Crossing Number Problem Research Articles

Crossing Minimal Edge‐Constrained Layout Planning using Benders Decomposition

We present a new crossing number problem, which we refer to as the edge‐constrained weighted two‐layer crossing number problem (ECW2CN). The ECW2CN arises in layout planning of hose coupling stations at BASF, where the challenge is to find a crossing minimal assignment of tube‐connected units to given positions on two opposing layers. This allows the use of robots in an effort to reduce the probability of operational disruptions and to increase human safety. Physical limitations imply maximal length and maximal curvature conditions on the tubes as well as spatial constraints imposed by the surrounding walls. This is the major difference of ECW2CN to all known variants of the crossing number problem. Such as many variants of the crossing number problem, ECW2CN is [Formula: see text]‐hard. Because the optimization model grows fast with respect to the input data, we face out‐of‐memory errors for the monolithic model. Therefore, we develop two solution methods. In the first method, we tailor Benders decomposition toward the problem. The Benders subproblems are solved analytically and the Benders master problem is strengthened by additional cuts. Furthermore, we combine this Benders decomposition with ideas borrowed from fix‐and‐relax heuristics to design the Dynamic Fix‐and‐Relax Pump (DFRP). Based on an initial solution, DFRP improves successively feasible points by solving dynamically sampled smaller problems with Benders decomposition. Because the optimization model is a surrogate model for its time‐dependent formulation, we evaluate the obtained solutions for different choices of the objective function via a simulation model. All algorithms are implemented efficiently using advanced features of the GuRoBi‐Python API, such as callback functions and lazy constraints. We present a case study for BASF using real data and make the real‐world data openly available.

The Implementation of Bipartite Graph To Minimize Crossing Number Problem of Crossroads in Manado

The Implementation Of Bipartite Graph To Minimize Crossing Number Problem Of Crossroads In Manado. Supervised by BENNY PINONTAN as main supervisor and CHRIESTIE E. J. C. MONTOLALU as co-supervisor. In general, the crossroads are the meeting points of two-way roads from four different places. This causes cross direction at that point. There are various methods that can be used to minimize the crossing number problem crossroad, for example graph theory. The ability of graph theory can later help describe crossroads in Manado into graph, with nodes and lines. In this case, the crossing number problems will solve by bipartite graph. Bipartite graph is a graph that does not have an odd cycle, and can be partitioned into two parts of a set of vertices. Based on results of this research, the appropriate form of the bipartite graph is and in two different form. First, with an isolated vertex, and second, without isolated vertex. In the case of crossroads, Bipartite graph turns out to be one method that is very suitable and helps determine the crossing number and its solution quickly.

The Complexity of Computing the Cylindrical and the $t$-Circle Crossing Number of a Graph

A plane drawing of a graph is cylindrical if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The cylindrical crossing number of a graph \(G\) is the minimum number of crossings in a cylindrical drawing of \(G\). In his influential survey on the variants of the definition of the crossing number of a graph, Schaefer lists the complexity of computing the cylindrical crossing number of a graph as an open question. In this paper, we prove that the problem of deciding whether a given graph admits a cylindrical embedding is NP-complete, and as a consequence we show that the \(t\)-cylindrical crossing number problem is also NP-complete. Moreover, we show an analogous result for the natural generalization of the cylindrical crossing number, namely the \(t\)-crossing number.

On the rectilinear crossing number of complete uniform hypergraphs

In this paper, we consider a generalized version of the rectilinear crossing number problem of drawing complete graphs on a plane. The minimum number of crossing pairs of hyperedges among all d-dimensional rectilinear drawings of a d-uniform hypergraph is known as the d-dimensional rectilinear crossing number of the hypergraph. The currently best-known lower bound on the d-dimensional rectilinear crossing number of a complete d-uniform hypergraph with n vertices in general position in Rd is Ω(2ddlog⁡d)(n2d). In this paper, we improve this lower bound to Ω(2d)(n2d). We also consider the special case when all the vertices of a d-uniform hypergraph are placed on the d-dimensional moment curve. For such d-dimensional rectilinear drawings of the complete d-uniform hypergraph with n vertices, we show that the number of crossing pairs of hyperedges is Θ(4dd)(n2d).

A tighter insertion-based approximation of the crossing number

Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of $$G+F$$ with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F. We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee—depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph $$G+F$$ , while computing the crossing number of $$G+F$$ exactly is NP-hard already when $$|F|=1$$ . Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.

1-Planarity of Graphs with a Rotation System

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. 1-planarity is known NP-hard, even for graphs of bounded bandwidth, pathwidth, or treewidth, and for near-planar graphs in which an edge is added to a planar graph. On the other hand, there is a linear time 1-planarity testing algorithm for maximal 1-planar graphs with a given rotation system. In this work, we show that 1-planarity remains NP-hard even for 3-connected graphs with (or without) a rotation system. Moreover, the crossing number problem remains NP-hard for 3-connected 1-planar graphs with (or without) a rotation system.

Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný.

Advances in the Planarization Method: Effective Multiple Edge Insertions

The planarization method is the strongest known method to heuristically find good solutions to the general crossing number problem in graphs: starting from a planar subgraph, one iteratively inserts edges, representing crossings via dummy nodes. In the recent years, several improvements both from the practical and the theoretical point of view have been made. We review these advances and conduct an extensive study of the algorithms' practical implications. Thereby, we present the first implementation of an approximation algorithm for the crossing number problem of general graphs, and compare the obtained results with known exact crossing number solutions.

Vertex insertion approximates the crossing number of apex graphs

An apex graph is a graph G from which only one vertex v has to be removed to make it planar. We show that the crossing number of such G can be approximated up to a factor of Δ ( G − v ) ⋅ d ( v ) / 2 by solving the vertex inserting problem, i.e. inserting a vertex plus incident edges into an optimally chosen planar embedding of a planar graph. Since the latter problem can be solved in polynomial time, this establishes the first polynomial fixed-factor approximation algorithm for the crossing number problem of apex graphs with bounded degree. Furthermore, we extend this result by showing that the optimal solution for inserting multiple edges or vertices into a planar graph also approximates the crossing number of the resulting graph.

Guest Editor’s Foreword: Algorithms, Combinatorics, & Geometry

Knowledge of mathematics and combinatorics is central to the design of combinatorial algorithms. In addition, applications-related concepts may also give rise to interesting mathematical problems, or may even be used to solve mathematical problems. The workshop on Algorithms, Combinatorics, and Geometry (ACG) was held during November 29–December 1, 2007, at the University of North Texas (UNT), Denton, Texas. The ACG workshop was intended to provide a deeper understanding of some important concepts which are fundamental to different disciplines, and to bring together researchers in different areas of discrete mathematics and computer science and provide an opportunity for them to interact. The ACG workshop did not have a proceedings, but an abstract for each talk was posted online. See acg.unt.edu for details. This special issue of Algorithmica presents a collection of 10 invited papers which are related to the main theme of the workshop. Preliminary versions of some of these papers may have been published in conferences. Nonetheless, all papers appearing in this issue have been thoroughly refereed and are revised so that they are in a suitable form for publication in Algorithmica. We have included papers that cover a broad range of important concepts which are of interest to researchers in graph theory and graph algorithms, combinatorial and computational geometry, computer security, graph drawing, and randomized algorithms. While the main results in some papers may appear purely structural, all results have computational consequences. Cabello and Mohar study the crossing number problem for those graphs that become planar after removal of one edge. They develop min-max formulas, and hence, efficiently computable lower and upper bounds that give rise to approximation algorithms that improve the best known previous results.

Facets in the Crossing Number Polytope

In the last years, several integer linear programming (ILP) formulations for the crossing number problem arose. While they all contain a common conceptual core, the properties of the corresponding polytopes have never been investigated. In this paper, we formally establish the crossing number polytope and show several facet-defining constraint classes.

Experiments on exact crossing minimization using column generation

The crossing number of a graph G is the smallest number of edge crossings in any drawing of G into the plane. Recently, the first branch-and-cut approach for solving the crossing number problem has been presented in Buchheim et al. [2005]. Its major drawback was the huge number of variables out of which only very few were actually used in the optimal solution. This restricted the algorithm to rather small graphs with low crossing number. In this article, we discuss two column generation schemes; the first is based on traditional algebraic pricing, and the second uses combinatorial arguments to decide whether and which variables need to be added. The main focus of this article is the experimental comparison between the original approach and these two schemes. In addition, we evaluate the quality achieved by the best-known crossing number heuristic by comparing the new results with the results of the heuristic.

Hopfield Neural Network Based on Estimation of Distribution for Two-Page Crossing Number Problem

This paper presents a Hopfield neural network (HNN) combined with estimation of distribution (EDA) for the two-page crossing number problem. In the proposed algorithm, once the network is trapped in local minima, the perturbation based on EDA can generate a new starting point for the HNN for further search, which is in a promising area characterized by a probability model and is not far away from the best solution found so far. The proposed algorithm can escape from local minima and further search better results. Simulation results show that the proposed algorithm is better than previous methods.

