Networks are amongst the central building blocks of many systems. Given a graph of a network, methods from graph theory enable a precise investigation of its properties. Software for the analysis of graphs is widely available and has been applied to study various types of networks. In some applications, graph acquisition is relatively simple. However, for many networks data collection relies on images where graph extraction requires domain-specific solutions. Here we introduce NEFI, a tool that extracts graphs from images of networks originating in various domains. Regarding previous work on graph extraction, theoretical results are fully accessible only to an expert audience and ready-to-use implementations for non-experts are rarely available or insufficiently documented. NEFI provides a novel platform allowing practitioners to easily extract graphs from images by combining basic tools from image processing, computer vision and graph theory. Thus, NEFI constitutes an alternative to tedious manual graph extraction and special purpose tools. We anticipate NEFI to enable time-efficient collection of large datasets. The analysis of these novel datasets may open up the possibility to gain new insights into the structure and function of various networks. NEFI is open source and available at http://nefi.mpi-inf.mpg.de.

We present a certifying algorithm that tests graphs for 3-edge-connectivity; the algorithm works in linear time. If the input graph is not 3-edge-connected, the algorithm returns a 2-edge-cut. If it is 3-edge-connected, it returns a construction sequence that constructs the input graph from the graph with two vertices and three parallel edges using only operations that (obviously) preserve 3-edge-connectivity. Additionally, we show how compute and certify the 3-edge-connected components and a cactus representation of the 2-cuts in linear time. For 3-vertex-connectivity, we show how to compute the 3-vertex-connected components of a 2-connected graph.Comment: 29 pages in Algorithmica, 201

Abstract. In the online checkpointing problem, the task is to continuously maintain a set of k checkpoints that allow to rewind an ongoing computation faster than by a full restart. The only operation allowed is to remove an old checkpoint and to store the current state instead. Our aim are checkpoint placement strategies that minimize rewinding cost, i.e., such that at all times T when requested to rewind to some time t ≤ T the number of computation steps that need to be redone to get to t from a checkpoint before t is as small as possible. In particular, we want that the closest checkpoint earlier than t is not further away from t than p k times the ideal distance T /(k + 1), where p k is a small constant. Improving over earlier work showing 1 + 1/k ≤ p k ≤ 2, we show that p k can be chosen less than 2 uniformly for all k. More precisely, we show the uniform bound p k ≤ 1.7 for all k, and present algorithms with asymptotic performance p k ≤ 1.59 + o(1) valid for all k and p k ≤ ln(4) + o(1) ≤ 1.39 + o(1) valid for k being a power of two. For small values of k, we show how to use a linear programming approach to compute good checkpointing algorithms. This gives performances of less than 1.53 for k ≤ 10. One the more theoretical side, we show the first lower bound that is asymptotically more than one, namely p k ≥ 1.30 − o(1). We also show that optimal algorithms (yielding the infimum performance) exist for all k.

We consider a robust variant of the classical k-median problem, introduced by Anthony et al. [2]. In the Robust k-Median problem, we are given an n-vertex metric space (V, d) and m client setsThe objective is to open a set F ⊆ V of k facilities such that the worst case connection cost over all client sets is minimized; in other words, minimize max i v∈Si d (F, v). Anthony et al. showed an O(log m) approximation algorithm for any metric and APX-hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing Ω(log m/ log log m) approximation hardness, unless NP ⊆ δ>0 DTIME(2 n δ ). This hardness result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances.

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