Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study the exact fully dynamic shortest paths problem. For real-weighted directed graphs, we show a deterministic fully dynamic data structure with Õ(mn^{4/5}) worst-case update time processing arbitrary s,t-distance queries in Õ(n^{4/5}) time. This constitutes the first non-trivial update/query tradeoff for this problem in the regime of sparse weighted directed graphs.
Moreover, we give a Monte Carlo randomized fully dynamic reachability data structure processing single-edge updates in Õ(n√m) worst-case time and queries in O(√m) time. For sparse digraphs, such a tradeoff has only been previously described with amortized update time [Roditty and Zwick, SIAM J. Comp. 2008].

Adam Karczmarz and Piotr Sankowski. Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 84:1-84:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{karczmarz_et_al:LIPIcs.ICALP.2023.84, author = {Karczmarz, Adam and Sankowski, Piotr}, title = {{Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {84:1--84:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.84}, URN = {urn:nbn:de:0030-drops-181363}, doi = {10.4230/LIPIcs.ICALP.2023.84}, annote = {Keywords: dynamic shortest paths, dynamic reachability, dynamic transitive closure} }

Document

**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension d. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for d = 2.
The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the A^* search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra’s algorithm. Specifically, if the n corresponding points of a weighted unit-disk graph G are picked from a unit square uniformly at random, and the connectivity radius is r ∈ (0,1), A^* finds a shortest path in G in O(n) expected time when r = Ω(√{log n/n}), even though G has Θ((nr)²) edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges.
In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to A^*. Specifically, we show that, if we are able to report the set of all k points of G from an arbitrary rectangular region of the plane in O(k + t(n)) time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in O(1/r² + t(n)) expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all r = Ω(√{log n/n}) and O(√n) expected bound for r = Ω(n^{-1/4}) after only near-linear preprocessing of the point set.
Our approach naturally generalizes to higher dimensions d ≥ 3 and yields sublinear expected bounds for all d = O(1) and sufficiently large r.

Adam Karczmarz, Jakub Pawlewicz, and Piotr Sankowski. Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 46:1-46:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{karczmarz_et_al:LIPIcs.SoCG.2021.46, author = {Karczmarz, Adam and Pawlewicz, Jakub and Sankowski, Piotr}, title = {{Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {46:1--46:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.46}, URN = {urn:nbn:de:0030-drops-138454}, doi = {10.4230/LIPIcs.SoCG.2021.46}, annote = {Keywords: unit-disk graphs, shortest paths, distance oracles} }

Document

**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

In this paper we give an O~((nm)^(2/3) log C) time algorithm for computing min-cost flow (or min-cost circulation) in unit capacity planar multigraphs where edge costs are integers bounded by C. For planar multigraphs, this improves upon the best known algorithms for general graphs: the O~(m^(10/7) log C) time algorithm of Cohen et al. [SODA 2017], the O(m^(3/2) log(nC)) time algorithm of Gabow and Tarjan [SIAM J. Comput. 1989] and the O~(sqrt(n) m log C) time algorithm of Lee and Sidford [FOCS 2014]. In particular, our result constitutes the first known fully combinatorial algorithm that breaks the Omega(m^(3/2)) time barrier for min-cost flow problem in planar graphs.
To obtain our result we first give a very simple successive shortest paths based scaling algorithm for unit-capacity min-cost flow problem that does not explicitly operate on dual variables. This algorithm also runs in O~(m^(3/2) log C) time for general graphs, and, to the best of our knowledge, it has not been described before. We subsequently show how to implement this algorithm faster on planar graphs using well-established tools: r-divisions and efficient algorithms for computing (shortest) paths in so-called dense distance graphs.

Adam Karczmarz and Piotr Sankowski. Min-Cost Flow in Unit-Capacity Planar Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 66:1-66:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{karczmarz_et_al:LIPIcs.ESA.2019.66, author = {Karczmarz, Adam and Sankowski, Piotr}, title = {{Min-Cost Flow in Unit-Capacity Planar Graphs}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {66:1--66:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.66}, URN = {urn:nbn:de:0030-drops-111878}, doi = {10.4230/LIPIcs.ESA.2019.66}, annote = {Keywords: minimum-cost flow, minimum-cost circulation, planar graphs} }