A branch-and-cut approach to the crossing number problem

The crossing number of a graph is the minimum number of edge crossings in any drawing of the graph in the plane. Extensive research has produced bounds on the crossing number and exact formulae for special graph classes, yet the crossing numbers of graphs such as K 11 or K 9 , 11 are still unknown. Finding the crossing number is NP-hard for general graphs and no practical algorithm for its computation has been published so far. We present an integer linear programming formulation that is based on a reduction of the general problem to a restricted version of the crossing number problem in which each edge may be crossed at most once. We also present cutting plane generation heuristics and a column generation scheme. As we demonstrate in a computational study, a branch-and-cut algorithm based on these techniques as well as recently published preprocessing algorithms can be used to successfully compute the crossing number for small- to medium-sized general graphs for the first time.

Approximating the fixed linear crossing number

We present a randomized polynomial-time approximation algorithm for the fixed linear crossing number problem (FLCNP). In this problem, the vertices of a graph are placed in a fixed order along a horizontal “node line” in the plane, each edge is drawn as an arc in one of the two half-planes (pages), and the objective is to minimize the number of edge crossings. FLCNP is NP-hard, and no previous polynomial-time approximation algorithms are known. We show that the problem can be generalized to k pages and transformed to the maximum k-cut problem which admits a randomized polynomial-time approximation. For the 2-page case, our approach leads to a randomized polynomial time 0.878 + 0.122 ρ approximation algorithm for FLCNP, where ρ is the ratio of the number of conflicting pairs (pairs of edges that cross if drawn in the same page) to the crossing number. We further investigate this performance ratio on the random graph family G n , 1 / 2 , where each edge of the complete graph K n occurs with probability 1 2 . We show that a longstanding conjecture for the crossing number of K n implies that with probability at least 1 - 4 e - λ 2 , the expected performance bound of the algorithm on a random graph from G n , 1 / 2 is 1.366 + O ( λ / n ) . A series of experiments is performed to compare the algorithm against two other leading heuristics on a set of test graphs. The results indicate that the randomized algorithm yields near-optimal solutions and outperforms the other heuristics overall.

An Improved Neural Network Model for the Two-Page Crossing Number Problem

The simplest graph drawing method is that of putting the vertices of a graph on a line and drawing the edges as half-circles either above or below the line. Such drawings are called two-page book drawings. The smallest number of crossings over all two-page drawings of a graph G is called the two-page crossing number of G. Cimikowski and Shope have solved the two-page crossing number problem for an n-vertex and m-edge graph by using a Hopfield network with 2 m neurons. We present here an improved Hopfield model with m neurons. The new model achieves much better performance in the quality of solutions and is more efficient than the model of Cimikowski and Shope for all graphs tested. The parallel time complexity of the algorithm, without considering the crossing number calculations, is O(m) for the new Hopfield model with m processors clearly outperforming the previous algorithm.

Parallelisation of genetic algorithms for the 2-page crossing number problem

Genetic algorithms (GAs) have been applied to solve the 2-page crossing number problem successfully, but since they work with one global population, the search time and space are limited. Parallelisation provides an attractive prospect to improve the efficiency and solution quality of GAs. This paper investigates the complexity of parallel genetic algorithms (PGAs) based on two evaluation measures: computation time to communication time and population size to chromosome size. Moreover, the paper unifies the framework of PGA models with the function PGA ( subpopulation size, cluster size, migration period, topology), and explores the performance of PGAs for the 2-page crossing number problem.

An analysis of some linear graph layout heuristics

The worst-case performances of some heuristics for the fixed linear crossing number problem (FLCNP) are analyzed. FLCNP is similar to the 2-page book crossing number problem in which the vertices of a graph are optimally placed on a horizontal "node line" in the plane, each edge is drawn as an arc in one half-plane (page), and the objective is to minimize the number of edge crossings. In FLCNP, the order of the vertices along the node line is predetermined and fixed. FLCNP belongs to the class of NP-hard optimization problems Masuda et al., 1990. In this paper we show that for each of the heuristics described, there exist classes of n-vertex, m-edge graphs which force it to obtain a number of crossings which is a function of n or m when the optimal number is a small constant. This leaves open the problem of finding a heuristic with a constant error bound for the problem.

Crossing number is hard for cubic graphs

It was proved by [M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods 4 (1983) 312–316] that computing the crossing number of a graph is an NP-hard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NP-hard to determine the crossing number of a simple 3-connected cubic graph. In particular, this implies that the minor-monotone version of the crossing number problem is also NP-hard, which has been open till now.

Crossing Minimisation Heuristics for 2-page Drawings

Abstract The minimisation of edge crossings in a book drawing of a graph G is one of important goals for a linear VLSI design, and the two-page crossing number of a graph G provides an upper bound for the standard planar crossing number. We propose several new heuristics for the 2-page drawing problem and test them on benchmark test sets like Rome graphs, Random Connected Graphs and some typical graphs. We get exact results of some structural graphs, and compare some of the experimental results with the one in paper [Cimikowski, Robert: Algorithms for the fixed linear crossing number problem, Discrete Applied Mathematics 122 (2002), 93–115].