Document

Front Matter

**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

Front Matter, Table of Contents, Preface, Conference Organization

13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 0:i-0:xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{paul_et_al:LIPIcs.IPEC.2018.0, author = {Paul, Christophe and Pilipczuk, Michal}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {0:i--0:xiv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.0}, URN = {urn:nbn:de:0030-drops-102012}, doi = {10.4230/LIPIcs.IPEC.2018.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

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

We consider two optimization problems in planar graphs. In {Maximum Weight Independent Set of Objects} we are given a graph G and a family D of {objects}, each being a connected subgraph of G with a prescribed weight, and the task is to find a maximum-weight subfamily of D consisting of pairwise disjoint objects. In {Minimum Weight Distance Set Cover} we are given an edge-weighted graph G, two sets D,C of vertices of G, where vertices of D have prescribed weights, and a nonnegative radius r. The task is to find a minimum-weight subset of D such that every vertex of C is at distance at most r from some selected vertex. Via simple reductions, these two problems generalize a number of geometric optimization tasks, notably {Maximum Weight Independent Set} for polygons in the plane and {Weighted Geometric Set Cover} for unit disks and unit squares. We present {quasi-polynomial time approximation schemes} (QPTASs) for both of the above problems in planar graphs: given an accuracy parameter epsilon>0 we can compute a solution whose weight is within multiplicative factor of (1+epsilon) from the optimum in time 2^{poly(1/epsilon,log |D|)}* n^{O(1)}, where n is the number of vertices of the input graph. Our main technical contribution is to transfer the techniques used for recursive approximation schemes for geometric problems due to Adamaszek, Har-Peled, and Wiese [Adamaszek and Wiese, 2013; Adamaszek and Wiese, 2014; Sariel Har-Peled, 2014] to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods.

Michal Pilipczuk, Erik Jan van Leeuwen, and Andreas Wiese. Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{pilipczuk_et_al:LIPIcs.ESA.2018.65, author = {Pilipczuk, Michal and van Leeuwen, Erik Jan and Wiese, Andreas}, title = {{Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {65:1--65:13}, 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.2018.65}, URN = {urn:nbn:de:0030-drops-95282}, doi = {10.4230/LIPIcs.ESA.2018.65}, annote = {Keywords: QPTAS, planar graphs, Voronoi diagram} }

Document

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

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.

Jakub Gajarský, Stephan Kreutzer, Jaroslav Nesetril, Patrice Ossona de Mendez, Michal Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk. First-Order Interpretations of Bounded Expansion Classes. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 126:1-126:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gajarsky_et_al:LIPIcs.ICALP.2018.126, author = {Gajarsk\'{y}, Jakub and Kreutzer, Stephan and Nesetril, Jaroslav and Ossona de Mendez, Patrice and Pilipczuk, Michal and Siebertz, Sebastian and Torunczyk, Szymon}, title = {{First-Order Interpretations of Bounded Expansion Classes}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {126:1--126:14}, 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.2018.126}, URN = {urn:nbn:de:0030-drops-91300}, doi = {10.4230/LIPIcs.ICALP.2018.126}, annote = {Keywords: Logical interpretations/transductions, structurally sparse graphs, bounded expansion} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Consider the Maximum Weight Independent Set problem for rectangles: given a family of weighted axis-parallel rectangles in the plane, find a maximum-weight subset of non-overlapping rectangles. The problem is notoriously hard both in the approximation and in the parameterized setting. The best known polynomial-time approximation algorithms achieve super-constant approximation ratios [Chalermsook & Chuzhoy, Proc. SODA 2009; Chan & Har-Peled, Discrete & Comp. Geometry, 2012], even though there is a (1+epsilon)-approximation running in quasi-polynomial time [Adamaszek & Wiese, Proc. FOCS 2013; Chuzhoy & Ene, Proc. FOCS 2016]. When parameterized by the target size of the solution, the problem is W[1]-hard even in the unweighted setting [Marx, ESA 2005].
To achieve tractability, we study the following shrinking model: one is allowed to shrink each input rectangle by a multiplicative factor 1-delta for some fixed delta > 0, but the performance is still compared against the optimal solution for the original, non-shrunk instance. We prove that in this regime, the problem admits an EPTAS with running time f(epsilon,delta) n^{O(1)}, and an FPT algorithm with running time f(k,delta) n^{O(1)}, in the setting where a maximum-weight solution of size at most k is to be computed. This improves and significantly simplifies a PTAS given earlier for this problem [Adamaszek, Chalermsook & Wiese, Proc. APPROX/RANDOM 2015], and provides the first parameterized results for the shrinking model. Furthermore, we explore kernelization in the shrinking model, by giving efficient kernelization procedures for several variants of the problem when the input rectangles are squares.

