
Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton Unbounded Discrepancy of Deterministic Random Walks on Grids. SIAM Journal on Discrete Mathematics 2018
Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs, called rotorrouter model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave remarkably similar: Starting with the same initial configuration, the number of tokens in the rotorrouter model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer (2006) showed that on \(\mathbb{Z}^{d}\) the single vertex discrepancy is only a constant \(c_d\). Other authors also determined the precise value of \(c_d\) for \(d=1,2\). All these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph \(\mathbb{Z}^{d}\). We show that this assumption is crucial by proving that otherwise the single vertex discrepancy can become arbitrarily large. For all dimensions \(d\geq1\) and arbitrary discrepancies~\(\ell \geq 0\), we construct configurations that reach a discrepancy of at least \(\ell\).

Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton Hyperbolic Embeddings for NearOptimal Greedy Routing. Algorithm Engineering and Experiments (ALENEX) 2018: 199208
Greedy routing computes paths between nodes in a network by successively moving to the neighbor closest to the target with respect to coordinates given by an embedding into some metric space. Its advantage is that only local information is used for routing decisions. We present different algorithms for generating graph embeddings into the hyperbolic plane that are well suited for greedy routing. In particular our embeddings guarantee that greedy routing always succeeds in reaching the target and we try to minimize the lengths of the resulting greedy paths. We evaluate our algorithm on multiple generated and real wold networks. For networks that are generally assumed to have a hidden underlying hyperbolic geometry, such as the Internet graph, we achieve nearoptimal results, i.e., the resulting greedy paths are only slightly longer than the corresponding shortest paths. In the case of the Internet graph, they are only \(6\%\) longer when using our best algorithm, which greatly improves upon the previous best known embedding, whose creation required substantial manual intervention.

Bläsius, Thomas; Freiberger, Cedric; Friedrich, Tobias; Katzmann, Maximilian; MontenegroRetana, Felix; Thieffry, Marianne Efficient Shortest Paths in ScaleFree Networks with Underlying Hyperbolic Geometry. International Colloquium on Automata, Languages, and Programming (ICALP) 2018
A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scalefree realworld networks. Such networks typically have a heterogeneous degree distribution (e.g., a powerlaw distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is \(O(n^{2  1/ α} + n^{1/(2α)} + \delta_{\max})\) with high probability, where \(α ∈ (0.5, 1)\) controls the powerlaw exponent of the degree distribution, and \(\delta_{\max}\) is the maximum degree. This bound is sublinear, improving the obvious worstcase linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.

Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton; Striebel, Jonathan Towards a Systematic Evaluation of Generative Network Models. Workshop on Algorithms and Models for the Web Graph (WAW) 2018: 99114
Generative graph models play an important role in network science. Unlike realworld networks, they are accessible for mathematical analysis and the number of available networks is not limited. The explanatory power of results on generative models, however, heavily depends on how realistic they are. We present a framework that allows for a systematic evaluation of generative network models. It is based on the question whether realworld networks can be distinguished from generated graphs with respect to certain graph parameters. As a proof of concept, we apply our framework to four popular random graph models (ErdősRényi, BarabásiAlbert, ChungLu, and hyperbolic random graphs). Our experiments for example show that all four models are bad representations for Facebook's social networks, while ChungLu and hyperbolic random graphs are good representations for other networks, with different strengths and weaknesses.

Katzmann, Maximilian; Komusiewicz, Christian Systematic Exploration of Larger Local Search Neighborhoods for the Minimum Vertex Cover Problem. Association for the Advancement of Artificial Intelligence (AAAI) 2017: 846852
We investigate the potential of exhaustively exploring larger neighborhoods in local search algorithms for Minimum Vertex Cover. More precisely, we study whether, for moderate values of \(k\), it is feasible and worthwhile to determine, given a graph \(G\) with vertex cover \(C\), if there is a \(k\)swap \(S\) such that \((C \setminus S) cup (S \setminus C)\) is a smaller vertex cover of \(G\). First, we describe an algorithm running in \(\Delta^{O(k)} \cdot n\) time for searching the \(k\)swap neighborhood on \(n\)vertex graphs with maximum degree \(\Delta\). Then, we demonstrate that, by devising additional pruning rules that decrease the size of the search space, this algorithm can be implemented so that it solves the problem quickly for \(k \approx 20\). Finally, we show that it is worthwhile to consider moderatelysized \(k\)swap neighborhoods. For our benchmark data set, we show that when combining our algorithm with a hillclimbing approach, the solution quality improves quickly with the radius \(k\) of the local search neighborhood and that in most cases optimal solutions can be found by setting \(k = 21\).

Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton Unbounded Discrepancy of Deterministic Random Walks on Grids. International Symposium on Algorithms and Computation (ISAAC) 2015: 212222
Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs, called rotorrouter model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave remarkably similar: Starting with the same initial configuration, the number of tokens in the rotorrouter model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer (2006) showed that on \(\mathbb{Z}^d\) the single vertex discrepancy is only a constant \(c_d\). Other authors also determined the precise value of \(c_d\) for \(d=1,2\). All these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph \(\mathbb{Z}^d\). We show that this assumption is crucial by proving that otherwise the single vertex discrepancy can become arbitrarily large. For all dimensions \(d \ge 1\) and arbitrary discrepancies \(\ell \ge 0\), we construct configurations that reach a discrepancy of at least \(\ell\).