Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios however the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality \(k\) without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to \(k\). Further, we derive some polynomial-time solvable cases for \(k = 2\) with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of \(k\).
Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M.The Benefit of Recombination in Noisy Evolutionary Search. International Symposium of Algorithms and Computation (ISAAC) 2015: 140-150
Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings meta-heuristics are a popular choice for deriving good optimization algorithms, most notably evolutionary algorithms which mimic evolution in nature. Empirical evidence suggests that genetic recombination is useful in uncertain environments because it can stabilize a noisy fitness signal. With this paper we want to support this claim with mathematical rigor. The setting we consider is that of noisy optimization. We study a simple noisy fitness function that is derived by adding Gaussian noise to a monotone function. First, we show that a classical evolutionary algorithm that does not employ sexual recombination (the \((\mu+1)\)-EA) cannot handle the noise efficiently, regardless of the population size. Then we show that an evolutionary algorithm which does employ sexual recombination (the Compact Genetic Algorithm, short: cGA) can handle the noise using a graceful scaling of the population.
Friedrich, Tobias; Katzmann, Maximilian; Krohmer, AntonUnbounded Discrepancy of Deterministic Random Walks on Grids. International Symposium on Algorithms and Computation (ISAAC) 2015: 212-222
Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs, called rotor-router 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 rotor-router 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 \(\ell\).
Friedrich, Tobias; Krohmer, AntonParameterized Clique on Scale-Free Networks. International Symposium on Algorithms and Computation (ISAAC) 2012: 659-668
Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. However, in typical applications like social networks or protein-protein interaction networks, the considered graphs are scale-free, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more efficiently. We prove that on inhomogeneous random graphs with \(n\) nodes and power law exponent \(\gamma\), cliques of size \(k\) can be found in time \(O(n^2)\) for \(\gamma \ge 3\) and in time \(O(n, exp(k^4))\) for \(2<\gamma <3\).
Friedrich, Tobias; Sauerwald, Thomas; Stauffer, AlexandreDiameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions. International Symposium on Algorithms and Computation (ISAAC) 2011: 190-199
A random geometric graph (RGG) is defined by placing \(n\) points uniformly at random in \([0,n^{1/d}]^d\), and joining two points by an edge whenever their Euclidean distance is at most some fixed \(r\). We assume that \(r\) is larger than the critical value for the emergence of a connected component with \(\Omega(n)\) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a Euclidean distance of \(\omega(\log n / r^{d-1})\), their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is \(\Theta(n^{1/d}/r)\) w.h.p. We also prove that the condition on the Euclidean distance above is essentially tight. We also analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that w.h.p. this algorithm informs every node in the largest connected component of an RGG within \(\Theta(n^1/d/r+\log n)\) rounds.
Friedrich, Tobias; Hebbinghaus, NilsAverage Update Times for Fully-Dynamic All-Pairs Shortest Paths. International Symposium of Algorithms and Computation (ISAAC) 2008: 692-703
We study the fully-dynamic all pairs shortest path problem for graphs with arbitrary non-negative edge weights. It is known for digraphs that an update of the distance matrix costs \(O(n^{2.75})\) worst-case time Thorup, STOC '05 and \(O(n^2)\) amortized time Demetrescu and Italiano, J.ACM '04 where \(n\) is the number of vertices. We present the first average-case analysis of the undirected problem. For a random update we show that the expected time per update is bounded by \(O(n^{4/3 + \epsilon)\) for all \(\epsilon > 0\).
Bringmann, Karl; Friedrich, TobiasApproximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects. International Symposium on Algorithms and Computation (ISAAC) 2008: 436-447
We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where one can: (1) test whether a given point lies inside the object, (2) sample a point uniformly, (3) calculate the volume of the object in polynomial time. All three oracles can be weak, that is, just approximate. This implies that Klee's measure problem and the hypervolume indicator can be approximated efficiently even though they are #P-hard and hence cannot be solved exactly in time polynomial in the number of dimensions unless P=NP. Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of the intersection of high-dimensional geometric objects we prove #P-hardness for boxes and show that there is no multiplicative polynomial-time \(2^{d^{1-\epsilon}}\)-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time \(\epsilon\)-approximation.
Ajwani, Deepak; Friedrich, TobiasAverage-Case Analysis of Online Topological Ordering. International Symposium on Algorithms and Computation (ISAAC) 2007: 464-475
Many applications like pointer analysis and incremental compilation require maintaining a topological ordering of the nodes of a directed acyclic graph (DAG) under dynamic updates. All known algorithms for this problem are either only analyzed for worst-case insertion sequences or only evaluated experimentally on random DAGs. We present the first average-case analysis of online topological ordering algorithms. We prove an expected runtime of \(O(n^2 \text{polylog(n))\) under insertion of the edges of a complete DAG in a random order for the algorithms of Alpern et al. (SODA, 1990), Katriel and Bodlaender (TALG, 2006), and Pearce and Kelly (JEA, 2006). This is much less than the best known worst-case bound \(O(n^{2.75})\) for this problem.
Doerr, Benjamin; Friedrich, TobiasDeterministic Random Walks on the Two-Dimensional Grid. International Symposium on Algorithms and Computation (ISAAC) 2006: 474-483
Deterministic and randomized balancing schemes are used to distribute workload evenly in networks. In this paper, we compare two very general ones: The random walk and the (deterministic) Propp machine. Roughly speaking, we show that on the two-dimensional grid, the Propp machine always has the same number of tokens on a node as does the random walk in expectation, apart from an additive error of less than eight. This constant is independent of the total number of tokens and the runtime of the two processes. However, we also show that it makes a difference whether the Propp machine serves the neighbors in a circular or non-circular order.
Algorithm Engineering
Our research focus is on theoretical computer science and algorithm engineering. We are equally interested in the mathematical foundations of algorithms and developing efficient algorithms in practice. A special focus is on random structures and methods.