Michal Pilipczuk, Erik Jan van Leeuwen, and Andreas Wiese. Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pilipczuk_et_al:LIPIcs.MFCS.2017.42, author = {Pilipczuk, Michal and van Leeuwen, Erik Jan and Wiese, Andreas}, title = {{Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {42:1--42:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.42}, URN = {urn:nbn:de:0030-drops-80917}, doi = {10.4230/LIPIcs.MFCS.2017.42}, annote = {Keywords: Combinatorial optimization, Approximation algorithms, Fixed-parameter algorithms} }

Document

Invited Talk

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

This is an overview of the invited talk delivered at the 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017).

Michal Pilipczuk. On Definable and Recognizable Properties of Graphs of Bounded Treewidth (Invited Talk). In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 82:1-82:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pilipczuk:LIPIcs.MFCS.2017.82, author = {Pilipczuk, Michal}, title = {{On Definable and Recognizable Properties of Graphs of Bounded Treewidth}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {82:1--82:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.82}, URN = {urn:nbn:de:0030-drops-81384}, doi = {10.4230/LIPIcs.MFCS.2017.82}, annote = {Keywords: monadic second-order logic, treewidth, recognizability} }

Document

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

In the multicoloring problem, also known as (a:b)-coloring or b-fold coloring, we are given a graph G and a set of a colors, and the task is to assign a subset of b colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the b=1 case) is equivalent to finding a homomorphism to the Kneser graph KG_{a,b}, and gives relaxations approaching the fractional chromatic number.
We study the complexity of determining whether a graph has an (a:b)-coloring. Our main result is that this problem does not admit an algorithm with running time f(b) * 2^{o(log b) n}, for any computable f(b), unless the Exponential Time Hypothesis (ETH) fails. A (b+1)^n * poly(n)-time algorithm due to Nederlof [2008] shows that this is tight. A direct corollary of our result is that the graph homomorphism problem does not admit a 2^O(n+h) algorithm unless ETH fails, even if the target graph is required to be a Kneser graph. This refines the understanding given by the recent lower bound of Cygan et al. [SODA 2016].
The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindström [Canad. Math. Bull., 1965], which is a combinatorial tool that, to the best of our knowledge, has not yet been used for proving complexity lower bounds. As a side result, we prove that the running time of the algorithms of Abasi et al. [MFCS 2014] and of Gabizon et al. [ESA 2015] for the r-monomial detection problem are optimal under ETH.

Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, Arkadiusz Socala, and Marcin Wrochna. Tight Lower Bounds for the Complexity of Multicoloring. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bonamy_et_al:LIPIcs.ESA.2017.18, author = {Bonamy, Marthe and Kowalik, Lukasz and Pilipczuk, Michal and Socala, Arkadiusz and Wrochna, Marcin}, title = {{Tight Lower Bounds for the Complexity of Multicoloring}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {18:1--18:14}, 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.2017.18}, URN = {urn:nbn:de:0030-drops-78527}, doi = {10.4230/LIPIcs.ESA.2017.18}, annote = {Keywords: multicoloring, Kneser graph homomorphism, ETH lower bound} }

