Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph with bounded overlap, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex v means to remove it and to add two new copies of v and to make each previous neighbor of v adjacent with at least one of the copies. In this work, we study underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem is NP -complete. We then make structural observations that show that the covering viewpoint and the vertex-splitting viewpoint are equivalent, yielding NP-hardness for the vertex-splitting problem. On the positive side, we show that splitting at most k vertices to obtain a cluster graph has a problem kernel with O ( k ) vertices. Finally, we observe that combining our hardness results with structural observations and a so-called critical-clique lemma yields a simple alternative NP-hardness proof for the Cluster Editing With Vertex Splitting problem, where we add or delete edges and split vertices to obtain a cluster graph.

Word clouds are a popular text visualization technique that summarize an input text by displaying its most important words in a compact image. The traditional layout methods do not take proximity effects between words into account; this has been improved in semantic word clouds, where relative word placement is controlled by edges in a word similarity graph. We introduce MySemCloud, a new human-in-the-loop tool to visualize and edit semantic word clouds. MySemCloud lets users perform computer-assisted local moves of words, which improve or at least retain the semantic quality. To achieve this, we construct a word similarity graph on which a system of forces is applied to generate a compact initial layout with good semantic quality. The force system also allows us to maintain these attributes after each user interaction, as well as preserve the user’s mental map. The tool provides algorithmic support for the editing operations to help the user enhance the semantic quality of the visualization, while adjusting it to their personal preference. We show that MySemCloud provides high user satisfaction as well as permits users to create layouts of higher quality than state-of-the-art semantic word cloud generation tools.

We present a novel vertex-splitting approach with which to iteratively resolve edge crossings in order to improve the readability of graph drawings. Dense graphs, even when small in size (10 to 15 nodes in size) quickly become difficult to read with increasing numbers of edges, and form so-called “hairballs”. The readability of a graph drawing is measured using many different quantitative aesthetic metrics. One such metric of particular importance is the number of edge crossings. Classical approaches to improving readability, such as the minimization of the number of edge crossings, focus on providing overviews of the input graph by aggregating or sampling vertices and/or edges. However, this simplification of the graph drawing does not allow for detailed views into the data, as not all vertices or edges are rendered, and also requires sophisticated interaction approaches to perform well. To avoid this, our locally optimal vertex splitting approach aims to minimize the number of remaining edge crossings while also minimizing the number of vertices that need to be split. In each iteration, we identify the vertex contributing the largest number of edge crossings, remove it, locate the embedding locations of said vertex’s two split copies, and determine each copy’s unique adjacency. We conduct a user study with 52 participants to evaluate whether vertex splitting affects users’ abilities to conduct a set of graph analytical tasks on graphs 12 nodes in size. Users were tasked with identifying a vertex’s adjacency, determining the shared neighbors of two vertices, and checking the validity of a set of paths. We ultimately conclude that within the context of small, dense graphs, systematic vertex splitting is preferred by participants and even positively impacts user performance, though at the cost of the time taken per task.

Bounded expansion and nowhere-dense classes of graphs capture the theoretical tractability for several important algorithmic problems. These classes of graphs can be characterized by the so-called weak coloring numbers of graphs, which generalize the well-known graph invariant degeneracy (also called k-core number). Being NP-hard, weak-coloring numbers were previously computed on real-world graphs mainly via incremental heuristics. We study whether it is feasible to augment such heuristics with exponential-time subprocedures that kick in when a desired upper bound on the weak coloring number is breached. We provide hardness and tractability results on the corresponding computational subproblems. We implemented several of the resulting algorithms and show them to be competitive with previous approaches on a previously studied set of benchmark instances containing 86 graphs with up to 183831 edges. We obtain improved weak coloring numbers for over half of the instances.

We study a variant of the geometric multicut problem, where we are given a set $\mathcal{P}$ of colored and pairwise interior-disjoint polygons in the plane. The objective is to compute a set of simple closed polygon boundaries (fences) that separate the polygons in such a way that any two polygons that are enclosed by the same fence have the same color, and the total number of links of all fences is minimized. We call this the minimum link fencing (MLF) problem and consider the natural case of bounded minimum link fencing (BMLF), where $\mathcal{P}$ contains a polygon $Q$ that is unbounded in all directions and can be seen as an outer polygon. We show that BMLF is NP-hard in general and that it is XP-time solvable when each fence contains at most two polygons and the number of segments per fence is the parameter. Finally, we present an $O(n \log n)$-time algorithm for the case that the convex hull of $\mathcal{P} \setminus \{Q\}$ does not intersect $Q$.

More and more networks are becoming reconfigurable: not just the routing can be programmed, but the physical layer itself as well. Various technologies enable this programmability, ranging from optical circuit switches to beamformed wireless connections and free-space optical interconnects. Existing reconfigurable network topologies are typically hybrid in nature, consisting of static and a reconfigurable links. However, even though the static and reconfigurable links form a joint structure, routing policies are artificially segregated and hence do not fully exploit the network resources: the state of the art is to route large elephant flows on direct reconfigurable links, whereas the remaining traffic is left to the static network topology. Recent work showed that such artificial segregation is inefficient, but did not provide the tools to actually leverage the benefits on non-segregated routing. In this paper, we provide several algorithms which take advantage of non-segregated routing, by jointly optimizing topology and routing. We compare our algorithms to segregated routing policies and also evaluate their performance in workload-driven simulations, based on real-world traffic traces. We find that our algorithms do not only outperform segregated routing policies, in various settings, but also come close to the optimal solution, computed by a integer linear program formulation, also presented in this paper. Finally, we also provide insights into the complexity of the underlying combinatorial optimization problem, by deriving approximation hardness results.

More and more networks are becoming reconfigurable: not just the routing can be programmed, but the physical layer itself as well. Various technologies enable this programmability, ranging from optical circuit switches to beamformed wireless connections and free-space optical interconnects. Existing reconfigurable network topologies are typically hybrid in nature, consisting of static and a reconfigurable links. However, even though the static and reconfigurable links form a joint structure, routing policies are artificially segregated and hence do not fully exploit the network resources: the state of the art is to route large elephant flows on direct reconfigurable links, whereas the remaining traffic is left to the static network topology. Recent work showed that such artificial segregation is inefficient, but did not provide the tools to actually leverage the benefits on non-segregated routing. In this paper, we provide several algorithms which take advantage of non-segregated routing, by jointly optimizing topology and routing. We compare our algorithms to segregated routing policies and also evaluate their performance in workload-driven simulations, based on real-world traffic traces. We find that our algorithms do not only outperform segregated routing policies, in various settings, but also come close to the optimal solution, computed by a mixed integer program formulation, also presented in this paper. Finally, we also provide insights into the complexity of the underlying combinatorial optimization problem, by deriving approximation hardness results.