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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We study the computational complexity of several polynomial-time-solvable graph problems parameterized by vertex integrity, a measure of a graph’s vulnerability to vertex removal in terms of connectivity. Vertex integrity is the smallest number ι such that there is a set S of ι' ≤ ι vertices such that every connected component of G-S contains at most ι-ι' vertices. It is known that the vertex integrity lies between the well-studied parameters vertex cover number and tree-depth. Our work follows similar studies for vertex cover number [Alon and Yuster, ESA 2007] and tree-depth [Iwata, Ogasawara, and Ohsaka, STACS 2018].
Alon and Yuster designed algorithms for graphs with small vertex cover number using fast matrix multiplications. We demonstrate that fast matrix multiplication can also be effectively used when parameterizing by vertex integrity ι by developing efficient algorithms for problems including an O(ι^{ω-1}n)-time algorithm for Maximum Matching and an O(ι^{(ω-1)/2}n²) ⊆ O(ι^{0.687} n²)-time algorithm for All-Pairs Shortest Paths. These algorithms can be faster than previous algorithms parameterized by tree-depth, for which fast matrix multiplication is not known to be effective.

Matthias Bentert, Klaus Heeger, and Tomohiro Koana. Fully Polynomial-Time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bentert_et_al:LIPIcs.ESA.2023.16, author = {Bentert, Matthias and Heeger, Klaus and Koana, Tomohiro}, title = {{Fully Polynomial-Time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.16}, URN = {urn:nbn:de:0030-drops-186692}, doi = {10.4230/LIPIcs.ESA.2023.16}, annote = {Keywords: FPT in P, Algebraic Algorithms, Adaptive Algorithms, Subgraph Detection, Matching, APSP} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

For solving NP-hard problems there is often a huge gap between theoretical guarantees and observed running times on real-world instances. As a first step towards tackling this issue, we propose an approach to quantify the correlation between theoretical and observed running times.
We use two NP-hard problems related to finding large "cliquish" subgraphs in a given graph as demonstration of this measure. More precisely, we focus on finding maximum s-clubs and s-plexes, i. e., graphs of diameter s and graphs where each vertex is adjacent to all but s vertices. Preprocessing based on Turing kernelization is a standard tool to tackle these problems, especially on sparse graphs. We provide a parameterized analysis for the Turing kernelization and demonstrate their usefulness in practice. Moreover, we demonstrate that our measure indeed captures the correlation between these new theoretical and the observed running times.

Aleksander Figiel, Tomohiro Koana, André Nichterlein, and Niklas Wünsche. Correlating Theory and Practice in Finding Clubs and Plexes. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{figiel_et_al:LIPIcs.ESA.2023.47, author = {Figiel, Aleksander and Koana, Tomohiro and Nichterlein, Andr\'{e} and W\"{u}nsche, Niklas}, title = {{Correlating Theory and Practice in Finding Clubs and Plexes}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {47:1--47:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.47}, URN = {urn:nbn:de:0030-drops-187000}, doi = {10.4230/LIPIcs.ESA.2023.47}, annote = {Keywords: Preprocessing, Turing kernelization, Pearson correlation coefficient} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We study the α-Fixed Cardinality Graph Partitioning (α-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0 ≤ α ≤ 1, the question is whether there is a set S ⊆ V of size k with a specified coverage function cov_α(S) at least p (or at most p for the minimization version). The coverage function cov_α(⋅) counts edges with exactly one endpoint in S with weight α and edges with both endpoints in S with weight 1 - α. α-FCGP generalizes a number of fundamental graph problems such as Densest k-Subgraph, Max k-Vertex Cover, and Max (k,n-k)-Cut.
A natural question in the study of α-FCGP is whether the algorithmic results known for its special cases, like Max k-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Max k-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greed vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for α > 0 and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with α > 1/3 and minimization with α < 1/3.