Document

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

Suppose F is a finite family of graphs. We consider the following meta-problem, called F-Immersion Deletion: given a graph G and an integer k, decide whether the deletion of at most k edges of G can result in a graph that does not contain any graph from F as an immersion. This problem is a close relative of the F-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from F.
We prove that whenever all graphs from F are connected and at least one graph of F is planar and subcubic, then the F-Immersion Deletion problem admits:
- a constant-factor approximation algorithm running in time O(m^3 n^3 log m)
- a linear kernel that can be computed in time O(m^4 n^3 log m) and
- a O(2^{O(k)} + m^4 n^3 log m)-time fixed-parameter algorithm,
where n,m count the vertices and edges of the input graph. Our findings mirror those of Fomin et al. [FOCS 2012], who obtained similar results for F-Minor Deletion, under the assumption that at least one graph from F is planar.
An important difference is that we are able to obtain a linear kernel for F-Immersion Deletion, while the exponent of the kernel of Fomin et al. depends heavily on the family F. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of F-Immersion Deletion is quite different than that of F-Minor Deletion.

Archontia C. Giannopoulou, Michal Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, and Marcin Wrochna. Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{giannopoulou_et_al:LIPIcs.ICALP.2017.57, author = {Giannopoulou, Archontia C. and Pilipczuk, Michal and Raymond, Jean-Florent and Thilikos, Dimitrios M. and Wrochna, Marcin}, title = {{Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {57:1--57:15}, 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.2017.57}, URN = {urn:nbn:de:0030-drops-73891}, doi = {10.4230/LIPIcs.ICALP.2017.57}, annote = {Keywords: Kernelization, Approximation, Immersion, Protrusion, Tree-cut width} }

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

We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r,eps)|A|^(1+eps), where r is any positive integer, eps is any positive real, and f is a function that depends only on the class C. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by [Reidl et al., CoRR, 2016]. As an algorithmic application of the above result, we show that for every fixed integer r, the parameterized Distance-r Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by [Drange et al., STACS 2016], and shows that the limit of parameterized tractability of Distance-r Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.

Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michal Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{eickmeyer_et_al:LIPIcs.ICALP.2017.63, author = {Eickmeyer, Kord and Giannopoulou, Archontia C. and Kreutzer, Stephan and Kwon, O-joung and Pilipczuk, Michal and Rabinovich, Roman and Siebertz, Sebastian}, title = {{Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {63:1--63:14}, 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.2017.63}, URN = {urn:nbn:de:0030-drops-74288}, doi = {10.4230/LIPIcs.ICALP.2017.63}, annote = {Keywords: Graph Structure Theory, Nowhere Dense Graphs, Parameterized Complexity, Kernelization, Dominating Set} }

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

A simple digraph is semi-complete if for any two of its vertices u and v, at least one of the arcs (u,v) and (v,u) is present. We study the complexity of computing two layout parameters of semi-complete digraphs: cutwidth and optimal linear arrangement (OLA). We prove that:
-Both parameters are NP-hard to compute and the known exact and parameterized algorithms for them have essentially optimal running times, assuming the Exponential Time Hypothesis.
- The cutwidth parameter admits a quadratic Turing kernel, whereas it does not admit any polynomial kernel unless coNP/poly contains NP. By contrast, OLA admits a linear kernel.
These results essentially complete the complexity analysis of computing cutwidth and OLA on semi-complete digraphs. Our techniques can be also used to analyze the sizes of minimal obstructions for having small cutwidth under the induced subdigraph relation.

Florian Barbero, Christophe Paul, and Michal Pilipczuk. Exploring the Complexity of Layout Parameters in Tournaments and Semi-Complete Digraphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 70:1-70:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{barbero_et_al:LIPIcs.ICALP.2017.70, author = {Barbero, Florian and Paul, Christophe and Pilipczuk, Michal}, title = {{Exploring the Complexity of Layout Parameters in Tournaments and Semi-Complete Digraphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {70:1--70:13}, 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.2017.70}, URN = {urn:nbn:de:0030-drops-74537}, doi = {10.4230/LIPIcs.ICALP.2017.70}, annote = {Keywords: cutwidth, OLA, tournament, semi-complete digraph} }

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

The classic algorithm of Bodlaender and Kloks solves the following problem in linear fixed-parameter time: given a tree decomposition of a graph of (possibly suboptimal) width k, compute an optimum-width tree decomposition of the graph. In this work, we prove that this problem can also be solved in MSO in the following sense: for every positive integer k, there is an MSO transduction from tree decompositions of width k to tree decompositions of optimum width. Together with our recent results, this implies that for every k there exists an MSO transduction which inputs a graph of treewidth k, and nondeterministically outputs its tree decomposition of optimum width.

Mikolaj Bojanczyk and Michal Pilipczuk. Optimizing Tree Decompositions in MSO. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bojanczyk_et_al:LIPIcs.STACS.2017.15, author = {Bojanczyk, Mikolaj and Pilipczuk, Michal}, title = {{Optimizing Tree Decompositions in MSO}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {15:1--15:13}, 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.2017.15}, URN = {urn:nbn:de:0030-drops-70173}, doi = {10.4230/LIPIcs.STACS.2017.15}, annote = {Keywords: tree decomposition, treewidth, transduction, monadic second-order logic} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k. We prove that every minimal immersion obstruction for cutwidth at most k has size at most 2^O(k^3*log(k)).
As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2^O(k^2*log(k))*n, where k is the optimum width and n is the number of vertices. While being slower by a log k-factor in the exponent than the fastest known algorithm, due to Thilikos, Bodlaender, and Serna [J. Algorithms 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.

Archontia C. Giannopoulou, Michal Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, and Marcin Wrochna. Cutwidth: Obstructions and Algorithmic Aspects. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{giannopoulou_et_al:LIPIcs.IPEC.2016.15, author = {Giannopoulou, Archontia C. and Pilipczuk, Michal and Raymond, Jean-Florent and Thilikos, Dimitrios M. and Wrochna, Marcin}, title = {{Cutwidth: Obstructions and Algorithmic Aspects}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.15}, URN = {urn:nbn:de:0030-drops-69306}, doi = {10.4230/LIPIcs.IPEC.2016.15}, annote = {Keywords: cutwidth, obstructions, immersions, fixed-parameter tractability} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

In the EDGE BIPARTIZATION problem one is given an undirected graph G and an integer k, and the question is whether k edges can be deleted from G so that it becomes bipartite. In 2006, Guo et al. [J. Comput. Syst. Sci., 72(8):1386-1396, 2006] proposed an algorithm solving this problem in time O(2^k m^2); today, this algorithm is a textbook example of an application of the iterative compression technique. Despite extensive progress in the understanding of the parameterized complexity of graph separation problems in the recent years, no significant improvement upon this result has been yet reported. We present an algorithm for Edge Bipartization that works in time O(1.977^k nm), which is the first algorithm with the running time dependence on the parameter better than 2^k. To this end, we combine the general iterative compression strategy of Guo et al. [J. Comput. Syst. Sci., 72(8):1386-1396, 2006], the technique proposed by Wahlström [SODA'14] of using a polynomial-time solvable relaxation in the form of a Valued Constraint Satisfaction Problem to guide a bounded-depth branching algorithm, and an involved Measure&Conquer analysis of the recursion tree.

Marcin Pilipczuk, Michal Pilipczuk, and Marcin Wrochna. Edge Bipartization Faster Than 2^k. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 26:1-26:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pilipczuk_et_al:LIPIcs.IPEC.2016.26, author = {Pilipczuk, Marcin and Pilipczuk, Michal and Wrochna, Marcin}, title = {{Edge Bipartization Faster Than 2^k}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {26:1--26:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.26}, URN = {urn:nbn:de:0030-drops-69285}, doi = {10.4230/LIPIcs.IPEC.2016.26}, annote = {Keywords: edge bipartization, FPT algorithm} }

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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 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The generalised colouring numbers adm_r(G), col_r(G), and wcol_r(G) were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Nesetril and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of r. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed parameter tractable on classes of graphs with excluded topological minors.

