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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

We present a method for solving the shortest transshipment problem - also known as uncapacitated minimum cost flow - up to a multiplicative error of (1 + epsilon) in undirected graphs with non-negative integer edge weights using a tailored gradient descent algorithm. Our gradient descent algorithm takes epsilon^(-3) polylog(n) iterations, and in each iteration it needs to solve an instance of the transshipment problem up to a multiplicative error of polylog(n), where n is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a careful white-box analysis, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve prior work by obtaining the following results:
(1) Broadcast CONGEST model: (1 + epsilon)-approximate SSSP using ~O((sqrt(n) + D) epsilon^(-O(1))) rounds, where D is the (hop) diameter of the network.
(2) Broadcast congested clique model: (1 + epsilon)-approximate shortest transshipment and SSSP using ~O(epsilon^(-O(1))) rounds.
(3) Multipass streaming model: (1 + epsilon)-approximate shortest transshipment and SSSP using ~O(n) space and ~O(epsilon^(-O(1))) passes.
The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in n; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs for traversing an edge in opposite directions, running times scale with the maximum ratio between the costs of both directions over all edges.

Ruben Becker, Andreas Karrenbauer, Sebastian Krinninger, and Christoph Lenzen. Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{becker_et_al:LIPIcs.DISC.2017.7, author = {Becker, Ruben and Karrenbauer, Andreas and Krinninger, Sebastian and Lenzen, Christoph}, title = {{Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {7:1--7:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.7}, URN = {urn:nbn:de:0030-drops-80031}, doi = {10.4230/LIPIcs.DISC.2017.7}, annote = {Keywords: Shortest Paths, Shortest Transshipment, Undirected Min-cost Flow, Gradient Descent, Spanner} }

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Brief Announcement

**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

We revisit the hardness of approximating the diameter of a network. In the CONGEST model, ~Omega(n) rounds are necessary to compute the diameter [Frischknecht et al. SODA'12]. Abboud et al. [DISC 2016] extended this result to sparse graphs and, at a more fine-grained level, showed that, for any integer 1 <= l <= polylog(n) , distinguishing between networks of diameter 4l + 2 and 6l + 1 requires ~Omega(n) rounds. We slightly tighten this result by showing that even distinguishing between diameter 2l + 1 and 3l + 1 requires ~Omega(n) rounds. The reduction of Abboud et al. is inspired by recent conditional lower bounds in the RAM model, where the orthogonal vectors problem plays a pivotal role. In our new lower bound, we make the connection to orthogonal vectors explicit, leading to a conceptually more streamlined exposition. This is suited for teaching both the lower bound in the CONGEST model and the conditional lower bound in the RAM model.

Karl Bringmann and Sebastian Krinninger. Brief Announcement: A Note on Hardness of Diameter Approximation. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 44:1-44:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bringmann_et_al:LIPIcs.DISC.2017.44, author = {Bringmann, Karl and Krinninger, Sebastian}, title = {{Brief Announcement: A Note on Hardness of Diameter Approximation}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {44:1--44:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.44}, URN = {urn:nbn:de:0030-drops-79874}, doi = {10.4230/LIPIcs.DISC.2017.44}, annote = {Keywords: diameter, fine-grained reductions, conditional lower bounds} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We introduce a new dynamic data structure for maintaining the strongly connected components (SCCs) of a directed graph (digraph) under edge deletions, so as to answer a rich repertoire of connectivity queries. Our main technical contribution is a decremental data structure that supports sensitivity queries of the form "are u and v strongly connected in the graph G \ w?", for any triple of vertices u, v, w, while G undergoes deletions of edges. Our data structure processes a sequence of edge deletions in a digraph with $n$ vertices in O(m n log n) total time and O(n^2 log n) space, where m is the number of edges before any deletion, and answers the above queries in constant time. We can leverage our data structure to obtain decremental data structures for many more types of queries within the same time and space complexity. For instance for edge-related queries, such as testing whether two query vertices u and v are strongly connected in G \ e, for some query edge e.
As another important application of our decremental data structure, we provide the first nontrivial algorithm for maintaining the dominator tree of a flow graph under edge deletions. We present an algorithm that processes a sequence of edge deletions in a flow graph in O(m n log n) total time and O(n^2 log n) space. For reducible flow graphs we provide an O(mn)-time and O(m + n)-space algorithm. We give a conditional lower bound that provides evidence that these running times may be tight up to subpolynomial factors.

