Showing papers by "David Eppstein published in 2019"

Track Layouts, Layered Path Decompositions, and Leveled Planarity

Abstract: We investigate two types of graph layouts, track layouts and layered path decompositions, and the relations between their associated parameters track-number and layered pathwidth. We use these two types of layouts to characterize leveled planar graphs, which are the graphs with planar leveled drawings with no dummy vertices. It follows from the known NP-completeness of leveled planarity that track-number and layered pathwidth are also NP-complete, even for the smallest constant parameter values that make these parameters nontrivial. We prove that the graphs with bounded layered pathwidth include outerplanar graphs, Halin graphs, and squaregraphs, but that (despite having bounded track-number) series–parallel graphs do not have bounded layered pathwidth. Finally, we investigate the parameterized complexity of these layouts, showing that past methods used for book layouts do not work to parameterize the problem by treewidth or almost-tree number but that the problem is (non-uniformly) fixed-parameter tractable for tree-depth.

NC Algorithms for Computing a Perfect Matching, the Number of Perfect Matchings, 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 K3,3-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in K3,3-free graphs." In this paper, we finally settle this 30-year-old open problem. Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and by Sankowski, we obtain NC algorithms for perfect matching in any minor-closed graph family that forbids a one-crossing graph. This result applies to several well-studied graph families including the K3,3-minor-free graphs and K5-minor-free graphs. Graphs in these families not only have unbounded genus, but can have genus as high as O(n). Our method applies as well to several other problems related to perfect matching. In particular, we obtain NC algorithms for the following problems in any family of graphs (or networks) with a one-crossing forbidden minor: - Determining whether a given graph has a perfect matching and if so, finding one. - Finding a minimum weight perfect matching in the graph, assuming that the edge weights are polynomially bounded. - Computing the number of perfect matchings in the graph. - Finding a maximum st-flow in the network, with arbitrary capacities. The main new idea enabling our results is the definition and use of matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.

Tracking Paths in Planar Graphs

Abstract: We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et. al. [3]. Given an undirected graph with a source $s$ and a destination $t$, find the smallest subset of vertices whose intersection with any $s-t$ path results in a unique sequence. In this paper, we show that this problem remains NP-complete when the graph is planar and we give a 4-approximation algorithm in this setting. We also show, via Courcelle's theorem, that it can be solved in linear time for graphs of bounded-clique width, when its clique decomposition is given in advance.

Reconfiguring Undirected Paths

Abstract: We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We show that this problem may be solved in linear time in trees, and is fixed-parameter tractable when parameterized either by the cyclomatic number of the input graph or by the length of the path. However, it is \(\mathsf {PSPACE}\)-complete for paths of unbounded length in graphs of bounded bandwidth.

Maximum Plane Trees in Multipartite Geometric Graphs

Abstract: A geometric graph is a graph whose vertices are points in the plane and whose edges are straight-line segments between the points. A plane spanning tree in a geometric graph is a spanning tree that is non-crossing. Let R and B be two disjoint sets of points in the plane where the points of R are colored red and the points of B are colored blue, and let \(n=|R\cup B|\). A bichromatic plane spanning tree is a plane spanning tree in the complete bipartite geometric graph with bipartition (R, B). In this paper we consider the maximum bichromatic plane spanning tree problem, which is the problem of computing a bichromatic plane spanning tree of maximum total edge length.

Tracking Paths in Planar Graphs

Abstract: We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et. al. [3]. Given an undirected graph with a source $s$ and a destination $t$, find the smallest subset of vertices whose intersection with any $s-t$ path results in a unique sequence. In this paper, we show that this problem remains NP-complete when the graph is planar and we give a 4-approximation algorithm in this setting. We also show, via Courcelle's theorem, that it can be solved in linear time for graphs of bounded-clique width, when its clique decomposition is given in advance.