Stephan Kreutzer, Michal Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. The Generalised Colouring Numbers on Classes of Bounded Expansion. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 85:1-85:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kreutzer_et_al:LIPIcs.MFCS.2016.85, author = {Kreutzer, Stephan and Pilipczuk, Michal and Rabinovich, Roman and Siebertz, Sebastian}, title = {{The Generalised Colouring Numbers on Classes of Bounded Expansion}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {85:1--85:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.85}, URN = {urn:nbn:de:0030-drops-64937}, doi = {10.4230/LIPIcs.MFCS.2016.85}, annote = {Keywords: graph structure theory, nowhere dense graphs, generalised colouring numbers, splitter game, first-order model-checking} }

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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 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We prove that for every positive integer r and for every graph class G of bounded expansion, the r-DOMINATING SET problem admits a linear kernel on graphs from G. Moreover, in the more general case when G is only assumed to be nowhere dense, we give an almost linear kernel on G for the classic DOMINATING SET problem, i.e., for the case r=1. These results generalize a line of previous research on finding linear kernels for DOMINATING SET and r-DOMINATING SET (Alber et al., JACM 2004, Bodlaender et al., FOCS 2009, Fomin et al., SODA 2010, Fomin et al., SODA 2012, Fomin et al., STACS 2013). However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related CONNECTED DOMINATING SET problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on H-topological-minor-free graphs (Fomin et al., STACS 2013). Also, we prove that for any somewhere dense class G, there is some r for which r-DOMINATING SET is W[2]-hard on G. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of r-DOMINATING SET on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.

Pål Grønås Drange, Markus Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, Felix Reidl, Fernando Sánchez Villaamil, Saket Saurabh, Sebastian Siebertz, and Somnath Sikdar. Kernelization and Sparseness: the Case of Dominating Set. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{drange_et_al:LIPIcs.STACS.2016.31, author = {Drange, P\r{a}l Gr{\o}n\r{a}s and Dregi, Markus and Fomin, Fedor V. and Kreutzer, Stephan and Lokshtanov, Daniel and Pilipczuk, Marcin and Pilipczuk, Michal and Reidl, Felix and S\'{a}nchez Villaamil, Fernando and Saurabh, Saket and Siebertz, Sebastian and Sikdar, Somnath}, title = {{Kernelization and Sparseness: the Case of Dominating Set}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.31}, URN = {urn:nbn:de:0030-drops-57327}, doi = {10.4230/LIPIcs.STACS.2016.31}, annote = {Keywords: kernelization, dominating set, bounded expansion, nowhere dense} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition's width. Following the work of Allender et al. [Theory of Computing, 2014], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [J. Comput. Syst. Sci., 2003], we prove that the question is closely related to a conjecture that the Longest Common Subsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we complete the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth. We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition's depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non- deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.

Michal Pilipczuk and Marcin Wrochna. On Space Efficiency of Algorithms Working on Structural Decompositions of Graphs. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{pilipczuk_et_al:LIPIcs.STACS.2016.57, author = {Pilipczuk, Michal and Wrochna, Marcin}, title = {{On Space Efficiency of Algorithms Working on Structural Decompositions of Graphs}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {57:1--57:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.57}, URN = {urn:nbn:de:0030-drops-57588}, doi = {10.4230/LIPIcs.STACS.2016.57}, annote = {Keywords: tree decomposition, LCS, tree-depth, NAuxSA, Savitch’s theorem} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

In the k-Leaf Out-Branching and k-Internal Out-Branching problems we are given a directed graph D with a designated root r and a nonnegative integer k. The question is to determine the existence of an outbranching rooted at r that has at least k leaves, or at least k internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k^2) vertices are known on general graphs. In this work we show that k-Leaf Out-Branching admits a kernel with O(k) vertices on H-minor-free graphs, for any fixed H, whereas k-Internal Out-Branching admits a kernel with O(k) vertices on any graph class of bounded expansion.

Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, and Arkadiusz Socala. Linear Kernels for Outbranching Problems in Sparse Digraphs. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 199-211, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bonamy_et_al:LIPIcs.IPEC.2015.199, author = {Bonamy, Marthe and Kowalik, Lukasz and Pilipczuk, Michal and Socala, Arkadiusz}, title = {{Linear Kernels for Outbranching Problems in Sparse Digraphs}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {199--211}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.199}, URN = {urn:nbn:de:0030-drops-55839}, doi = {10.4230/LIPIcs.IPEC.2015.199}, annote = {Keywords: FPT algorithm, kernelization, outbranching, sparse graphs} }

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

Let F be a family of graphs. In the F-Completion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k edges can be added to G so that the resulting graph does not contain a graph from F as an induced subgraph. It appeared recently that special cases of F-Completion, the problem of completing into a chordal graph known as "Minimum Fill-in", corresponding to the case of F={C_4,C_5,C_6,...}, and the problem of completing into a split graph, i.e., the case of F={C_4,2K_2,C_5}, are solvable in parameterized subexponential time. The exploration of this phenomenon is the main motivation for our research on F-Completion.
In this paper we prove that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms by showing that:
- The problem Trivially Perfect Completion is solvable in parameterized subexponential time, that is F-Completion for F={C_4,P_4}, a cycle and a path on four vertices.
- The problems known in the literature as Pseudosplit Completion, the case where F={2K_2,C_4}, and Threshold Completion, where F={2K_2,P_4,C_4}, are also solvable in subexponential time.
We complement our algorithms for $F$-Completion with the following lower bounds:
- For F={2K_2}, F={C_4}, F={P_4}, and F={2K_2,P_4}, F-Completion cannot be solved in time 2^o(k).n^O(1) unless the Exponential Time Hypothesis (ETH) fails.
Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of F-Completion problems for F contained inside {2K_2,C_4,P_4}.

Pal Gronas Drange, Fedor V. Fomin, Michal Pilipczuk, and Yngve Villanger. Exploring Subexponential Parameterized Complexity of Completion Problems. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 288-299, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{drange_et_al:LIPIcs.STACS.2014.288, author = {Drange, Pal Gronas and Fomin, Fedor V. and Pilipczuk, Michal and Villanger, Yngve}, title = {{Exploring Subexponential Parameterized Complexity of Completion Problems}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {288--299}, 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.2014.288}, URN = {urn:nbn:de:0030-drops-44659}, doi = {10.4230/LIPIcs.STACS.2014.288}, annote = {Keywords: edge completion, modification, subexponential parameterized complexity} }

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

Given two graphs H and G, the Subgraph Isomorphism problem asks if H is isomorphic to a subgraph of G. While NP-hard in general, algorithms exist for various parameterized versions of the problem. However, the literature contains very little guidance on which combinations of parameters can or cannot be exploited algorithmically. Our goal is to systematically investigate the possible parameterized algorithms that can exist for Subgraph Isomorphism.
We develop a framework involving 10 relevant parameters for each of H and G (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time f1_(p_1,p_2,...,p_l).n^f_2(p_(l+1),...,p_k) exists, where each of p_1,...,p_k is one of the 10 parameters depending only on H or G. We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper.
On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of G gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when H is a bounded-degree tree of constant pathwidth and G is a planar graph of bounded pathwidth.

Dániel Marx and Michal Pilipczuk. Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask). In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 542-553, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{marx_et_al:LIPIcs.STACS.2014.542, author = {Marx, D\'{a}niel and Pilipczuk, Michal}, title = {{Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {542--553}, 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.2014.542}, URN = {urn:nbn:de:0030-drops-44863}, doi = {10.4230/LIPIcs.STACS.2014.542}, annote = {Keywords: parameterized complexity, subgraph isomorphism} }

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

In the Correlation Clustering problem, also known as Cluster Editing, we are given an undirected graph G and a positive integer k; the task is to decide whether G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, that is, by adding or deleting at most k edges. The motivation of the problem stems from various tasks in computational biology (Ben-Dor et al., Journal of Computational Biology 1999) and machine learning (Bansal et al., Machine Learning 2004). Although in general Correlation Clustering is APX-hard (Charikar et al., FOCS 2003), the version of the problem where the number of cliques may not exceed a prescribed constant p admits a PTAS (Giotis and Guruswami, SODA 2006).
We study the parameterized complexity of Correlation Clustering with this restriction on the number of cliques to be created. We give an algorithm that - in time O(2^{O(sqrt{pk})} + n+m) decides whether a graph G on n vertices and m edges can be transformed into a cluster graph with exactly p cliques by changing at most k adjacencies.
We complement these algorithmic findings by the following, surprisingly tight lower bound on the asymptotic behavior of our algorithm. We show that unless the Exponential Time Hypothesis (ETH) fails - for any constant 0 <= sigma <= 1, there is p = Theta(k^sigma) such that there is no algorithm deciding in time 2^{o(sqrt{pk})} n^{O(1)} whether an n-vertex graph G can be transformed into a cluster graph with at most p cliques by changing at most k adjacencies.
Thus, our upper and lower bounds provide an asymptotically tight analysis of the multivariate parameterized complexity of the problem for the whole range of values of p from constant to a linear function of k.

Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Yngve Villanger. Tight bounds for Parameterized Complexity of Cluster Editing. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 32-43, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2013.32, author = {Fomin, Fedor V. and Kratsch, Stefan and Pilipczuk, Marcin and Pilipczuk, Michal and Villanger, Yngve}, title = {{Tight bounds for Parameterized Complexity of Cluster Editing}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {32--43}, 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.2013.32}, URN = {urn:nbn:de:0030-drops-39209}, doi = {10.4230/LIPIcs.STACS.2013.32}, annote = {Keywords: parameterized complexity, cluster editing, correlation clustering, subexponential algorithms, tight bounds} }

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

The notions of cutwidth and pathwidth of digraphs play a central role in the containment theory for tournaments, or more generally semi-complete digraphs, developed in a recent series of papers by Chudnovsky, Fradkin, Kim, Scott, and Seymour (Maria Chudnovsky, Alexandra Fradkin, and Paul Seymour, 2012; Maria Chudnovsky, Alex Scott, and Paul Seymour, 2011; Maria Chudnovsky and Paul D. Seymour, 2011; Alexandra Fradkin and Paul Seymour, 2010; Alexandra Fradkin and Paul Seymour, 2011; Ilhee Kim and Paul Seymour, 2012). In this work we introduce a new approach to computing these width measures on semi-complete digraphs, via degree orderings. Using the new technique we are able to reprove the main results of (Maria Chudnovsky, Alexandra Fradkin, and Paul Seymour, 2012; Alexandra Fradkin and Paul Seymour, 2011) in a unified and significantly simplified way, as well as obtain new results. First, we present polynomial-time approximation algorithms for both cutwidth and pathwidth, faster and simpler than the previously known ones; the most significant improvement is in case of pathwidth, where instead of previously known O(OPT)-approximation in fixed-parameter tractable time (Fedor V. Fomin and Michal Pilipczuk, 2013) we obtain a constant-factor approximation in polynomial time. Secondly, by exploiting the new set of obstacles for cutwidth and pathwidth, we show that topological containment and immersion in semi-complete digraphs can be tested in single-exponential fixed-parameter tractable time. Finally, we present how the new approach can be used to obtain exact fixed-parameter tractable algorithms for cutwidth and pathwidth, with single-exponential running time dependency on the optimal width.

Michal Pilipczuk. Computing cutwidth and pathwidth of semi-complete digraphs via degree orderings. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 197-208, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{pilipczuk:LIPIcs.STACS.2013.197, author = {Pilipczuk, Michal}, title = {{Computing cutwidth and pathwidth of semi-complete digraphs via degree orderings}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {197--208}, 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.2013.197}, URN = {urn:nbn:de:0030-drops-39340}, doi = {10.4230/LIPIcs.STACS.2013.197}, annote = {Keywords: semi-complete digraph, tournament, pathwidth, cutwidth} }

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)

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@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} }

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