A Reduction System for Optimal 1-Planar Graphs

There is a graph reduction system so that every optimal 1-planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if it can be drawn in the plane with at most one crossing per edge and is optimal if it has the maximum of 4n-8 edges. We show that the reduction system is context-sensitive so that the preservation of the graph class can be granted by local conditions which can be tested in constant time. Every optimal 1-planar graph G can be reduced to every extended wheel graph whose size is in a range from the (second) smallest one to some upper bound that depends on G. There is a reduction to the smallest extended wheel graph if G is not 5-connected, but not conversely. The reduction system has side effects and is non-deterministic and non-confluent. Nevertheless, reductions can be computed in linear time.Comment: 21 pages, 16 figure

An annotated bibliography on 1-planarity

The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs. It reviews the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations. As an additional contribution, we offer a list of open problems on 1-planar graphs

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 Hlin\v{e}n\'y.Comment: 27 pages, 10 figures. Part of the results appeared in Proceedings of the 26th Annual Symposium on Computational Geometry (SoCG), 68-76, 201

Sparsification and subexponential approximation

Instance sparsification is well-known in the world of exact computation since it is very closely linked to the Exponential Time Hypothesis. In this paper, we extend the concept of sparsification in order to capture subexponential time approximation. We develop a new tool for inapproximability, called approximation preserving sparsification and use it in order to get strong inapproximability results in subexponential time for several fundamental optimization problems as Max Independent Set, Min Dominating Set, Min Feedback Vertex Set, and Min Set Cover.Comment: 16 page

Crossing Minimization within Graph Embeddings

We propose a novel optimization-based approach to embedding heterogeneous high-dimensional data characterized by a graph. The goal is to create a two-dimensional visualization of the graph structure such that edge-crossings are minimized while preserving proximity relations between nodes. This paper provides a fundamentally new approach for addressing the crossing minimization criteria that exploits Farkas' Lemma to re-express the condition for no edge-crossings as a system of nonlinear inequality constraints. The approach has an intuitive geometric interpretation closely related to support vector machine classification. While the crossing minimization formulation can be utilized in conjunction with any optimization-based embedding objective, here we demonstrate the approach on multidimensional scaling by modifying the stress majorization algorithm to include penalties for edge crossings. The proposed method is used to (1) solve a visualization problem in tuberculosis molecular epidemiology and (2) generate embeddings for a suite of randomly generated graphs designed to challenge the algorithm. Experimental results demonstrate the efficacy of the approach. The proposed edge-crossing constraints and penalty algorithm can be readily adapted to other supervised and unsupervised optimization-based embedding or dimensionality reduction methods. The constraints can be generalized to remove overlaps between any graph components represented as convex polyhedrons including node-edge and node-node intersections.Comment: Previous versions of this paper are at http://cseweb.ucsd.edu/~lvdmaaten/workshops/nips2010/papers/bennett.pdf and http://www.cs.rpi.edu/research/pdf/11-03.pd

Crossing Number is Hard for Kernelization

The graph crossing number problem, cr(G)<=k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixed- parameter tractable for the parameter k [Grohe]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G)<=k can be in polynomial time reduced to an equivalent instance of size polynomial in k (and independent of |G|). We answer this question in the negative. Along the proof we show that the tile crossing number problem of twisted planar tiles is NP-hard, which has been an open problem for some time, too, and then employ the complexity technique of cross-composition. Our result holds already for the special case of graphs obtained from planar graphs by adding one edge

Relaxed Connected Dominating Set Problem with Application to Secure Power Network Design

This paper investigates a combinatorial optimization problem motived from a secure power network design application in [D\'{a}n and Sandberg 2010]. Two equivalent graph optimization formulations are derived. One of the formulations is a relaxed version of the connected dominating set problem, and hence the considered problem is referred to as relaxed connected dominating set (RCDS) problem. The RCDS problem is shown to be NP-hard, even for planar graphs. A mixed integer linear programming formulation is presented. In addition, for planar graphs a fixed parameter polynomial time solution methodology based on sphere-cut decomposition and dynamic programming is presented. The computation cost of the sphere-cut decomposition based approach grows linearly with problem instance size, provided that the branchwidth of the underlying graph is fixed and small. A case study with IEEE benchmark power networks verifies that small branchwidth are not uncommon in practice. The case study also indicates that the proposed methods show promise in computation efficiency

On 4-Map Graphs and 1-Planar Graphs and their Recognition Problem

We establish a one-to-one correspondence between 1-planar graphs and general and hole-free 4-map graphs and show that 1-planar graphs can be recognized in polynomial time if they are crossing-augmented, fully triangulated, and maximal 1-planar, respectively, with a polynomial of degree 120, 3, and 5, respectively

Approximation algorithms and hardness for domination with propagation

The power dominating set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes $S$ that power dominates all the nodes, where a node $v$ is power dominated if (1) $v$ is in $S$ or $v$ has a neighbor in $S$, or (2) $v$ has a neighbor $w$ such that $w$ and all of its neighbors except $v$ are power dominated. We show a hardness of approximation threshold of $2^{\log^{1-\epsilon}{n}}$ in contrast to the logarithmic hardness for the dominating set problem. We give an $O(\sqrt{n})$ approximation algorithm for planar graphs, and show that our methods cannot improve on this approximation guarantee. Finally, we initiate the study of PDS on directed graphs, and show the same hardness threshold of $2^{\log^{1-\epsilon}{n}}$ for directed \emph{acyclic} graphs. Also we show that the directed PDS problem can be solved optimally in linear time if the underlying undirected graph has bounded tree-width

Intractability of Optimal Multi-Robot Path Planning on Planar Graphs

We study the computational complexity of optimally solving multi-robot path planning problems on planar graphs. For four common time- and distance-based objectives, we show that the associated path optimization problems for multiple robots are all NP-complete, even when the underlying graph is planar. Establishing the computational intractability of optimal multi-robot path planning problems on planar graphs has important practical implications. In particular, our result suggests the preferred approach toward solving such problems, when the number of robots is large, is to augment the planar environment to reduce the sharing of paths among robots traveling in opposite directions on those paths. Indeed, such efficiency boosting structures, such as highways and elevated intersections, are ubiquitous in robotics and transportation applications.Comment: Updated draf