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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

In the minimum common string partition problem (MCSP), one gets two strings and is asked to find the minimum number of cuts in the first string such that the second string can be obtained by rearranging the resulting pieces. It is a difficult algorithmic problem having applications in computational biology, text processing, and data compression. MCSP has been studied extensively from various algorithmic angles: there are many papers studying approximation, heuristic, and parameterized algorithms. At the same time, almost nothing is known about its exact complexity. In this paper, we present new results in this direction. We improve the known 2ⁿ upper bound (where n is the length of input strings) to ϕⁿ where ϕ ≈ 1.618... is the golden ratio. The algorithm uses Fibonacci numbers to encode subsets as monomials of a certain implicit polynomial and extracts one of its coefficients using the fast Fourier transform. Then, we show that the case of constant size alphabet can be solved in subexponential time 2^{O(nlog log n/log n)} by a hybrid strategy: enumerate all long pieces and use dynamic programming over histograms of short pieces. Finally, we prove almost matching lower bounds assuming the Exponential Time Hypothesis.

Marek Cygan, Alexander S. Kulikov, Ivan Mihajlin, Maksim Nikolaev, and Grigory Reznikov. Minimum Common String Partition: Exact Algorithms. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cygan_et_al:LIPIcs.ESA.2021.35, author = {Cygan, Marek and Kulikov, Alexander S. and Mihajlin, Ivan and Nikolaev, Maksim and Reznikov, Grigory}, title = {{Minimum Common String Partition: Exact Algorithms}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {35:1--35:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.35}, URN = {urn:nbn:de:0030-drops-146167}, doi = {10.4230/LIPIcs.ESA.2021.35}, annote = {Keywords: similarity measure, string distance, exact algorithms, upper bounds, lower bounds} }

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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 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Given a traveling salesman problem (TSP) tour H in graph G, a k-move is an operation which removes k edges from H, and adds k edges of G so that a new tour H' is formed. The popular k-opt heuristic for TSP finds a local optimum by starting from an arbitrary tour H and then improving it by a sequence of k-moves.
Until 2016, the only known algorithm to find an improving k-move for a given tour was the naive solution in time O(n^k). At ICALP'16 de Berg, Buchin, Jansen and Woeginger showed an O(n^{floor(2/3k)+1})-time algorithm.
We show an algorithm which runs in O(n^{(1/4 + epsilon_k)k}) time, where lim_{k -> infinity} epsilon_k = 0. It improves over the state of the art for every k >= 5. For the most practically relevant case k=5 we provide a slightly refined algorithm running in O(n^{3.4}) time. We also show that for the k=4 case, improving over the O(n^3)-time algorithm of de Berg et al. would be a major breakthrough: an O(n^{3 - epsilon})-time algorithm for any epsilon > 0 would imply an O(n^{3 - delta})-time algorithm for the All Pairs Shortest Paths problem, for some delta>0.

Marek Cygan, Lukasz Kowalik, and Arkadiusz Socala. Improving TSP Tours Using Dynamic Programming over Tree Decompositions. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cygan_et_al:LIPIcs.ESA.2017.30, author = {Cygan, Marek and Kowalik, Lukasz and Socala, Arkadiusz}, title = {{Improving TSP Tours Using Dynamic Programming over Tree Decompositions}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {30:1--30:14}, 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.2017.30}, URN = {urn:nbn:de:0030-drops-78539}, doi = {10.4230/LIPIcs.ESA.2017.30}, annote = {Keywords: TSP, treewidth, local search, XP algorithm, hardness in P} }

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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:** Dagstuhl Reports, Volume 7, Issue 1 (2017)

Dagstuhl Seminar 17041 "Randomization in Parameterized Complexity" took place from January 22nd to January 27th 2017 with the objective to bridge the gap between randomization and parameterized complexity theory. This report documents the talks held during the seminar as well as the open questions arised in the discussion sessions.