Document

**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

In this paper we study three previously unstudied variants of the online Facility Location problem, considering an intrinsic scenario when the clients and facilities are not only allowed to arrive to the system, but they can also depart at any moment.
We begin with the study of a natural fully-dynamic online uncapacitated model where clients can be both added and removed. When a client arrives, then it has to be assigned either to an existing facility or to a new facility opened at the client's location. However, when a client who has been also one of the open facilities is to be removed, then our model has to allow to reconnect all clients that have been connected to that removed facility. In this model, we present an optimal O(log(n_{act}) / log log(n_{act}))-competitive algorithm, where n_{act} is the number of active clients at the end of the input sequence.
Next, we turn our attention to the capacitated Facility Location problem. We first note that if no deletions are allowed, then one can achieve an optimal competitive ratio of O(log(n) / log(log n)), where n is the length of the sequence. However, when deletions are allowed, the capacitated version of the problem is significantly more challenging than the uncapacitated one. We show that still, using a more sophisticated algorithmic approach, one can obtain an online O(log N + log c log n)-competitive algorithm for the capacitated Facility Location problem in the fully dynamic model, where N is number of points in the input metric and c is the capacity of any open facility.

Marek Cygan, Artur Czumaj, Marcin Mucha, and Piotr Sankowski. Online Facility Location with Deletions. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 21:1-21:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{cygan_et_al:LIPIcs.ESA.2018.21, author = {Cygan, Marek and Czumaj, Artur and Mucha, Marcin and Sankowski, Piotr}, title = {{Online Facility Location with Deletions}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.21}, URN = {urn:nbn:de:0030-drops-94843}, doi = {10.4230/LIPIcs.ESA.2018.21}, annote = {Keywords: online algorithms, facility location, fully-dynamic online algorithms} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Consider a planar graph G=(V,E) with polynomially bounded edge weight function w:E -> [0, poly(n)]. The main results of this paper are NC algorithms for finding minimum weight perfect matching in G. In order to solve this problems we develop a new relatively simple but versatile framework that is combinatorial in spirit. It handles the combinatorial structure of matchings directly and needs to only know weights of appropriately defined matchings from algebraic subroutines.
Moreover, using novel planarity preserving reductions, we show how to find: maximum weight matching in G when G is bipartite; maximum multiple-source multiple-sink flow in G where c:E -> [1, poly(n)] is a polynomially bounded edge capacity function; minimum weight f-factor in G where f:V -> [1, poly(n)]; min-cost flow in G where c:E -> [1, poly(n)] is a polynomially bounded edge capacity function and b:V -> [1, poly(n)] is a polynomially bounded vertex demand function. There have been no known NC algorithms for these problems previously.

Piotr Sankowski. NC Algorithms for Weighted Planar Perfect Matching and Related Problems. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 97:1-97:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{sankowski:LIPIcs.ICALP.2018.97, author = {Sankowski, Piotr}, title = {{NC Algorithms for Weighted Planar Perfect Matching and Related Problems}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {97:1--97:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.97}, URN = {urn:nbn:de:0030-drops-91011}, doi = {10.4230/LIPIcs.ICALP.2018.97}, annote = {Keywords: planar graph, NC algorithms, maximum cardinality matching, maximum weight matching, min-cost flow, maximum multiple-source multiple-sink flow, f-factors} }

Document

**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

In this paper, we report progress on answering the open problem presented by Pagh [11], who considered the near neighbor search without false negatives for the Hamming distance. We show new data structures for solving the c-approximate near neighbors problem without false negatives for Euclidean high dimensional space
\mathcal{R}^d. These data structures work for any c = \omega(\sqrt{\log{\log{n}}}), where n is the number of points in the input set, with poly-logarithmic query time and polynomial
pre-processing time. This improves over the known algorithms, which require c to be \Omega(\sqrt{d}).
This improvement is obtained by applying a sequence of reductions, which are interesting on their own. First, we reduce the problem to d instances of dimension logarithmic in n. Next, these instances are reduced to a number of c-approximate near neighbor search without false negatives instances in \big(\Rspace^k\big)^L space equipped with metric m(x,y) = \max_{1 \le i \leL}(\dist{x_i - y_i}_2).