Cubic Planar Graphs that cannot be Drawn on few Lines

Abstract: For every integer l, we construct a cubic 3-vertex-connected planar bipartite graph G with O(l^3) vertices such that there is no planar straight-line drawing of G whose vertices all lie on l lines. This strengthens previous results on graphs that cannot be drawn on few lines, which constructed significantly larger maximal planar graphs. We also find apex-trees and cubic bipartite series-parallel graphs that cannot be drawn on a bounded number of lines.

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.

Reconfiguring Undirected Paths

Abstract: We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We show that this problem may be solved in linear time in trees, and is fixed-parameter tractable when parameterized either by the cyclomatic number of the input graph or by the length of the path. However, it is PSPACE-complete for paths of unbounded length in graphs of bounded bandwidth.

Euclidean TSP, Motorcycle Graphs, and Other New Applications of Nearest-Neighbor Chains.

Abstract: We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We apply it to a diverse class of geometric problems: we construct the greedy multi-fragment tour for Euclidean TSP in $O(n\log n)$ time in any fixed dimension and for Steiner TSP in planar graphs in $O(n\sqrt{n}\log n)$ time; we compute motorcycle graphs (which are a central part in straight skeleton algorithms) in $O(n^{4/3+\varepsilon})$ time for any $\varepsilon>0$; we introduce a narcissistic variant of the $k$-attribute stable matching model, and solve it in $O(n^{2-4/(k(1+\varepsilon)+2)})$ time; we give a linear-time $2$-approximation for a 1D geometric set cover problem with applications to radio station placement.

Some Polycubes Have No Edge-Unzipping

Abstract: It is unknown whether or not every polycube has an edge-unfolding. A polycube is an object constructed by gluing cubes face-to-face. An edge-unfolding cuts edges on the surface and unfolds it to a net, a non-overlapping polygon in the plane. Here we explore the more restricted edge-unzippings where the cut edges form a path. We construct two different polycubes neither of which has an edge-unzipping.

Finding maximal sets of laminar 3-separators in planar graphs in linear time

Abstract: We consider decomposing a 3-connected planar graph G using laminar separators of size three. We show how to find a maximal set of laminar 3-separators in such a graph in linear time. We also discuss how to find maximal laminar set of 3-separators from special families. For example we discuss non-trivial cuts, ie. cuts which split G into two components of size at least two. For any vertex v, we also show how to find a maximal set of 3-separators disjoint from v which are laminar and satisfy: every vertex in a separator X has two neighbours not in the unique component of G − X containing v. In all cases, we show how to construct a corresponding tree decomposition of adhesion three. Our new algorithms form an important component of recent methods for finding disjoint paths in nonplanar graphs.

Cubic Planar Graphs that cannot be Drawn on few Lines

Abstract: For every integer $\ell$, we construct a cubic 3-vertex-connected planar bipartite graph $G$ with $O(\ell^3)$ vertices such that there is no planar straight-line drawing of $G$ whose vertices all lie on $\ell$ lines. This strengthens previous results on graphs that cannot be drawn on few lines, which constructed significantly larger maximal planar graphs. We also find apex-trees and cubic bipartite series-parallel graphs that cannot be drawn on a bounded number of lines.

Homotopy Height, Grid-Major Height and Graph-Drawing Height

Abstract: It is well-known that both the pathwidth and the outer-planarity of a graph can be used to obtain lower bounds on the height of a planar straight-line drawing of a graph. But both bounds fall short for some graphs. In this paper, we consider two other parameters, the (simple) homotopy height and the (simple) grid-minor height. We discuss the relationship between them and to the other parameters, and argue that they give lower bounds on the straight-line drawing height that are never worse than the ones obtained from pathwidth and outer-planarity.

Bipartite and Series-Parallel Graphs Without Planar Lombardi Drawings.