Marek Cygan, Fedor V. Fomin, Danny Hermelin, and Magnus Wahlström. Randomization in Parameterized Complexity (Dagstuhl Seminar 17041). In Dagstuhl Reports, Volume 7, Issue 1, pp. 103-128, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{cygan_et_al:DagRep.7.1.103, author = {Cygan, Marek and Fomin, Fedor V. and Hermelin, Danny and Wahlstr\"{o}m, Magnus}, title = {{Randomization in Parameterized Complexity (Dagstuhl Seminar 17041)}}, pages = {103--128}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {7}, number = {1}, editor = {Cygan, Marek and Fomin, Fedor V. and Hermelin, Danny and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.1.103}, URN = {urn:nbn:de:0030-drops-72479}, doi = {10.4230/DagRep.7.1.103}, annote = {Keywords: fixed-parameter tractability, intractability, parameterized complexity, randomness} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

The H-free Edge Deletion problem asks, for a given graph G and integer k, whether it is possible to delete at most k edges from G to make it H-free, that is, not containing H as an induced subgraph. The H-free Edge Completion problem is defined similarly, but we add edges instead of deleting them. The study of these two problem families has recently been the subject of intensive studies from the point of view of parameterized complexity and kernelization. In particular, it was shown that the problems do not admit polynomial kernels (under plausible complexity assumptions) for almost all graphs H, with several important exceptions occurring when the class of H-free graphs exhibits some structural properties.
In this work we complement the parameterized study of edge modification problems to H-free graphs by considering their approximability. We prove that whenever H is 3-connected and has at least two non-edges, then both H-free Edge Deletion and H-free Edge Completion are very hard to approximate: they do not admit poly(OPT)-approximation in polynomial time, unless P=NP, or even in time subexponential in OPT, unless the Exponential Time Hypothesis fails. The assumption of the existence of two non-edges appears to be important: we show that whenever H is a complete graph without one edge, then H-free Edge Deletion is tightly connected to the \minhorn problem, whose approximability is still open. Finally, in an attempt to extend our hardness results beyond 3-connected graphs, we consider the cases of H being a path or a cycle, and we achieve an almost complete dichotomy there.

Ivan Bliznets, Marek Cygan, Pawel Komosa, and Michal Pilipczuk. Hardness of Approximation for H-Free Edge Modification Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bliznets_et_al:LIPIcs.APPROX-RANDOM.2016.3, author = {Bliznets, Ivan and Cygan, Marek and Komosa, Pawel and Pilipczuk, Michal}, title = {{Hardness of Approximation for H-Free Edge Modification Problems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {3:1--3:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.3}, URN = {urn:nbn:de:0030-drops-66260}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.3}, annote = {Keywords: hardness of approximation, parameterized complexity, kernelization, edge modification problems} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

In the Closest String problem one is given a family S of equal-length strings over some fixed alphabet, and the task is to find a string y that minimizes the maximum Hamming distance between y and a string from S. While polynomial-time approximation schemes (PTASes) for this problem are known for a long time [Li et al.; J. ACM'02], no efficient polynomial-time approximation scheme (EPTAS) has been proposed so far. In this paper, we prove that the existence of an EPTAS for Closest String is in fact unlikely, as it would imply that FPT=W[1], a highly unexpected collapse in the hierarchy of parameterized complexity classes. Our proof also shows that the existence of a PTAS for Closest String with running time f(eps) n^o(1/eps), for any computable function f, would contradict the Exponential Time Hypothesis.

Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Lower Bounds for Approximation Schemes for Closest String. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 12:1-12:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cygan_et_al:LIPIcs.SWAT.2016.12, author = {Cygan, Marek and Lokshtanov, Daniel and Pilipczuk, Marcin and Pilipczuk, Michal and Saurabh, Saket}, title = {{Lower Bounds for Approximation Schemes for Closest String}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {12:1--12:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.12}, URN = {urn:nbn:de:0030-drops-60232}, doi = {10.4230/LIPIcs.SWAT.2016.12}, annote = {Keywords: closest string, PTAS, efficient PTAS} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

In the Upper Degree-Constrained Partial Orientation (UDPO) problem we are given an undirected graph G=(V,E), together with two degree constraint functions d^-,d^+:V -> N. The goal is to orient as many edges as possible, in such a way that for each vertex v in V the number of arcs entering v is at most d^-(v), whereas the number of arcs leaving v is at most d^+(v). This problem was introduced by Gabow [SODA'06], who proved it to be MAXSNP-hard (and thus APX-hard). In the same paper Gabow presented an LP-based iterative rounding 4/3-approximation algorithm.
As already observed by Gabow, the problem in question is a special case of the classic 3-Dimensional Matching, which in turn is a special case of the k-Set Packing problem. Back in 2006 the best known polynomial time approximation algorithm for 3-Dimensional Matching was a simple local search by Hurkens and Schrijver [SIDMA'89], the approximation ratio of which is (3+epsilon)/2; hence the algorithm of Gabow was an improvement over the approach brought from the more general problems.
In this paper we show that the UDPO problem when cast as 3-Dimensional Matching admits a special structure, which is obliviously exploited by the known approximation algorithms for k-Set Packing. In fact, we show that already the local-search routine of Hurkens and Schrijver gives (4+epsilon)/3-approximation when used for the instances coming from UDPO. Moreover, the recent approximation algorithm for 3-Set Packing [Cygan, FOCS'13] turns out to be a (5+epsilon)/4-approximation for UDPO. This improves over 4/3 as the best ratio known up to date for UDPO.