Piotr Sankowski and Piotr Wygocki. Approximate Nearest Neighbors Search Without False Negatives For l_2 For c>sqrt{loglog{n}}. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 63:1-63:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{sankowski_et_al:LIPIcs.ISAAC.2017.63, author = {Sankowski, Piotr and Wygocki, Piotr}, title = {{Approximate Nearest Neighbors Search Without False Negatives For l\underline2 For c\ranglesqrt\{loglog\{n\}\}}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {63:1--63:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.63}, URN = {urn:nbn:de:0030-drops-82189}, doi = {10.4230/LIPIcs.ISAAC.2017.63}, annote = {Keywords: locality sensitive hashing, approximate near neighbor search, high- dimensional, similarity search} }

Document

**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We present a data structure that can maintain a simple planar graph under edge contractions in linear total time. The data structure supports adjacency queries and provides access to neighbor lists in O(1) time. Moreover, it can report all the arising self-loops and parallel edges.
By applying the data structure, we can achieve optimal running times for decremental bridge detection, 2-edge connectivity, maximal 3-edge connected components, and the problem of finding a unique perfect matching for a static planar graph. Furthermore, we improve the running times of algorithms for several planar graph problems, including decremental 2-vertex and 3-edge connectivity, and we show that using our data structure in a black-box manner, one obtains conceptually simple optimal algorithms for computing MST and 5-coloring in planar graphs.

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, Eva Rotenberg, and Piotr Sankowski. Contracting a Planar Graph Efficiently. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 50:1-50:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{holm_et_al:LIPIcs.ESA.2017.50, author = {Holm, Jacob and Italiano, Giuseppe F. and Karczmarz, Adam and Lacki, Jakub and Rotenberg, Eva and Sankowski, Piotr}, title = {{Contracting a Planar Graph Efficiently}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {50:1--50:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.50}, URN = {urn:nbn:de:0030-drops-78755}, doi = {10.4230/LIPIcs.ESA.2017.50}, annote = {Keywords: planar graphs, algorithms, data structures, connectivity, coloring} }

Document

**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Consider an unweighted, directed graph G with the diameter D. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time O-tilde(n^omega). The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the following problems efficiently.
* All Nodes Shortest Cycles - for every node return the length of the shortest cycle containing it. We give an O-tilde(n^omega) algorithm that improves the previous O-tilde(n^((omega + 3)/2)) algorithm for unweighted digraphs.
* We show how to compute all D sets of vertices lying on cycles of length c in {1, ..., D} in randomized time O-tilde(n^omega). It improves upon an algorithm by Cygan where algorithm that computes a single set is presented.
* We present a functional improvement of distance queries for directed, unweighted graphs.
* All Pairs All Walks - we show almost optimal O-tilde(n^3) time algorithm for all walks counting problem. We improve upon the naive O(D n^omega) time algorithm.

Piotr Sankowski and Karol Wegrzycki. Improved Distance Queries and Cycle Counting by Frobenius Normal Form. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 56:1-56:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{sankowski_et_al:LIPIcs.STACS.2017.56, author = {Sankowski, Piotr and Wegrzycki, Karol}, title = {{Improved Distance Queries and Cycle Counting by Frobenius Normal Form}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {56:1--56:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.56}, URN = {urn:nbn:de:0030-drops-69773}, doi = {10.4230/LIPIcs.STACS.2017.56}, annote = {Keywords: Frobenius Normal Form, Graph Algorithms, All Nodes Shortest Cycles} }

Document

Complete Volume

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

LIPIcs, Volume 57, ESA'16, Complete Volume

Piotr Sankowski and Christos Zaroliagis. LIPIcs, Volume 57, ESA'16, Complete Volume. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@Proceedings{sankowski_et_al:LIPIcs.ESA.2016, title = {{LIPIcs, Volume 57, ESA'16, Complete Volume}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, 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}, URN = {urn:nbn:de:0030-drops-65854}, doi = {10.4230/LIPIcs.ESA.2016}, annote = {Keywords: Data Structures, Nonnumerical Algorithms and Problems, Optimization, Discrete Mathematics, Mathematical Software, Algorithms, Problem Solving, Control Methods and Search, Computational Geometry and Object Modeling} }

Document

Front Matter

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

Front Matter, Table of Contents, Preface, Programm Commitee, External Reviewers

Piotr Sankowski and Christos Zaroliagis. Front Matter, Table of Contents, Preface, Programm Commitee, External Reviewers. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 0:i-0:xxiv, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{sankowski_et_al:LIPIcs.ESA.2016.0, author = {Sankowski, Piotr and Zaroliagis, Christos}, title = {{Front Matter, Table of Contents, Preface, Programm Commitee, External Reviewers}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {0:i--0:xxiv}, 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.2016.0}, URN = {urn:nbn:de:0030-drops-63429}, doi = {10.4230/LIPIcs.ESA.2016.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Programm Commitee, External Reviewers} }

Document

**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form 'Are vertices u and v connected with a path?' in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in O(n) time. This results improves over previously known O(n \log n) time algorithm.