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Tomohiro Koana. FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 46:1-46:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fomin_et_al:LIPIcs.MFCS.2023.46, author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Koana, Tomohiro}, title = {{FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {46:1--46:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.46}, URN = {urn:nbn:de:0030-drops-185806}, doi = {10.4230/LIPIcs.MFCS.2023.46}, annote = {Keywords: Partial Vertex Cover, Approximation Algorithms, Max Cut} }

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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

In this work, we study the Induced Matching problem: Given an undirected graph G and an integer 𝓁, is there an induced matching M of size at least 𝓁? An edge subset M is an induced matching in G if M is a matching such that there is no edge between two distinct edges of M. Our work looks into the parameterized complexity of Induced Matching with respect to "below guarantee" parameterizations. We consider the parameterization u - 𝓁 for an upper bound u on the size of any induced matching. For instance, any induced matching is of size at most n/2 where n is the number of vertices, which gives us a parameter n/2 - 𝓁. In fact, there is a straightforward 9^{n/2 - 𝓁} ⋅ n^O(1)-time algorithm for Induced Matching [Moser and Thilikos, J. Discrete Algorithms]. Motivated by this, we ask: Is Induced Matching FPT for a parameter smaller than n/2 - 𝓁? In search for such parameters, we consider MM(G) - 𝓁 and IS(G) - 𝓁, where MM(G) is the maximum matching size and IS(G) is the maximum independent set size of G. We find that Induced Matching is presumably not FPT when parameterized by MM(G) - 𝓁 or IS(G) - 𝓁. In contrast to these intractability results, we find that taking the average of the two helps - our main result is a branching algorithm that solves Induced Matching in 49^{(MM(G) + IS(G))/ 2 - 𝓁} ⋅ n^O(1) time. Our algorithm makes use of the Gallai-Edmonds decomposition to find a structure to branch on.

Tomohiro Koana. Induced Matching Below Guarantees: Average Paves the Way for Fixed-Parameter Tractability. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{koana:LIPIcs.STACS.2023.39, author = {Koana, Tomohiro}, title = {{Induced Matching Below Guarantees: Average Paves the Way for Fixed-Parameter Tractability}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {39:1--39:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.39}, URN = {urn:nbn:de:0030-drops-176914}, doi = {10.4230/LIPIcs.STACS.2023.39}, annote = {Keywords: Parameterized Complexity, Below Guarantees, Induced Matching, Gallai-Edmonds Decomposition} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem.

Leon Kellerhals, Tomohiro Koana, and Pascal Kunz. Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kellerhals_et_al:LIPIcs.IPEC.2022.19, author = {Kellerhals, Leon and Koana, Tomohiro and Kunz, Pascal}, title = {{Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.19}, URN = {urn:nbn:de:0030-drops-173758}, doi = {10.4230/LIPIcs.IPEC.2022.19}, annote = {Keywords: parameterized complexity, vertex cover, feedback vertex set, above guarantee parameterization} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We analyze the (parameterized) computational complexity of "fair" variants of bipartite many-to-one matching, where each vertex from the "left" side is matched to exactly one vertex and each vertex from the "right" side may be matched to multiple vertices. We want to find a "fair" matching, in which each vertex from the right side is matched to a "fair" set of vertices. Assuming that each vertex from the left side has one color modeling its attribute, we study two fairness criteria. In one of them, we deem a vertex set fair if for any two colors, the difference between the numbers of their occurrences does not exceed a given threshold. Fairness is relevant when finding many-to-one matchings between students and colleges, voters and constituencies, and applicants and firms. Here colors may model sociodemographic attributes, party memberships, and qualifications, respectively.
We show that finding a fair many-to-one matching is NP-hard even for three colors and maximum degree five. Our main contribution is the design of fixed-parameter tractable algorithms with respect to the number of vertices on the right side. Our algorithms make use of a variety of techniques including color coding. At the core lie integer linear programs encoding Hall like conditions. To establish the correctness of our integer programs, we prove a new separation result, inspired by Frank’s separation theorem [Frank, Discrete Math. 1982], which may also be of independent interest. We further obtain complete complexity dichotomies regarding the number of colors and the maximum degree of each side.

