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

We initiate a study of the complexity of MSM-Median, the problem of computing a median of a set of k real-valued time series under the move-split-merge distance. This distance measure is based on three operations: moves, which may shift a data point in a time series; splits, which replace one data point in a time series by two consecutive data points of the same value; and merges, which replace two consecutive data points of equal value by a single data point of the same value. The cost of a move operation is the difference of the data point value before and after the operation, the cost of split and merge operations is defined via a given constant c.
Our main results are as follows. First, we show that MSM-Median is NP-hard and W[1]-hard with respect to k for time series with at most three distinct values. Under the Exponential Time Hypothesis (ETH) our reduction implies that a previous dynamic programming algorithm with running time |I|^𝒪(k) [Holznigenkemper et al., Data Min. Knowl. Discov. '23] is essentially optimal. Here, |I| denotes the total input size. Second, we show that MSM-Median can be solved in 2^𝒪(d/c)⋅|I|^𝒪(1) time where d is the total distance of the median to the input time series.

Jana Holznigenkemper, Christian Komusiewicz, Nils Morawietz, and Bernhard Seeger. On the Complexity of Computing Time Series Medians Under the Move-Split-Merge Metric. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 54:1-54:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{holznigenkemper_et_al:LIPIcs.MFCS.2023.54, author = {Holznigenkemper, Jana and Komusiewicz, Christian and Morawietz, Nils and Seeger, Bernhard}, title = {{On the Complexity of Computing Time Series Medians Under the Move-Split-Merge Metric}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {54:1--54:15}, 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.2023.54}, URN = {urn:nbn:de:0030-drops-185889}, doi = {10.4230/LIPIcs.MFCS.2023.54}, annote = {Keywords: Parameterized Complexity, Median String, Time Series, ETH} }

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**Published in:** LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

We present a new simple graph-theoretic formulation of the exploratory blockmodeling problem on undirected and unweighted one-mode networks. Our formulation takes as input the network G and the maximum number t of blocks for the solution model. The task is to find a minimum-size set of edge insertions and deletions that transform the input graph G into a graph G' with at most t neighborhood classes. Herein, a neighborhood class is a maximal set of vertices with the same neighborhood. The neighborhood classes of G' directly give the blocks and block interactions of the computed blockmodel.
We analyze the classic and parameterized complexity of the exploratory blockmodeling problem, provide a branch-and-bound algorithm, an ILP formulation and several heuristics. Finally, we compare our exact algorithms to previous ILP-based approaches and show that the new algorithms are faster for t ≥ 4.

Alexander Bille, Niels Grüttemeier, Christian Komusiewicz, and Nils Morawietz. A Graph-Theoretic Formulation of Exploratory Blockmodeling. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bille_et_al:LIPIcs.SEA.2023.14, author = {Bille, Alexander and Gr\"{u}ttemeier, Niels and Komusiewicz, Christian and Morawietz, Nils}, title = {{A Graph-Theoretic Formulation of Exploratory Blockmodeling}}, booktitle = {21st International Symposium on Experimental Algorithms (SEA 2023)}, pages = {14:1--14:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-279-2}, ISSN = {1868-8969}, year = {2023}, volume = {265}, editor = {Georgiadis, Loukas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.14}, URN = {urn:nbn:de:0030-drops-183648}, doi = {10.4230/LIPIcs.SEA.2023.14}, annote = {Keywords: Clustering, Exact Algorithms, ILP-Formulation, Branch-and-Bound, Social Networks} }

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**Published in:** LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

Maximum Parsimony is the problem of computing a most parsimonious phylogenetic tree for a taxa set X from character data for X. A common strategy to attack this notoriously hard problem is to perform a local search over the phylogenetic tree space. Here, one is given a phylogenetic tree T and wants to find a more parsimonious tree in the neighborhood of T. We study the complexity of this problem when the neighborhood contains all trees within distance k for several classic distance functions. For the nearest neighbor interchange (NNI), subtree prune and regraft (SPR), tree bisection and reconnection (TBR), and edge contraction and refinement (ECR) distances, we show that, under the exponential time hypothesis, there are no algorithms with running time |I|^o(k) where |I| is the total input size. Hence, brute-force algorithms with running time |X|^𝒪(k) ⋅ |I| are essentially optimal.
In contrast to the above distances, we observe that for the sECR-distance, where the contracted edges are constrained to form a subtree, a better solution within distance k can be found in k^𝒪(k) ⋅ |I|^𝒪(1) time.

