Computer Science

We study the planar orthogonal drawing style within the framework of partial representation extension. Let (G,H, Γ H ) be a partial orthogonal drawing, i.e., G is a graph, H⊆G is a subgraph and Γ H is a planar orthogonal drawing of H. We show that the existence of an orthogonal drawing Γ G of G that extends Γ H can be tested in linear time. If such a drawing exists, then there also is one that uses O(|V(H)|) bends per edge. On the other hand, we show that it is NP-complete to find an extension that minimizes the number of bends or has a fixed number of bends per edge.

A de Bruijn sequence of order n over a k-symbol alphabet is a circular sequence where each length-n sequence occurs exactly once. We present a way of extending de Bruijn sequences by adding a new symbol to the alphabet: the extension is performed by embedding a given de Bruijn sequence into another one of the same order, but over the alphabet with one more symbol, while ensuring that there are no long runs without the new symbol. Our solution is based on auxiliary graphs derived from the de Bruijn graph and solving a problem of maximum flow.

Cutting planes are a key ingredient to successfully solve mixed-integer linear programs. For specific problems, their strength is often theoretically assessed by showing that they are facet-defining for the corresponding mixed-integer hull. In this paper we experimentally investigate the dimensions of faces induced by general-purpose cutting planes generated by a state-of-the-art solver. Therefore, we relate the dimension of each cutting plane to its impact in a branch-and-bound algorithm.

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Thereby we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both {\sc Partition} and {\sc Independent Set}. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We can show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).

The dynamics of fake news and rumor spreading is investigated using a model with three kinds of agents who are respectively the Seeds, the Agnostics and the Others. While Seeds are the ones who start spreading the rumor being adamantly convinced of its truth, Agnostics reject any kind of rumor and do not believe in conspiracy theories. In between, the Others constitute the main part of the community. While Seeds are always Believers and Agnostics are always Indifferents, Others can switch between being Believer and Indifferent depending on who they are discussing with. The underlying driving dynamics is implemented via local updates of randomly formed groups of agents. In each group, an Other turns into a Believer as soon as m or more Believers are present in the group. However, since some Believers may lose interest in the rumor as time passes by, we add a flipping fixed rate 0<d<1 from Believers into Indifferents. Rigorous analysis of the associated dynamics reveals that switching from m=1 to m≥2 triggers a drastic qualitative change in the spreading process. When m=1 even a small group of Believers may manage to convince a large part of the community very quickly. In contrast, for m≥2 , even a substantial fraction of Believers does not prevent the rumor dying out after a few update rounds. Our results provide an explanation on why a given rumor spreads within a social group and not in another, and also why some rumors will not spread in neither groups.

A graph is \emph{fan-crossing free} if it has a drawing in the plane so that each edge is crossed by independent edges, that is the crossing edges have distinct vertices. On the other hand, it is \emph{fan-crossing} if the crossing edges have a common vertex, that is they form a fan. Both are prominent examples for beyond-planar graphs. Further well-known beyond-planar classes are the k -planar, k -gap-planar, quasi-planar, and right angle crossing graphs. We use the subdivision, node-to-circle expansion and path-addition operations to distinguish all these graph classes. In particular, we show that the 2-subdivision and the node-to-circle expansion of any graph is fan-crossing free, which does not hold for fan-crossing and k -(gap)-planar graphs, respectively. Thereby, we obtain graphs that are fan-crossing free and neither fan-crossing nor k -(gap)-planar. Finally, we show that some graphs have a unique fan-crossing free embedding, that there are thinned maximal fan-crossing free graphs, and that the recognition problem for fan-crossing free graphs is NP-complete.

Here we consider an approach for fast computing the algebraic degree of Boolean functions. It combines fast computing the ANF (known as ANF transform) and thereafter the algebraic degree by using the weight-lexicographic order (WLO) of the vectors of the n -dimensional Boolean cube. Byte-wise and bitwise versions of a search based on the WLO and their implementations are discussed. They are compared with the usual exhaustive search applied in computing the algebraic degree. For Boolean functions of n variables, the bitwise implementation of the search by WLO has total time complexity O(n .2 n ) . When such a function is given by its truth table vector and its algebraic degree is computed by the bitwise versions of the algorithms discussed, the total time complexity is Θ((9n−2) .2 n−7 )=Θ(n .2 n ) . All algorithms discussed have time complexities of the same type, but with big differences in the constants hidden in the Θ -notation. The experimental results after numerous tests confirm the theoretical results - the running times of the bitwise implementation are dozens of times better than the running times of the byte-wise algorithms.

We consider the problem of completely covering an unknown discrete environment with a swarm of asynchronous, frequently-crashing autonomous mobile robots. We represent the environment by a discrete graph, and task the robots with occupying every vertex and with constructing an implicit distributed spanning tree of the graph. The robotic agents activate independently at random exponential waiting times of mean 1 and enter the graph environment over time from a source location. They grow the environment's coverage by 'settling' at empty locations and aiding other robots' navigation from these locations. The robots are identical and make decisions driven by the same simple and local rule of behaviour. The local rule is based only on the presence of neighbouring robots, and on whether a settled robot points to the current location. Whenever a robot moves, it may crash and disappear from the environment. Each vertex in the environment has limited physical space, so robots frequently obstruct each other. Our goal is to show that even under conditions of asynchronicity, frequent crashing, and limited physical space, the simple mobile robots complete their mission in linear time asymptotically almost surely, and time to completion degrades gracefully with the frequency of the crashes. Our model and analysis are based on the well-studied "totally asymmetric simple exclusion process" in statistical mechanics.

Computing an array of all pairs of geodesic distances between the pixels of an image is time consuming. In the sequel, we introduce new methods exploiting the redundancy of geodesic propagations and compare them to an existing one. We show that our method in which the source point of geodesic propagations is chosen according to its minimum number of distances to the other points, improves the previous method up to 32% and the naive method up to 50% in terms of reduction of the number of operations.

Many consensus string problems are based on Hamming distance. We replace Hamming distance by the more flexible (e.g., easily coping with different input string lengths) dynamic time warping distance, best known from applications in time series mining. Doing so, we study the problem of finding a mean string that minimizes the sum of (squared) dynamic time warping distances to a given set of input strings. While this problem is known to be NP-hard (even for strings over a three-element alphabet), we address the binary alphabet case which is known to be polynomial-time solvable. We significantly improve on a previously known algorithm in terms of worst-case running time. Moreover, we also show the practical usefulness of one of our algorithms in experiments with real-world and synthetic data. Finally, we identify special cases solvable in linear time (e.g., finding a mean of only two binary input strings) and report some empirical findings concerning combinatorial properties of optimal means.