Niclas Boehmer and Tomohiro Koana. The Complexity of Finding Fair Many-To-One Matchings. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{boehmer_et_al:LIPIcs.ICALP.2022.27, author = {Boehmer, Niclas and Koana, Tomohiro}, title = {{The Complexity of Finding Fair Many-To-One Matchings}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {27:1--27:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.27}, URN = {urn:nbn:de:0030-drops-163680}, doi = {10.4230/LIPIcs.ICALP.2022.27}, annote = {Keywords: Graph theory, polynomial-time algorithms, NP-hardness, FPT, ILP, color coding, submodular and supermodular functions, algorithmic fairness} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed α between zero and one we are given a graph and two numbers k ∈ ℕ and t ∈ ℚ. The task is to find a vertex subset S of exactly k vertices that has value at least (resp. at most for minimization) t. Here, the value of a vertex set computes as α times the number of edges with exactly one endpoint in S plus 1-α times the number of edges with both endpoints in S. These two problems generalize many prominent graph problems, such as Densest k-Subgraph, Sparsest k-Subgraph, Partial Vertex Cover, and Max (k,n-k)-Cut.
In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max (k,n-k)-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms.

Tomohiro Koana, Christian Komusiewicz, André Nichterlein, and Frank Sommer. Covering Many (Or Few) Edges with k Vertices in Sparse Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{koana_et_al:LIPIcs.STACS.2022.42, author = {Koana, Tomohiro and Komusiewicz, Christian and Nichterlein, Andr\'{e} and Sommer, Frank}, title = {{Covering Many (Or Few) Edges with k Vertices in Sparse Graphs}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {42:1--42:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.42}, URN = {urn:nbn:de:0030-drops-158525}, doi = {10.4230/LIPIcs.STACS.2022.42}, annote = {Keywords: Parameterized Complexity, Kernelization, Partial Vertex Cover, Densest k-Subgraph, Max (k,n-k)-Cut, Degeneracy} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number c and weak closure number γ [Fox et al., SICOMP 2020] in addition to the standard parameter solution size k. The weak closure number γ of a graph is upper-bounded by the minimum of its closure number c and its degeneracy d. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching we obtain the first kernels of size k^𝒪(γ), k^𝒪(γ), and (γk)^𝒪(γ), respectively. This extends previous results on the kernelization of these problems on degenerate graphs. These kernels are essentially tight as these problems are unlikely to admit kernels of size k^o(γ) by previous results on their kernelization complexity in degenerate graphs [Cygan et al., ACM TALG 2017]. For Capacitated Vertex Cover, we show that even a kernel of size k^o(c) is unlikely. In contrast, for Connected Vertex Cover, we obtain a problem kernel with 𝒪(ck²) vertices. Moreover, we prove that searching for an induced subgraph of order at least k belonging to a hereditary graph class 𝒢 admits a kernel of size k^𝒪(γ) when 𝒢 contains all complete and all edgeless graphs. Finally, we provide lower bounds for the kernelization of Independent Set on graphs with constant closure number c and kernels for Dominating Set on weakly closed split graphs and weakly closed bipartite graphs.

Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Essentially Tight Kernels For (Weakly) Closed Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{koana_et_al:LIPIcs.ISAAC.2021.35, author = {Koana, Tomohiro and Komusiewicz, Christian and Sommer, Frank}, title = {{Essentially Tight Kernels For (Weakly) Closed Graphs}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {35:1--35:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.35}, URN = {urn:nbn:de:0030-drops-154681}, doi = {10.4230/LIPIcs.ISAAC.2021.35}, annote = {Keywords: Fixed-parameter tractability, kernelization, c-closure, weak \gamma-closure, Independent Set, Induced Matching, Connected Vertex Cover, Ramsey numbers, Dominating Set} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

The Parameterized Algorithms and Computational Experiments challenge (PACE) 2021 was devoted to engineer algorithms solving the NP-hard Cluster Editing problem, also known as Correlation Clustering: Given an undirected graph the task is to compute a minimum number of edges to insert or remove in a way that the resulting graph is a cluster graph, that is, a graph in which each connected component is a clique.
Altogether 67 participants from 21 teams, 11 countries, and 3 continents submitted their implementations to the competition. In this report, we describe the setup of the challenge, the selection of benchmark instances, and the ranking of the participating teams. We also briefly discuss the approaches used in the submitted solvers.