Christian Komusiewicz, Simone Linz, Nils Morawietz, and Jannik Schestag. On the Complexity of Parameterized Local Search for the Maximum Parsimony Problem. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{komusiewicz_et_al:LIPIcs.CPM.2023.18, author = {Komusiewicz, Christian and Linz, Simone and Morawietz, Nils and Schestag, Jannik}, title = {{On the Complexity of Parameterized Local Search for the Maximum Parsimony Problem}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {18:1--18:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-276-1}, ISSN = {1868-8969}, year = {2023}, volume = {259}, editor = {Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.18}, URN = {urn:nbn:de:0030-drops-179729}, doi = {10.4230/LIPIcs.CPM.2023.18}, annote = {Keywords: phylogenetic trees, parameterized complexity, tree distances, NNI, TBR} }

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

A k-swap W for a vertex cover S of a graph G is a vertex set of size at most k such that S' = (S ⧵ W) ∪ (W ⧵ S), the symmetric difference of S and W, is a vertex cover of G. If |S'| < |S|, then W is improving. In LS-Vertex Cover, one is given a vertex cover S of a graph G and wants to know if there is an improving k-swap for S in G. In applications of LS-Vertex Cover, k is a very small parameter that can be set by a user to determine the trade-off between running time and solution quality. Consequently, k can be considered to be a constant. Motivated by this and the fact that LS-Vertex Cover is W[1]-hard with respect to k, we aim for algorithms with running time 𝓁^f(k) ⋅ n^𝒪(1) where 𝓁 is a structural graph parameter upper-bounded by n. We say that such a running time grows mildly with respect to 𝓁 and strongly with respect to k. We obtain algorithms with such a running time for 𝓁 being the h-index of G, the treewidth of G, or the modular-width of G. In addition, we consider a novel parameter, the maximum degree over all quotient graphs in a modular decomposition of G. Moreover, we adapt these algorithms to the more general problem where each vertex is assigned a weight and where we want to find a d-improving k-swap, that is, a k-swap which decreases the weight of the vertex cover by at least d.

Christian Komusiewicz and Nils Morawietz. Parameterized Local Search for Vertex Cover: When Only the Search Radius Is Crucial. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{komusiewicz_et_al:LIPIcs.IPEC.2022.20, author = {Komusiewicz, Christian and Morawietz, Nils}, title = {{Parameterized Local Search for Vertex Cover: When Only the Search Radius Is Crucial}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {20:1--20:18}, 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.2022.20}, URN = {urn:nbn:de:0030-drops-173764}, doi = {10.4230/LIPIcs.IPEC.2022.20}, annote = {Keywords: Local Search, Structural parameterization, Fixed-parameter tractability} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

For PLS-complete local search problems, there is presumably no polynomial-time algorithm which finds a locally optimal solution, even though determining whether a solution is locally optimal and replacing it by a better one if this is not the case can be done in polynomial time.
We study local search for Weighted Independent Set and Weighted Dominating Set with the 3-swap neighborhood. The 3-swap neighborhood of a vertex set S in G is the set of vertex sets which can be obtained from S by exchanging at most three vertices. We prove the following dichotomy: On the negative side, the problem of finding a 3-swap-optimal independent set or dominating set is PLS-complete. On the positive side, locally optimal independent sets or dominating sets can be found in polynomial time when allowing all 3-swaps except a) the swaps that remove two vertices from the current solution and add one vertex to the solution or b) the swaps that remove one vertex from the current solution and add two vertices to the solution.

Christian Komusiewicz and Nils Morawietz. Finding 3-Swap-Optimal Independent Sets and Dominating Sets Is Hard. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{komusiewicz_et_al:LIPIcs.MFCS.2022.66, author = {Komusiewicz, Christian and Morawietz, Nils}, title = {{Finding 3-Swap-Optimal Independent Sets and Dominating Sets Is Hard}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {66:1--66:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.66}, URN = {urn:nbn:de:0030-drops-168644}, doi = {10.4230/LIPIcs.MFCS.2022.66}, annote = {Keywords: Local Search, Graph problems, 3-swap neighborhood, PLS-completeness} }

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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 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.

Petr A. Golovach, Christian Komusiewicz, Dieter Kratsch, and Van Bang Le. Refined Notions of Parameterized Enumeration Kernels with Applications to Matching Cut Enumeration. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{golovach_et_al:LIPIcs.STACS.2021.37, author = {Golovach, Petr A. and Komusiewicz, Christian and Kratsch, Dieter and Le, Van Bang}, title = {{Refined Notions of Parameterized Enumeration Kernels with Applications to Matching Cut Enumeration}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {37:1--37:18}, 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.2021.37}, URN = {urn:nbn:de:0030-drops-136823}, doi = {10.4230/LIPIcs.STACS.2021.37}, annote = {Keywords: enumeration problems, polynomial delay, output-sensitive algorithms, kernelization, structural parameterizations, matching cuts} }

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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 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

In a graph G = (V,E) with an edge coloring 𝓁:E → C and two distinguished vertices s and t, a colored (s,t)-cut is a set C̃ ⊆ C such that deleting all edges with some color c ∈ C̃ from G disconnects s and t. Motivated by applications in the design of robust networks, we introduce a family of problems called colored cut games. In these games, an attacker and a defender choose colors to delete and to protect, respectively, in an alternating fashion. It is the goal of the attacker to achieve a colored (s,t)-cut and the goal of the defender to prevent this. First, we show that for an unbounded number of alternations, colored cut games are PSPACE-complete. We then show that, even on subcubic graphs, colored cut games with a constant number i of alternations are complete for classes in the polynomial hierarchy whose level depends on i. To complete the dichotomy, we show that all colored cut games are polynomial-time solvable on graphs with degree at most two. Finally, we show that all colored cut games admit a polynomial kernel for the parameter k+κ_r where k denotes the total attacker budget and, for any constant r, κ_r is the number of vertex deletions that are necessary to transform G into a graph where the longest path has length at most r. In the case of r = 1, κ₁ is the vertex cover number vc of the input graph and we obtain a kernel with 𝒪(vc²k²) edges. Moreover, we introduce an algorithm solving the most basic colored cut game, Colored (s,t)-Cut, in 2^{vc + k}n^{𝒪(1)} time.

Nils Morawietz, Niels Grüttemeier, Christian Komusiewicz, and Frank Sommer. Colored Cut Games. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{morawietz_et_al:LIPIcs.FSTTCS.2020.30, author = {Morawietz, Nils and Gr\"{u}ttemeier, Niels and Komusiewicz, Christian and Sommer, Frank}, title = {{Colored Cut Games}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {30:1--30:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.30}, URN = {urn:nbn:de:0030-drops-132719}, doi = {10.4230/LIPIcs.FSTTCS.2020.30}, annote = {Keywords: Labeled Cut, Labeled Path, Network Robustness, Kernelization, PSPACE, Polynomial Hierarchy} }

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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 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Given an undirected graph G and integers c and k, the Maximum Edge-Colorable Subgraph problem asks whether we can delete at most k edges in G to obtain a graph that has a proper edge coloring with at most c colors. We show that Maximum Edge-Colorable Subgraph admits, for every fixed c, a linear-size problem kernel when parameterized by the edge deletion distance of G to a graph with maximum degree c-1. This parameterization measures the distance to instances that, due to Vizing’s famous theorem, are trivial yes-instances. For c≤ 4, we also provide a linear-size kernel for the same parameterization for Multi Strong Triadic Closure, a related edge coloring problem with applications in social network analysis. We provide further results for Maximum Edge-Colorable Subgraph parameterized by the vertex deletion distance to graphs where every component has order at most c and for the list-colored versions of both problems.