Marek Cygan and Tomasz Kociumaka. Approximating Upper Degree-Constrained Partial Orientations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 212-224, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cygan_et_al:LIPIcs.APPROX-RANDOM.2015.212, author = {Cygan, Marek and Kociumaka, Tomasz}, title = {{Approximating Upper Degree-Constrained Partial Orientations}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {212--224}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.212}, URN = {urn:nbn:de:0030-drops-53040}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.212}, annote = {Keywords: graph orientations, degree-constrained orientations, approximation algorithm, local search} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

The k-center problem is a classic facility location problem, where given an edge-weighted graph G=(V,E) one is to find a subset of k vertices S, such that each vertex in V is "close" to some vertex in S. The approximation status of this basic problem is well understood, as a simple 2-approximation algorithm is known to be tight. Consequently different extensions were studied.
In the capacitated version of the problem each vertex is assigned a capacity, which is a strict upper bound on the number of clients a facility can serve, when located at this vertex. A constant factor approximation for the capacitated k-center was obtained last year in [Cygan, Hajiaghayi and Khuller, FOCS'12], which was recently improved to a 9-approximation in [An, Bhaskara and Svensson, arXiv'13].
In a different generalization of the problem some clients (denoted as outliers) may be disregarded. Here we are additionally given an integer p and the goal is to serve exactly p clients, which the algorithm is free to choose. In [Charikar et al., SODA'01] the authors presented a 3-approximation for the k-center problem with outliers.
In this paper we consider a common generalization of the two extensions previously studied separately, i.e. we work with the capacitated k-center with outliers. We present the first constant factor approximation algorithm with approximation ratio of 25 even for the case of non-uniform hard capacities.

Marek Cygan and Tomasz Kociumaka. Constant Factor Approximation for Capacitated k-Center with Outliers. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 251-262, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{cygan_et_al:LIPIcs.STACS.2014.251, author = {Cygan, Marek and Kociumaka, Tomasz}, title = {{Constant Factor Approximation for Capacitated k-Center with Outliers}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {251--262}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.251}, URN = {urn:nbn:de:0030-drops-44625}, doi = {10.4230/LIPIcs.STACS.2014.251}, annote = {Keywords: approximation algorithms, k-center, capacities, outliers, LP rounding} }

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

Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f(.) is a subgraph H \subseteq G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function.
In this paper we study pairwise spanners, where we require to approximate the u-v distance only for pairs (u,v) in a given set P \subseteq V x V. Such P-spanners were studied before [Coppersmith,Elkin'05] only in the special case that f(.) is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers).
Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f(.)).
In more detail, for arbitrary P, we show that there exists a P-spanner of size O(n(|P|log n)^{1/4}) with f(d) = d + 4 log n. Alternatively, for any epsislon > 0, there exists a P-spanner of size O(n|P|^{1/4} sqrt{(log n) / epsilon}) with f(d) = (1 + epsilon)d + 4. We also consider the relevant special case that there is a critical set of nodes S \subseteq V, and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S x S)-spanner of size O(n sqrt{|S|}) with f(d) = d + 2, and an (S x V)-spanner of size O(n sqrt{|S| log n}) with f(d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time.

Marek Cygan, Fabrizio Grandoni, and Telikepalli Kavitha. On Pairwise Spanners. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 209-220, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{cygan_et_al:LIPIcs.STACS.2013.209, author = {Cygan, Marek and Grandoni, Fabrizio and Kavitha, Telikepalli}, title = {{On Pairwise Spanners}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {209--220}, 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.2013.209}, URN = {urn:nbn:de:0030-drops-39353}, doi = {10.4230/LIPIcs.STACS.2013.209}, annote = {Keywords: Undirected graphs, shortest paths, additive spanners, distance preservers} }

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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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