Leon Kellerhals, Tomohiro Koana, André Nichterlein, and Philipp Zschoche. The PACE 2021 Parameterized Algorithms and Computational Experiments Challenge: Cluster Editing. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kellerhals_et_al:LIPIcs.IPEC.2021.26, author = {Kellerhals, Leon and Koana, Tomohiro and Nichterlein, Andr\'{e} and Zschoche, Philipp}, title = {{The PACE 2021 Parameterized Algorithms and Computational Experiments Challenge: Cluster Editing}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {26:1--26:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.26}, URN = {urn:nbn:de:0030-drops-154096}, doi = {10.4230/LIPIcs.IPEC.2021.26}, annote = {Keywords: Correlation Clustering, Cluster Editing, Algorithm Engineering, FPT, Kernelization, Heuristics} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

We thoroughly study a novel but basic combinatorial matrix completion problem: Given a binary incomplete matrix, fill in the missing entries so that the resulting matrix has a specified maximum diameter (that is, upper-bounding the maximum Hamming distance between any two rows of the completed matrix) as well as a specified minimum Hamming distance between any two of the matrix rows. This scenario is closely related to consensus string problems as well as to recently studied clustering problems on incomplete data.
We obtain an almost complete picture concerning the complexity landscape (P vs NP) regarding the diameter constraints and regarding the number of missing entries per row of the incomplete matrix. We develop polynomial-time algorithms for maximum diameter three, which are based on Deza’s theorem [Discret. Math. 1973, J. Comb. Theory, Ser. B 1974] from extremal set theory. In this way, we also provide one of the rare links between sunflower techniques and stringology. On the negative side, we prove NP-hardness for diameter at least four. For the number of missing entries per row, we show polynomial-time solvability when there is only one missing entry and NP-hardness when there can be at least two missing entries. In general, our algorithms heavily rely on Deza’s theorem and the correspondingly identified sunflower structures pave the way towards solutions based on computing graph factors and solving 2-SAT instances.

Tomohiro Koana, Vincent Froese, and Rolf Niedermeier. Binary Matrix Completion Under Diameter Constraints. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{koana_et_al:LIPIcs.STACS.2021.47, author = {Koana, Tomohiro and Froese, Vincent and Niedermeier, Rolf}, title = {{Binary Matrix Completion Under Diameter Constraints}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {47:1--47:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.47}, URN = {urn:nbn:de:0030-drops-136925}, doi = {10.4230/LIPIcs.STACS.2021.47}, annote = {Keywords: sunflowers, binary matrices, Hamming distance, stringology, consensus problems, complexity dichotomy, combinatorial algorithms, graph factors, 2-Sat, Hamming radius} }

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

A vertex set S of a graph G is geodetic if every vertex of G lies on a shortest path between two vertices in S. Given a graph G and k ∈ ℕ, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size at most k. Complementing various works on Geodetic Set restricted to special graph classes, we initiate a parameterized complexity study of Geodetic Set and show, on the negative side, that Geodetic Set is W[1]-hard when parameterized by feedback vertex number, path-width, and solution size, combined. On the positive side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph.

