More Like this

1. Simple Topological Drawings of k-Planar Graphs.

   https://doi.org/10.1007/978-3-030-68766-3_31  

   Hoffmann, Michael ; Liu, Chih-Hung ; Reddy, Meghana M. ; Toth, Csaba D. ( February 2021 , Graph Drawing and Network Visualization)

   null (Ed.)

   Every finite graph admits a simple (topological) drawing, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, k-planar graphs are those graphs that can be drawn so that every edge has at most k crossings (i.e., they admit a k-plane drawing). It is known that for k≤3 , every k-planar graph admits a k-plane simple drawing. But for k≥4 , there exist k-planar graphs that do not admit a k-plane simple drawing. Answering a question by Schaefer, we show that there exists a function Open image in new window such that every k-planar graph admits an f(k)-plane simple drawing, for all Open image in new window. Note that the function f depends on k only and is independent of the size of the graph. Furthermore, we develop an algorithm to show that every 4-planar graph admits an 8-plane simple drawing. 

   more » « less
2. A Model for Ant Trail Formation and its Convergence Properties

   https://doi.org/10.4230/LIPIcs.ITCS.2021.85  

   Charikar, Moses ; Garg, Shivam ; Gordon, Deborah M. ; Shiragur, Kirankumar ( January 2021 , 12th Innovations in Theoretical Computer Science Conference (ITCS 2021))

   null (Ed.)

   We introduce a model for ant trail formation, building upon previous work on biologically feasible local algorithms that plausibly describe how ants maintain trail networks. The model is a variant of a reinforced random walk on a directed graph, where ants lay pheromone on edges as they traverse them and the next edge to traverse is chosen based on the level of pheromone; this pheromone decays with time. There is a bidirectional flow of ants in the network: the forward flow proceeds along forward edges from source (e.g. the nest) to sink (e.g. a food source), and the backward flow in the opposite direction. Some fraction of ants are lost as they pass through each node (modeling the loss of ants due to exploration observed in the field). We initiate a theoretical study of this model. We note that ant navigation has inspired the field of ant colony optimization, heuristics that have been applied to several combinatorial optimization problems; however the algorithms developed there are considerably more complex and not constrained to being biologically feasible. We first consider the linear decision rule, where the flow divides itself among the next set of edges in proportion to their pheromone level. Here, we show that the process converges to the path with minimum leakage when the forward and backward flows do not change over time. On the other hand, when the forward and backward flows increase over time (caused by positive reinforcement from the discovery of a food source, for example), we show that the process converges to the shortest path. These results are for graphs consisting of two parallel paths (a case that has been investigated before in experiments). Through simulations, we show that these results hold for more general graphs drawn from various random graph models; proving this convergence in the general case is an interesting open problem. Further, to understand the behaviour of other decision rules beyond the linear rule, we consider a general family of decision rules. For this family, we show that there is no advantage of using a non-linear decision rule, if the goal is to find the shortest or the minimum leakage path. We also show that bidirectional flow is necessary for convergence to such paths. Our results provide a plausible explanation for field observations, and open up new avenues for further theoretical and experimental investigation. 

   more » « less
3. Irreducibility of Recombination Markov Chains in the Triangular Lattice

   Cannon, Sarah ( May 2023 , SIAM Conference on Applied and Computational Discrete Algorithms (ACDA23))

   Berry, Jonathan ; Shmoys, David ; Cowen, Lenore ; Naumann, Uwe (Ed.)

   In the United States, regions (such as states or counties) are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can have a profound effect on who's elected, and drawing the districts to give an advantage to a certain group is known as gerrymandering. It can be surprisingly difficult to detect when gerrymandering is occurring, but one algorithmic method is to compare a current districting plan to a large number of randomly sampled plans to see whether it is an outlier. Recombination Markov chains are often used to do this random sampling: randomly choose two districts, consider their union, and split this union up in a new way. This approach works well in practice and has been widely used, including in litigation, but the theory behind it remains underdeveloped. For example, it's not known if recombination Markov chains are irreducible, that is, if recombination moves suffice to move from any districting plan to any other. Irreducibility of recombination Markov chains can be formulated as a graph problem: for a planar graph G, is the space of all partitions of G into κ connected subgraphs (κ districts) connected by recombination moves? While the answer is yes when districts can be as small as one vertex, this is not realistic in real-world settings where districts must have approximately balanced populations. Here we fix district sizes to be κ1 ± 1 vertices, κ2 ± 1 vertices,… for fixed κ1, κ2,…, a more realistic setting. We prove for arbitrarily large triangular regions in the triangular lattice, when there are three simply connected districts, recombination Markov chains are irreducible. This is the first proof of irreducibility under tight district size constraints for recombination Markov chains beyond small or trivial examples. The triangular lattice is the most natural setting in which to first consider such a question, as graphs representing states/regions are frequently triangulated. The proof uses a sweep-line argument, and there is hope it will generalize to more districts, triangulations satisfying mild additional conditions, and other redistricting Markov chains. 

   more » « less
4. Visualization of Bipartite Graphs in Limited Window Size

   Evans, William ; Koeck, Kassian ; Kobourov, Stephen ( February 2024 , 49th Intl. Conference on Current Trends in Theory and Practice of Computer Science (SofSem))

   Bipartite graphs are commonly used to visualize objects and their features. An object may possess several features and several objects may share a common feature. The standard visualization of bipartite graphs, with objects and features on two (say horizontal) parallel lines at integer coordinates and edges drawn as line segments, can often be difficult to work with. A common task in visualization of such graphs is to consider one object and all its features. This naturally defines a drawing window, defined as the smallest interval that contains the x-coordinates of the object and all its features. We show that if both objects and features can be reordered, minimizing the average window size is NP-hard. However, if the features are fixed, then we provide an efficient polynomial time algorithm for arranging the objects, so as to minimize the average window size. Finally, we introduce a different way of visualizing the bipartite graph, by placing the nodes of the two parts on two con- centric circles. For this setting we also show NP-hardness for the general case and a polynomial time algorithm when the features are fixed. 

   more » « less
5. C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

   https://doi.org/10.1007/s00453-021-00839-2  

   Da Lozzo, Giordano ; Eppstein, David ; Goodrich, Michael T. ; Gupta, Siddharth ( August 2021 , Algorithmica)

   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 Tóth, 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 Tóth. 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 $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances. 

   more » « less