GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs

We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale graphs and complex networks are particularly considered in its design. The problems currently supported include maximum clique, graph colouring, maximum independent set, minimum vertex clique covering, minimum dominating set, as well as the longest simple cycle problem. Suboptimal solutions and intervals for optimal objective values are estimated using scalable heuristics. The tool is designed with extensibility in mind, with the view of further problems and both new fast and high-performance heuristics to be added in the future. GraphCombEx has already been successfully used as a support tool in a number of recent research studies using combinatorial optimisation to analyse complex networks, indicating its promise as a research software tool

Hierarchical Partial Planarity

In this paper we consider graphs whose edges are associated with a degree of {\em importance}, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to construct layouts of these graphs in which the readability of an edge is proportional to its importance, that is, more important edges have fewer crossings. We formalize this problem and study the case in which there exist three different degrees of importance. We give a polynomial-time testing algorithm when the graph induced by the two most important sets of edges is biconnected. We also discuss interesting relationships with other constrained-planarity problems.Comment: Conference version appeared in WG201

Planarity of Streamed Graphs

In this paper we introduce a notion of planarity for graphs that are presented in a streaming fashion. A $\textit{streamed graph}$ is a stream of edges $e_1,e_2,...,e_m$ on a vertex set $V$. A streamed graph is $\omega$-$\textit{stream planar}$ with respect to a positive integer window size $\omega$ if there exists a sequence of planar topological drawings $\Gamma_i$ of the graphs $G_i=(V,\{e_j \mid i\leq j < i+\omega\})$ such that the common graph $G^{i}_\cap=G_i\cap G_{i+1}$ is drawn the same in $\Gamma_i$ and in $\Gamma_{i+1}$, for $1\leq i < m-\omega$. The $\textit{Stream Planarity}$ Problem with window size $\omega$ asks whether a given streamed graph is $\omega$-stream planar. We also consider a generalization, where there is an additional $\textit{backbone graph}$ whose edges have to be present during each time step. These problems are related to several well-studied planarity problems. We show that the $\textit{Stream Planarity}$ Problem is NP-complete even when the window size is a constant and that the variant with a backbone graph is NP-complete for all $\omega \ge 2$. On the positive side, we provide $O(n+\omega{}m)$-time algorithms for (i) the case $\omega = 1$ and (ii) all values of $\omega$ provided the backbone graph consists of one $2$-connected component plus isolated vertices and no stream edge connects two isolated vertices. Our results improve on the Hanani-Tutte-style $O((nm)^3)$-time algorithm proposed by Schaefer [GD'14] for $\omega=1$.Comment: 21 pages, 9 figures, extended version of "Planarity of Streamed Graphs" (9th International Conference on Algorithms and Complexity, 2015

Drawing Trees in a Streaming Model

We introduce a data stream model of computation for Graph Drawing, where a source produces a graph one edge at a time. When an edge is produced, it is immediately drawn and its drawing can not be altered. The drawing has an image persistence, that controls the lifetime of edges. If the persistence is k, an edge remains in the drawing for the time spent by the source to generate k edges, then it fades away. In this model we study the area requirement of planar straight-line grid drawings of trees, with different streaming orders, layout models, and quality criteria. We assess the output quality of the presented algorithms by computing the competitive ratio with respect to the best known offline algorithms