Leon Kellerhals and Tomohiro Koana. Parameterized Complexity of Geodetic Set. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kellerhals_et_al:LIPIcs.IPEC.2020.20, author = {Kellerhals, Leon and Koana, Tomohiro}, title = {{Parameterized Complexity of Geodetic Set}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.20}, URN = {urn:nbn:de:0030-drops-133237}, doi = {10.4230/LIPIcs.IPEC.2020.20}, annote = {Keywords: NP-hard graph problems, Shortest paths, Tree-likeness, Parameter hierarchy, Data reduction, Integer linear programming} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

A graph G is weakly γ-closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that |N(u)∩ N(v)| < γ. The weak closure γ(G) of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that G is weakly γ-closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and s-plexes, are fixed-parameter tractable with respect to γ(G). Moreover, we show that the problem of determining whether a weakly γ-closed graph G has a subgraph on at least k vertices that belongs to a graph class 𝒢 which is closed under taking subgraphs admits a kernel with at most γ k² vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by γ+k where k is the solution size.

Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Computing Dense and Sparse Subgraphs of Weakly Closed Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{koana_et_al:LIPIcs.ISAAC.2020.20, author = {Koana, Tomohiro and Komusiewicz, Christian and Sommer, Frank}, title = {{Computing Dense and Sparse Subgraphs of Weakly Closed Graphs}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {20:1--20:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.20}, URN = {urn:nbn:de:0030-drops-133646}, doi = {10.4230/LIPIcs.ISAAC.2020.20}, annote = {Keywords: Fixed-parameter tractability, c-closure, degeneracy, clique relaxations, bicliques, dominating set} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number c such that G is c-closed. Fox et al. [SIAM J. Comput. '20] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^𝒪(c), that Induced Matching admits a kernel with 𝒪(c⁷ k⁸) vertices, and that Irredundant Set admits a kernel with 𝒪(c^{5/2} k³) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show.

Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Exploiting c-Closure in Kernelization Algorithms for Graph Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{koana_et_al:LIPIcs.ESA.2020.65, author = {Koana, Tomohiro and Komusiewicz, Christian and Sommer, Frank}, title = {{Exploiting c-Closure in Kernelization Algorithms for Graph Problems}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {65:1--65:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.65}, URN = {urn:nbn:de:0030-drops-129316}, doi = {10.4230/LIPIcs.ESA.2020.65}, annote = {Keywords: Fixed-parameter tractability, kernelization, c-closure, Dominating Set, Induced Matching, Irredundant Set, Ramsey numbers} }

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**Published in:** LIPIcs, Volume 161, 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)

Considering matrices with missing entries, we study NP-hard matrix completion problems where the resulting completed matrix should have limited (local) radius. In the pure radius version, this means that the goal is to fill in the entries such that there exists a "center string" which has Hamming distance to all matrix rows as small as possible. In stringology, this problem is also known as Closest String with Wildcards. In the local radius version, the requested center string must be one of the rows of the completed matrix.
Hermelin and Rozenberg [CPM 2014, TCS 2016] performed a parameterized complexity analysis for Closest String with Wildcards. We answer one of their open questions, fix a bug concerning a fixed-parameter tractability result in their work, and improve some running time upper bounds. For the local radius case, we reveal a computational complexity dichotomy. In general, our results indicate that, although being NP-hard as well, this variant often allows for faster (fixed-parameter) algorithms.

Tomohiro Koana, Vincent Froese, and Rolf Niedermeier. Parameterized Algorithms for Matrix Completion with Radius Constraints. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{koana_et_al:LIPIcs.CPM.2020.20, author = {Koana, Tomohiro and Froese, Vincent and Niedermeier, Rolf}, title = {{Parameterized Algorithms for Matrix Completion with Radius Constraints}}, booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-149-8}, ISSN = {1868-8969}, year = {2020}, volume = {161}, editor = {G{\o}rtz, Inge Li and Weimann, Oren}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2020.20}, URN = {urn:nbn:de:0030-drops-121456}, doi = {10.4230/LIPIcs.CPM.2020.20}, annote = {Keywords: fixed-parameter tractability, consensus string problems, Closest String, Closest String with Wildcards} }

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