Jakub Lacki and Piotr Sankowski. Optimal Decremental Connectivity in Planar Graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 608-621, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

Copy BibTex To Clipboard

@InProceedings{lacki_et_al:LIPIcs.STACS.2015.608, author = {Lacki, Jakub and Sankowski, Piotr}, title = {{Optimal Decremental Connectivity in Planar Graphs}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {608--621}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.608}, URN = {urn:nbn:de:0030-drops-49457}, doi = {10.4230/LIPIcs.STACS.2015.608}, annote = {Keywords: decremental connectivity, planar graphs, dynamic connectivity, algorithms} }

Document

**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

The well-known bidimensionality theory provides a method for designing fast, subexponential-time parameterized algorithms for a vast number of NP-hard problems on sparse graph classes such as planar graphs, bounded genus graphs, or, more generally, graphs with a fixed excluded minor. However, in order to apply the bidimensionality framework the considered problem needs to fulfill a special density property. Some well-known problems do not have this property, unfortunately, with probably the most prominent and important example being the Steiner Tree problem. Hence the question whether a subexponential-time parameterized algorithm for Steiner Tree on planar graphs exists has remained open. In this paper, we answer this question positively and develop an algorithm running in O(2^{O((k log k)^{2/3})}n) time and polynomial space, where k is the size of the Steiner tree and n is the number of vertices of the graph.
Our algorithm does not rely on tools from bidimensionality theory or graph minors theory, apart from Baker's classical approach. Instead, we introduce new tools and concepts to the study of the parameterized complexity of problems on sparse graphs.

Marcin Pilipczuk, Michal Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Subexponential-Time Parameterized Algorithm for Steiner Tree on Planar Graphs. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 353-364, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

Copy BibTex To Clipboard

@InProceedings{pilipczuk_et_al:LIPIcs.STACS.2013.353, author = {Pilipczuk, Marcin and Pilipczuk, Michal and Sankowski, Piotr and van Leeuwen, Erik Jan}, title = {{Subexponential-Time Parameterized Algorithm for Steiner Tree on Planar Graphs}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {353--364}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.353}, URN = {urn:nbn:de:0030-drops-39471}, doi = {10.4230/LIPIcs.STACS.2013.353}, annote = {Keywords: planar graph, Steiner tree, subexponential-time algorithms} }

Document

**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph. Here, nodes represent requests whereas edges correspond to items.
In this paper, we initiate the study of a new family of multi-layer covering problems. Each such problem consists of a collection of h distinct instances of a standard covering problem (layers), with the constraint that all layers share the same set of requests. We identify two main subfamilies of these problems:
- in an union multi-layer problem, a request is satisfied if it is satisfied in at least one layer;
- in an intersection multi-layer problem, a request is satisfied if it is satisfied in all layers.
To see some natural applications, consider both generalizations of k-MST. Union k-MST can model a problem where we are asked to connect a set of users to at least one of two communication networks, e.g., a wireless and a wired network. On the other hand, Intersection k-MST can formalize the problem of providing both electricity and water to at least k users.

Marek Cygan, Fabrizio Grandoni, Stefano Leonardi, Marcin Mucha, Marcin Pilipczuk, and Piotr Sankowski. Approximation Algorithms for Union and Intersection Covering Problems. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 28-40, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

Copy BibTex To Clipboard

@InProceedings{cygan_et_al:LIPIcs.FSTTCS.2011.28, author = {Cygan, Marek and Grandoni, Fabrizio and Leonardi, Stefano and Mucha, Marcin and Pilipczuk, Marcin and Sankowski, Piotr}, title = {{Approximation Algorithms for Union and Intersection Covering Problems}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {28--40}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.28}, URN = {urn:nbn:de:0030-drops-33213}, doi = {10.4230/LIPIcs.FSTTCS.2011.28}, annote = {Keywords: Approximation algorithms, Partial covering problems} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail