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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

This year’s Parameterized Algorithms and Computational Experiments challenge (PACE 2020) was devoted to the problem of computing the treedepth of a given graph. Altogether 51 participants from 20 teams, 12 countries and 3 continents submitted their implementations to the competition.
In this report, we describe the setup of the challenge, the selection of benchmark instances and the ranking of the participating teams. We also briefly discuss the approaches used in the submitted solvers and the differences in their performance on our benchmark dataset.

Łukasz Kowalik, Marcin Mucha, Wojciech Nadara, Marcin Pilipczuk, Manuel Sorge, and Piotr Wygocki. The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kowalik_et_al:LIPIcs.IPEC.2020.37, author = {Kowalik, {\L}ukasz and Mucha, Marcin and Nadara, Wojciech and Pilipczuk, Marcin and Sorge, Manuel and Wygocki, Piotr}, title = {{The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {37:1--37:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.37}, URN = {urn:nbn:de:0030-drops-133404}, doi = {10.4230/LIPIcs.IPEC.2020.37}, annote = {Keywords: computing treedepth, contest, implementation challenge, FPT} }

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

In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey.
Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space.

Marcin Mucha, Jesper Nederlof, Jakub Pawlewicz, and Karol Węgrzycki. Equal-Subset-Sum Faster Than the Meet-in-the-Middle. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{mucha_et_al:LIPIcs.ESA.2019.73, author = {Mucha, Marcin and Nederlof, Jesper and Pawlewicz, Jakub and W\k{e}grzycki, Karol}, title = {{Equal-Subset-Sum Faster Than the Meet-in-the-Middle}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {73:1--73:16}, 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.2019.73}, URN = {urn:nbn:de:0030-drops-111946}, doi = {10.4230/LIPIcs.ESA.2019.73}, annote = {Keywords: Equal-Subset-Sum, Subset-Sum, meet-in-the-middle, enumeration technique, randomized algorithm} }

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**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)

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@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} }

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

In this paper, we construct a deterministic 4-competitive algorithm for the online file migration problem, beating the currently best 20-year old, 4.086-competitive MTLM algorithm by Bartal et al. (SODA 1997). Like MTLM, our algorithm also operates in phases, but it adapts their lengths dynamically depending on the geometry of requests seen so far. The improvement was obtained by carefully analyzing a linear model (factor-revealing LP) of a single phase of the algorithm. We also show that if an online algorithm operates in phases of fixed length and the adversary is able to modify the graph between phases, no algorithm can beat the competitive ratio of 4.086.

Marcin Bienkowski, Jaroslaw Byrka, and Marcin Mucha. Dynamic Beats Fixed: On Phase-Based Algorithms for File Migration. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bienkowski_et_al:LIPIcs.ICALP.2017.13, author = {Bienkowski, Marcin and Byrka, Jaroslaw and Mucha, Marcin}, title = {{Dynamic Beats Fixed: On Phase-Based Algorithms for File Migration}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {13:1--13:14}, 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.2017.13}, URN = {urn:nbn:de:0030-drops-73942}, doi = {10.4230/LIPIcs.ICALP.2017.13}, annote = {Keywords: file migration, factor-revealing linear programs, online algorithms, competitive analysis} }

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

In the recent years, significant progress has been made in explaining apparent hardness of improving over naive solutions for many fundamental polynomially solvable problems. This came in the form of conditional lower bounds -- reductions from a problem assumed to be hard. These include 3SUM, All-Pairs Shortest Paths, SAT and Orthogonal Vectors, and others.
In the (min,+)-convolution problem, the goal is to compute a sequence c, where c[k] = min_i a[i]+b[k-i], given sequences a and b. This can easily be done in O(n^2) time, but no O(n^{2-eps}) algorithm is known for eps > 0. In this paper we undertake a systematic study of the (min,+)-convolution problem as a hardness assumption.
As the first step, we establish equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The (min,+)-convolution has been used as a building block in algorithms for many problems, notably problems in stringology. It has also already appeared as an ad hoc hardness assumption. We investigate some of these connections and provide new reductions and other results.

Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk. On Problems Equivalent to (min,+)-Convolution. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cygan_et_al:LIPIcs.ICALP.2017.22, author = {Cygan, Marek and Mucha, Marcin and Wegrzycki, Karol and Wlodarczyk, Michal}, title = {{On Problems Equivalent to (min,+)-Convolution}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {22:1--22:15}, 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.2017.22}, URN = {urn:nbn:de:0030-drops-74216}, doi = {10.4230/LIPIcs.ICALP.2017.22}, annote = {Keywords: fine-grained complexity, knapsack, conditional lower bounds, (min,+)-convolution, subquadratic equivalence} }

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**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

In the Shortest Superstring problem (SS) one has to find a shortest string s containing given strings s_1,...,s_n as substrings. The problem is NP-hard, so a natural question is that of its approximability.
One natural approach to approximately solving SS is the following GREEDY heuristic: repeatedly merge two strings with the largest overlap until only a single string is left. This heuristic is conjectured to be a 2-approximation, but even after 30 years since the conjecture has been posed, we are still very far from proving it. The situation is better for non-greedy approximation algorithms, where several approaches yielding 2.5-approximation (and better) are known.
In this talk, we will survey the main results in the area, focusing on the fundamental ideas and intuitions.

Marcin Mucha. Shortest Superstring. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mucha:LIPIcs.CPM.2017.3, author = {Mucha, Marcin}, title = {{Shortest Superstring}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {3:1--3:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.3}, URN = {urn:nbn:de:0030-drops-73483}, doi = {10.4230/LIPIcs.CPM.2017.3}, annote = {Keywords: shortest superstring, approximation algorithms} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

The Travelling Salesman Problem is one of the most fundamental and most studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides's algorithm with approximation factor of 3/2, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only 4/3.
Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. (2011), and then by Momke and Svensson (2011). In this paper, we provide an improved analysis of the approach used by the latter, yielding a bound of 13/9 on the approximation factor, as well as a bound of 19/12+epsilon for any epsilon>0 for a more general Travelling Salesman Path Problem in graphic metrics.

Marcin Mucha. 13/9-approximation for Graphic TSP. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 30-41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{mucha:LIPIcs.STACS.2012.30, author = {Mucha, Marcin}, title = {{13/9-approximation for Graphic TSP}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {30--41}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph 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.2012.30}, URN = {urn:nbn:de:0030-drops-34025}, doi = {10.4230/LIPIcs.STACS.2012.30}, annote = {Keywords: approximation algorithms, travelling salesman problem} }

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**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)

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@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} }

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