Niels Grüttemeier, Christian Komusiewicz, and Nils Morawietz. Maximum Edge-Colorable Subgraph and Strong Triadic Closure Parameterized by Distance to Low-Degree Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gruttemeier_et_al:LIPIcs.SWAT.2020.26, author = {Gr\"{u}ttemeier, Niels and Komusiewicz, Christian and Morawietz, Nils}, title = {{Maximum Edge-Colorable Subgraph and Strong Triadic Closure Parameterized by Distance to Low-Degree Graphs}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {26:1--26:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.26}, URN = {urn:nbn:de:0030-drops-122731}, doi = {10.4230/LIPIcs.SWAT.2020.26}, annote = {Keywords: Graph coloring, social networks, parameterized complexity, kernelization} }

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

In the NP-hard Equality-Free String Factorization problem, we are given a string S and ask whether S can be partitioned into k factors that are pairwise distinct. We describe a randomized algorithm for Equality-Free String Factorization with running time 2^k⋅ k^{𝒪(1)}+𝒪(n) improving over previous algorithms with running time k^{𝒪(k)}+𝒪(n) [Schmid, TCS 2016; Mincu and Popa, Proc. SOFSEM 2020]. Our algorithm works for the generalization of Equality-Free String Factorization where equality can be replaced by an arbitrary polynomial-time computable equivalence relation on strings. We also consider two factorization problems to which this algorithm does not apply, namely Prefix-Free String Factorization where we ask for a factorization of size k such that no factor is a prefix of another factor and Substring-Free String Factorization where we ask for a factorization of size k such that no factor is a substring of another factor. We show that these two problems are NP-hard as well. Then, we show that Prefix-Free String Factorization with the prefix-free relation is fixed-parameter tractable with respect to k by providing a polynomial problem kernel. Finally, we show a generic ILP formulation for R-Free String Factorization where R is an arbitrary relation on strings. This formulation improves over a previous one for Equality-Free String Factorization in terms of the number of variables.

Niels Grüttemeier, Christian Komusiewicz, Nils Morawietz, and Frank Sommer. String Factorizations Under Various Collision Constraints. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gruttemeier_et_al:LIPIcs.CPM.2020.17, author = {Gr\"{u}ttemeier, Niels and Komusiewicz, Christian and Morawietz, Nils and Sommer, Frank}, title = {{String Factorizations Under Various Collision Constraints}}, booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)}, pages = {17:1--17:14}, 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.2020.17}, URN = {urn:nbn:de:0030-drops-121428}, doi = {10.4230/LIPIcs.CPM.2020.17}, annote = {Keywords: NP-hard problem, fixed-parameter algorithms, collision-aware string partitioning} }

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

In a graph, a matching cut is an edge cut that is a matching. Matching Cut, which is known to be NP-complete, is the problem of deciding whether or not a given graph G has a matching cut. In this paper we show that Matching Cut admits a quadratic-vertex kernel for the parameter distance to cluster and a linear-vertex kernel for the parameter distance to clique. We further provide an O^*(2^{dc(G)}) time and an O^*(2^{dc^-}(G)}) time FPT algorithm for Matching Cut, where dc(G) and dc^-(G) are the distance to cluster and distance to co-cluster, respectively. We also improve the running time of the best known branching algorithm to solve Matching Cut from O^*(1.4143^n) to O^*(1.3803^n). Moreover, we point out that, unless NP subseteq coNP/poly, Matching Cut does not admit a polynomial kernel when parameterized by treewidth.

