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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Cluster Editing, also known as correlation clustering, is a well-studied graph modification problem. In this problem, one is given a graph and allowed to perform up to k edge additions and deletions to transform it into a cluster graph, i.e., a graph consisting of a disjoint union of cliques. However, in real-world networks, clusters are often overlapping. For example, in social networks, a person might belong to several communities - e.g. those corresponding to work, school, or neighborhood. Another strong motivation comes from language networks where trying to cluster words with similar usage can be confounded by homonyms, that is, words with multiple meanings like "bat". The recently introduced operation of vertex splitting is one natural approach to incorporating such overlap into Cluster Editing. First used in the context of graph drawing, this operation allows a vertex v to be replaced by two vertices whose combined neighborhood is the neighborhood of v (and thus v can belong to more than one cluster). The problem of transforming a graph into a cluster graph using at most k edge additions, edge deletions, or vertex splits is called Cluster Editing with Vertex Splitting and is known to admit a polynomial kernel with respect to k and an O(9^{k²} + n + m)-time (parameterized) algorithm. However, it was not known whether the problem is NP-hard, a question which was originally asked by Abu-Khzam et al. [Combinatorial Optimization, 2018]. We answer this in the affirmative. We further give an improved algorithm running in O(2^{7klog k} + n + m) time.

Emmanuel Arrighi, Matthias Bentert, Pål Grønås Drange, Blair D. Sullivan, and Petra Wolf. Cluster Editing with Overlapping Communities. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arrighi_et_al:LIPIcs.IPEC.2023.2, author = {Arrighi, Emmanuel and Bentert, Matthias and Drange, P\r{a}l Gr{\o}n\r{a}s and Sullivan, Blair D. and Wolf, Petra}, title = {{Cluster Editing with Overlapping Communities}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {2:1--2:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara 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/LIPIcs.IPEC.2023.2}, URN = {urn:nbn:de:0030-drops-194218}, doi = {10.4230/LIPIcs.IPEC.2023.2}, annote = {Keywords: graph modification, correlation clustering, vertex splitting, NP-hardness, parameterized algorithm} }

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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

We show that the VC-dimension of a graph can be computed in time n^{⌈log d+1⌉} d^{O(d)}, where d is the degeneracy of the input graph. The core idea of our algorithm is a data structure to efficiently query the number of vertices that see a specific subset of vertices inside of a (small) query set. The construction of this data structure takes time O(d2^dn), afterwards queries can be computed efficiently using fast Möbius inversion.
This data structure turns out to be useful for a range of tasks, especially for finding bipartite patterns in degenerate graphs, and we outline an efficient algorithm for counting the number of times specific patterns occur in a graph. The largest factor in the running time of this algorithm is O(n^c), where c is a parameter of the pattern we call its left covering number.
Concrete applications of this algorithm include counting the number of (non-induced) bicliques in linear time, the number of co-matchings in quadratic time, as well as a constant-factor approximation of the ladder index in linear time.
Finally, we supplement our theoretical results with several implementations and run experiments on more than 200 real-world datasets - the largest of which has 8 million edges - where we obtain interesting insights into the VC-dimension of real-world networks.

Pål Grønås Drange, Patrick Greaves, Irene Muzi, and Felix Reidl. Computing Complexity Measures of Degenerate Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{drange_et_al:LIPIcs.IPEC.2023.14, author = {Drange, P\r{a}l Gr{\o}n\r{a}s and Greaves, Patrick and Muzi, Irene and Reidl, Felix}, title = {{Computing Complexity Measures of Degenerate Graphs}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {14:1--14:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara 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/LIPIcs.IPEC.2023.14}, URN = {urn:nbn:de:0030-drops-194333}, doi = {10.4230/LIPIcs.IPEC.2023.14}, annote = {Keywords: vc-dimension, datastructure, degeneracy, enumerating} }

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PACE Solver Description

**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

The graph parameter twin-width was recently introduced by Bonnet et al. Twin-width is a parameter that measures a graph’s similarity to a cograph, which is a graph that can be reduced to a single vertex by repeatedly contracting twins. This brief description introduces Zygosity, a heuristic for computing a low-width contraction sequence that achieved second place in the 2023 edition of Parameterized Algorithms and Computational Experiments Challenge (PACE). Zygosity starts by repeatedly contracting twins. Then, any attached trees are contracted down to a single pendant vertex. The remaining graph is then contracted using a randomized greedy algorithm.

Emmanuel Arrighi, Pål Grønås Drange, Kenneth Langedal, Farhad Vadiee, Martin Vatshelle, and Petra Wolf. PACE Solver Description: Zygosity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 39:1-39:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arrighi_et_al:LIPIcs.IPEC.2023.39, author = {Arrighi, Emmanuel and Drange, P\r{a}l Gr{\o}n\r{a}s and Langedal, Kenneth and Vadiee, Farhad and Vatshelle, Martin and Wolf, Petra}, title = {{PACE Solver Description: Zygosity}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {39:1--39:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara 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/LIPIcs.IPEC.2023.39}, URN = {urn:nbn:de:0030-drops-194583}, doi = {10.4230/LIPIcs.IPEC.2023.39}, annote = {Keywords: Twin-width, randomized greedy algorithm} }

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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 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

We study two clustering problems, Starforest Editing, the problem of adding and deleting edges to obtain a disjoint union of stars, and the generalization Bicluster Editing. We show that, in addition to being NP-hard, none of the problems can be solved in subexponential time unless the exponential time hypothesis fails.
Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the number of connected components in the solution should not make the problem easier: In particular, they argue that the subexponential time algorithm for editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J. Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p is a secondary parameter, bounding the number of components in the solution.
However, upon bounding the number of stars or bicliques in the solution, we obtain algorithms which run in time O(2^{3*sqrt(pk)} + n + m) for p-Starforest Editing and O(2^{O(p * sqrt(k) * log(pk))} + n + m) for p-Bicluster Editing. We obtain a similar result for the more general case of t-Partite p-Cluster Editing. This is subexponential in k for a fixed number of clusters, since p is then considered a constant.
Our results even out the number of multivariate subexponential time algorithms and give reasons to believe that this area warrants further study.

Pål Grønås Drange, Felix Reidl, Fernando Sánchez Villaamil, and Somnath Sikdar. Fast Biclustering by Dual Parameterization. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 402-413, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{drange_et_al:LIPIcs.IPEC.2015.402, author = {Drange, P\r{a}l Gr{\o}n\r{a}s and Reidl, Felix and S\'{a}nchez Villaamil, Fernando and Sikdar, Somnath}, title = {{Fast Biclustering by Dual Parameterization}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {402--413}, 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.2015.402}, URN = {urn:nbn:de:0030-drops-56004}, doi = {10.4230/LIPIcs.IPEC.2015.402}, annote = {Keywords: graph editing, subexponential algorithms, parameterized complexity} }

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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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