ vertices and a positive constant $\epsilon < 1$ finds a $(1+\epsilon)$-spanner with constant bounds on its maximum degree and its lightness using only 2-hop neighborhood information. This immediately improves the algorithm of Damian, Pandit, and Pemmaraju which runs in $\mathcal{O}(\log^*n)$ rounds but has a $\mathcal{O}(\log \Delta)$ bound on its lightness, where $\Delta$ is the ratio of the length of the longest edge in $G$ to the length of the shortest edge. We further study the problem in the two dimensional Euclidean plane and we provide a construction with similar properties that has a constant average number of edge intersection per node. This is the first distributed low-intersection topology control algorithm to the best of our knowledge. Our distributed algorithms rely on the maximal independent set algorithm of Schneider and Wattenhofer that runs in $\mathcal{O}(\log^*n)$ rounds of communication. If a maximal independent set is known beforehand, our algorithms run in constant number of rounds.

NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs

Abstract: In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this “opens ...

Angles of Arc-Polygons and Lombardi Drawings of Cacti

Abstract: We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are at most pi. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.

Rapid mixing of the hardcore Glauber dynamics and other Markov chains in bounded-treewidth graphs.

Abstract: We give a new rapid mixing result for a natural random walk on the independent sets of an input graph $G$. Rapid mixing is of interest in approximately sampling a structure, over some underlying set or graph, from some target distribution. In the case of independent sets, we show that when $G$ has bounded treewidth, this random walk -- known as the hardcore Glauber dynamics -- mixes rapidly for all values of the standard parameter $\lambda > 0$, giving a simple alternative to existing sampling algorithms for these structures. We also show rapid mixing for Markov chains on dominating sets and $b$-edge covers (for fixed $b\geq 1$ and $\lambda > 0$) in the case where treewidth is bounded, and for Markov chains on the $b$-matchings (for fixed $b \geq 1$ and $\lambda > 0$), the maximal independent sets, and the maximal $b$-matchings of a graph (for fixed $b \geq 1$), in the case where carving width is bounded. We prove our results by developing a divide-and-conquer framework using the well-known multicommodity flows technique. Using this technique, we additionally show that a similar dynamics on the $k$-angulations of a convex set of $n$ points mixes in quasipolynomial time for all $k \geq 3$. (McShine and Tetali gave a stronger result in the special case $k = 3$.) Our technique also allows us to strengthen existing results by Dyer, Goldberg, and Jerrum and by Heinrich for the Glauber dynamics on the $q$-colorings of $G$ on graphs of bounded carving width, when $q \geq \Delta + 2$ is bounded. Specifically, our technique yields an improvement in the dependence on treewidth when $\Delta < 2t$ or when $q < 4t$ and $\Delta < t^2$. We additionally show that the Glauber dynamics on the partial $q$-colorings of $G$ mix rapidly for all $\lambda > 0$ when $q \geq \Delta + 2$ is bounded.

The Graphs of Stably Matchable Pairs

Abstract: We study the graphs formed from instances of the stable matching problem by connecting pairs of elements with an edge when there exists a stable matching in which they are matched. Our results include the NP-completeness of recognizing these graphs, a recognition algorithm that is singly exponential in the number of edges of the given graph, and an algorithm whose time is linear in the number of vertices of the graph but exponential in a polynomial of its carving width.

Finding Relevant Points for Nearest-Neighbor Classification.

Abstract: In nearest-neighbor classification problems, a set of $d$-dimensional training points are given, each with a known classification, and are used to infer unknown classifications of other points by using the same classification as the nearest training point. A training point is relevant if its omission from the training set would change the outcome of some of these inferences. We provide a simple algorithm for thinning a training set down to its subset of relevant points, using as subroutines algorithms for finding the minimum spanning tree of a set of points and for finding the extreme points (convex hull vertices) of a set of points. The time bounds for our algorithm, in any constant dimension $d\ge 3$, improve on a previous algorithm for the same problem by Clarkson (FOCS 1994).

Egyptian Fractions with Denominators from Sequences Closed Under Doubling

Abstract: Resolving a conjecture of Zhi-Wei Sun, we prove that every rational number can be represented as a sum of distinct unit fractions whose denominators are practical numbers. The same method applies to allowed denominators that are closed under multiplication by two and include a multiple of every positive integer, including the odious numbers, evil numbers, Hardy-Ramanujan numbers, Jordan-Polya numbers, and fibbinary numbers.