Loukas Georgiadis, Thomas Dueholm Hansen, Giuseppe F. Italiano, Sebastian Krinninger, and Nikos Parotsidis. Decremental Data Structures for Connectivity and Dominators in Directed Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{georgiadis_et_al:LIPIcs.ICALP.2017.42, author = {Georgiadis, Loukas and Dueholm Hansen, Thomas and Italiano, Giuseppe F. and Krinninger, Sebastian and Parotsidis, Nikos}, title = {{Decremental Data Structures for Connectivity and Dominators in Directed Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {42:1--42:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.42}, URN = {urn:nbn:de:0030-drops-74455}, doi = {10.4230/LIPIcs.ICALP.2017.42}, annote = {Keywords: dynamic graph algorithms, decremental algorithms, dominator tree, strong connectivity under failures} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with n nodes and m edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in the context of quantitative verification. We focus on strongly polynomial algorithms to cover the use-case where the weights are relatively large compared to the size of the graph. Our main result is an algorithm with running time ~O(m^{3/4} n^{3/2}), which gives the first improvement over Megiddo's ~O(n^3) algorithm [JACM'83] for sparse graphs (We use the notation ~O(.) to hide factors that are polylogarithmic in n.) We further demonstrate how to obtain both an algorithm with running time n^3/2^{Omega(sqrt(log n)} on general graphs and an algorithm with running time ~O(n) on constant treewidth graphs. To obtain our main result, we develop a parallel algorithm for negative cycle detection and single-source shortest paths that might be of independent interest.

Karl Bringmann, Thomas Dueholm Hansen, and Sebastian Krinninger. Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 124:1-124:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bringmann_et_al:LIPIcs.ICALP.2017.124, author = {Bringmann, Karl and Dueholm Hansen, Thomas and Krinninger, Sebastian}, title = {{Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {124:1--124:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.124}, URN = {urn:nbn:de:0030-drops-74398}, doi = {10.4230/LIPIcs.ICALP.2017.124}, annote = {Keywords: quantitative verification and synthesis, parametric search, shortest paths, negative cycle detection} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

An alpha-spanner of a graph G is a subgraph H such that H preserves all distances of G within a factor of alpha. In this paper, we give fully dynamic algorithms for maintaining a spanner H of a graph G undergoing edge insertions and deletions with worst-case guarantees on the running time after each update. In particular, our algorithms maintain:
- a 3-spanner with ~O(n^{1+1/2}) edges with worst-case update time ~O(n^{3/4}), or
- a 5-spanner with ~O(n^{1+1/3}) edges with worst-case update time ~O (n^{5/9}).
These size/stretch tradeoffs are best possible (up to logarithmic factors). They can be extended to the weighted setting at very minor cost. Our algorithms are randomized and correct with high probability against an oblivious adversary. We also further extend our techniques to construct a 5-spanner with suboptimal size/stretch tradeoff, but improved worst-case update time.
To the best of our knowledge, these are the first dynamic spanner algorithms with sublinear worst-case update time guarantees. Since it is known how to maintain a spanner using small amortized}but large worst-case update time [Baswana et al. SODA'08], obtaining algorithms with strong worst-case bounds, as presented in this paper, seems to be the next natural step for this problem.

Greg Bodwin and Sebastian Krinninger. Fully Dynamic Spanners with Worst-Case Update Time. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bodwin_et_al:LIPIcs.ESA.2016.17, author = {Bodwin, Greg and Krinninger, Sebastian}, title = {{Fully Dynamic Spanners with Worst-Case Update Time}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {17:1--17:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.17}, URN = {urn:nbn:de:0030-drops-63688}, doi = {10.4230/LIPIcs.ESA.2016.17}, annote = {Keywords: Dynamic graph algorithms, spanners} }