Abstract: We find a family of planar bipartite graphs all of whose Lombardi drawings (drawings with circular arcs for edges, meeting at equal angles at the vertices) are nonplanar. We also find families of embedded series-parallel graphs and apex-trees (graphs formed by adding one vertex to a tree) for which there is no planar Lombardi drawing consistent with the given embedding.

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 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. ESA'95], has only been recently settled [Radoslav Fulek and Csaba D. 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 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.

Existence and hardness of conveyor belts

Abstract: An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary of each disk, possibly multiple times. We prove three main results. First, for unit disks whose centers are both $x$-monotone and $y$-monotone, or whose centers have $x$-coordinates that differ by at least two units, a conveyor belt always exists and can be found efficiently. Second, it is NP-complete to determine whether disks of varying radii have a conveyor belt, and it remains NP-complete when we constrain the belt to touch disks exactly once. Third, any disjoint set of $n$ disks of arbitrary radii can be augmented by $O(n)$ "guide" disks so that the augmented system has a conveyor belt touching each disk exactly once, answering a conjecture of Demaine, Demaine, and Palop.

Counting Polygon Triangulations is Hard

Abstract: We prove that it is $\#\mathsf{P}$-complete to count the triangulations of a (non-simple) polygon.

Some Polycubes Have No Edge Zipper Unfolding

Abstract: It is unknown whether every polycube (polyhedron constructed by gluing cubes face-to-face) has an edge unfolding, that is, cuts along edges of the cubes that unfolds the polycube to a single nonoverlapping polygon in the plane. Here we construct polycubes that have no *edge zipper unfolding* where the cut edges are further restricted to form a path.

New applications of nearest-neighbor chains: Euclidean TSP and motorcycle graphs

Abstract: We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We use it to construct the greedy multi-fragment tour for Euclidean TSP in O(n log n) time in any fixed dimension and for Steiner TSP in planar graphs in O(n sqrt(n)log n) time; we compute motorcycle graphs, a central step in straight skeleton algorithms, in O(n^(4/3+epsilon)) time for any epsilon>0.

Homotopy height, grid-major height and graph-drawing height.

Abstract: It is well-known that both the pathwidth and the outer-planarity of a graph can be used to obtain lower bounds on the height of a planar straight-line drawing of a graph. But both bounds fall short for some graphs. In this paper, we consider two other parameters, the (simple) homotopy height and the (simple) grid-major height. We discuss the relationship between them and to the other parameters, and argue that they give lower bounds on the straight-line drawing height that are never worse than the ones obtained from pathwidth and outer-planarity.

Face flips in origami tessellations

Abstract: Given a flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on a region of the plane) with a mountain-valley (MV) assignment $\mu:E\to\{-1,1\}$ indicating which creases in $E$ bend convexly (mountain) or concavely (valley), we may \emph{flip} a face $F$ of $G$ to create a new MV assignment $\mu_F$ which equals $\mu$ except for all creases $e$ bordering $F$, where we have $\mu_F(e)=-\mu(e)$. In this paper we explore the configuration space of face flips for a variety of crease patterns $G$ that are tilings of the plane, proving examples where $\mu_F$ results in a MV assignment that is either never, sometimes, or always flat-foldable for various choices of $F$. We also consider the problem of finding, given two foldable MV assignments $\mu_1$ and $\mu_2$ of a given crease pattern $G$, a minimal sequence of face flips to turn $\mu_1$ into $\mu_2$. We find polynomial-time algorithms for this in the cases where $G$ is either a square grid or the Miura-ori, and show that this problem is NP-hard in the case where $G$ is the triangle lattice.

Bipartite and Series-Parallel Graphs Without Planar Lombardi Drawings

Abstract: We find a family of planar bipartite graphs all of whose Lombardi drawings (drawings with circular arcs for edges, meeting at equal angles at the vertices) are nonplanar. We also find families of embedded series-parallel graphs and apex-trees (graphs formed by adding one vertex to a tree) for which there is no planar Lombardi drawing consistent with the given embedding.