Christian Komusiewicz, Dieter Kratsch, and Van Bang Le. Matching Cut: Kernelization, Single-Exponential Time FPT, and Exact Exponential Algorithms. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{komusiewicz_et_al:LIPIcs.IPEC.2018.19, author = {Komusiewicz, Christian and Kratsch, Dieter and Le, Van Bang}, title = {{Matching Cut: Kernelization, Single-Exponential Time FPT, and Exact Exponential Algorithms}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {19:1--19:13}, 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.2018.19}, URN = {urn:nbn:de:0030-drops-102207}, doi = {10.4230/LIPIcs.IPEC.2018.19}, annote = {Keywords: matching cut, decomposable graph, graph algorithm} }

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

A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G[A] and G[B] satisfy properties Pi_A and Pi_B, respectively. This so-called (Pi_A,Pi_B)-Recognition problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (Pi_A,Pi_B)-Recognition, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an "almost correct" bipartition (A',B'), and pushes appropriate vertices from A' to B' and vice versa to eventually arrive at a correct bipartition.
In this paper, we study whether (Pi_A,Pi_B)-Recognition problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where Pi_A is the set of P_3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and Pi_B is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP not subseteq coNP/poly, (Pi_A,Pi_B)-Recognition admits a polynomial kernel if and only if H contains a graph of order at most 2. In both the kernelization and the lower bound results, we make crucial use of the pushing process.

Iyad Kanj, Christian Komusiewicz, Manuel Sorge, and Erik Jan van Leeuwen. Solving Partition Problems Almost Always Requires Pushing Many Vertices Around. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kanj_et_al:LIPIcs.ESA.2018.51, author = {Kanj, Iyad and Komusiewicz, Christian and Sorge, Manuel and van Leeuwen, Erik Jan}, title = {{Solving Partition Problems Almost Always Requires Pushing Many Vertices Around}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {51:1--51:14}, 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.2018.51}, URN = {urn:nbn:de:0030-drops-95140}, doi = {10.4230/LIPIcs.ESA.2018.51}, annote = {Keywords: Fixed-parameter algorithms, Kernelization, Vertex-partition problems, Reduction rules, Cross-composition} }

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**Published in:** LIPIcs, Volume 105, 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)

Let G=(V,A) be a vertex-colored arc-weighted directed acyclic graph (DAG) rooted in some vertex r. The color hierarchy graph H(G) of G is defined as follows: the vertex set of H(G) is the color set C of G, and H(G) has an arc from c to c' if G has an arc from a vertex of color c to a vertex of color c'. We study the Maximum Colorful Arborescence (MCA) problem, which takes as input a DAG G such that H(G) is also a DAG, and aims at finding in G a maximum-weight arborescence rooted in r in which no color appears more than once. The MCA problem models the de novo inference of unknown metabolites by mass spectrometry experiments. Although the problem has been introduced ten years ago (under a different name), it was only recently pointed out that a crucial additional property in the problem definition was missing: by essence, H(G) must be a DAG. In this paper, we further investigate MCA under this new light and provide new algorithmic results for this problem, with a focus on fixed-parameter tractability (FPT) issues for different structural parameters of H(G). In particular, we develop an O^*(3^{{x_H}})-time algorithm for solving MCA, where {x_{H}} is the number of vertices of indegree at least two in H(G), thereby improving the O^*(3^{|C|})-time algorithm of Böcker et al. [Proc. ECCB '08]. We also prove that MCA is W[2]-hard with respect to the treewidth t_H of the underlying undirected graph of H(G), and further show that it is FPT with respect to t_H + l_{C}, where l_{C} := |V| - |C|.

