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AG Codes and Kryptographie Bildinformationen anzeigen

AG Codes and Kryptographie

Sascha Brauer


Codes und Kryptographie

Wissenschaftlicher Mitarbeiter

+49 5251 60-6627
Fürstenallee 11
33102 Paderborn
Warburger Str. 100
33098 Paderborn


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Complexity of single-swap heuristics for metric facility location and related problems

S. Brauer, Theoretical Computer Science (2018)



Complexity of Single-Swap Heuristics for Metric Facility Location and Related Problems

S. Brauer, in: Lecture Notes in Computer Science, Springer International Publishing, 2017, pp. 116-127


Metric facility location and K-means are well-known problems of combinatorial optimization. Both admit a fairly simple heuristic called single-swap, which adds, drops or swaps open facilities until it reaches a local optimum. For both problems, it is known that this algorithm produces a solution that is at most a constant factor worse than the respective global optimum. In this paper, we show that single-swap applied to the weighted metric uncapacitated facility location and weighted discrete K-means problem is tightly PLS-complete and hence has exponential worst-case running time.


A Theoretical Analysis of the Fuzzy K-Means Problem

J. Blömer, S. Brauer, K. Bujna, in: 2016 IEEE 16th International Conference on Data Mining (ICDM), IEEE, 2016, pp. 805-810


One of the most popular fuzzy clustering techniques is the fuzzy K-means algorithm (also known as fuzzy-c-means or FCM algorithm). In contrast to the K-means and K-median problem, the underlying fuzzy K-means problem has not been studied from a theoretical point of view. In particular, there are no algorithms with approximation guarantees similar to the famous K-means++ algorithm known for the fuzzy K-means problem. This work initiates the study of the fuzzy K-means problem from an algorithmic and complexity theoretic perspective. We show that optimal solutions for the fuzzy K-means problem cannot, in general, be expressed by radicals over the input points. Surprisingly, this already holds for simple inputs in one-dimensional space. Hence, one cannot expect to compute optimal solutions exactly. We give the first (1+eps)-approximation algorithms for the fuzzy K-means problem. First, we present a deterministic approximation algorithm whose runtime is polynomial in N and linear in the dimension D of the input set, given that K is constant, i.e. a polynomial time approximation scheme (PTAS) for fixed K. We achieve this result by showing that for each soft clustering there exists a hard clustering with similar properties. Second, by using techniques known from coreset constructions for the K-means problem, we develop a deterministic approximation algorithm that runs in time almost linear in N but exponential in the dimension D. We complement these results with a randomized algorithm which imposes some natural restrictions on the sought solution and whose runtime is comparable to some of the most efficient approximation algorithms for K-means, i.e. linear in the number of points and the dimension, but exponential in the number of clusters.

Hard-Clustering with Gaussian Mixture Models

J. Blömer, S. Brauer, K. Bujna, 2016

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