Guillaume Fertin, Julien Fradin, and Christian Komusiewicz. On the Maximum Colorful Arborescence Problem and Color Hierarchy Graph Structure. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fertin_et_al:LIPIcs.CPM.2018.17, author = {Fertin, Guillaume and Fradin, Julien and Komusiewicz, Christian}, title = {{On the Maximum Colorful Arborescence Problem and Color Hierarchy Graph Structure}}, booktitle = {29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-074-3}, ISSN = {1868-8969}, year = {2018}, volume = {105}, editor = {Navarro, Gonzalo and Sankoff, David and Zhu, Binhai}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2018.17}, URN = {urn:nbn:de:0030-drops-86939}, doi = {10.4230/LIPIcs.CPM.2018.17}, annote = {Keywords: Subgraph problem, computational complexity, algorithms, fixed-parameter tractability, kernelization} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

In this article, the Program Committee of the Second Parameterized Algorithms and Computational Experiments challenge (PACE 2017) reports on the second iteration of the PACE challenge. Track A featured the Treewidth problem and Track B the Minimum Fill-In problem. Over 44 participants on 17 teams from 11 countries submitted their implementations to the competition.

Holger Dell, Christian Komusiewicz, Nimrod Talmon, and Mathias Weller. The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 30:1-30:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dell_et_al:LIPIcs.IPEC.2017.30, author = {Dell, Holger and Komusiewicz, Christian and Talmon, Nimrod and Weller, Mathias}, title = {{The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {30:1--30:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.30}, URN = {urn:nbn:de:0030-drops-85582}, doi = {10.4230/LIPIcs.IPEC.2017.30}, annote = {Keywords: treewidth, minimum fill-in, contest, implementation challenge, FPT} }

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

In the Maximum-Duo Preservation String Mapping (Max-Duo PSM) problem, the input consists of two related strings A and B of length n and a nonnegative integer k. The objective is to determine whether there exists a mapping m from the set of positions of A to the set of positions of B that maps only to positions with the same character and preserves at least k duos, which are pairs of adjacent positions. We develop a randomized algorithm that solves Max-Duo PSM in time 4^k * n^{O(1)}, and a deterministic algorithm that solves this problem in time 6.855^k * n^{O(1)}. The previous best known (deterministic) algorithm for this problem has running time (8e)^{2k+o(k)} * n^{O(1)} [Beretta et al., Theor. Comput. Sci. 2016]. We also show that Max-Duo PSM admits a problem kernel of size O(k^3), improving upon the previous best known problem kernel of size O(k^6).

Christian Komusiewicz, Mateus de Oliveira Oliveira, and Meirav Zehavi. Revisiting the Parameterized Complexity of Maximum-Duo Preservation String Mapping. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{komusiewicz_et_al:LIPIcs.CPM.2017.11, author = {Komusiewicz, Christian and de Oliveira Oliveira, Mateus and Zehavi, Meirav}, title = {{Revisiting the Parameterized Complexity of Maximum-Duo Preservation String Mapping}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {11:1--11:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.11}, URN = {urn:nbn:de:0030-drops-73436}, doi = {10.4230/LIPIcs.CPM.2017.11}, annote = {Keywords: comparative genomics, parameterized complexity, kernelization} }

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

In Genomic Scaffold Filling, one aims at polishing in silico a draft genome, called scaffold. The scaffold is given in the form of an ordered set of gene sequences, called contigs. This is done by confronting the scaffold to an already complete reference genome from a close species. More precisely, given a scaffold S, a reference genome G and a score function f() between two genomes, the aim is to complete S by adding the missing genes from G so that the obtained complete genome S* optimizes f(S*, G). In this paper, we extend a model of Jiang et al. [CPM 2016] (i) by allowing the insertions of strings instead of single characters (i.e., some groups of genes may be forced to be inserted together) and (ii) by considering two alternative score functions: the first generalizes the notion of common adjacencies by maximizing the number of common k-mers between S* and G (k-Mer Scaffold Filling), the second aims at minimizing the number of breakpoints between S* and G (Min-Breakpoint Scaffold Filling). We study these problems from the parameterized complexity point of view, providing fixed-parameter (FPT) algorithms for both problems. In particular, we show that k-Mer Scaffold Filling is FPT wrt. parameter l, the number of additional k-mers realized by the completion of S—this answers an open question of Jiang et al. [CPM 2016]. We also show that Min-Breakpoint Scaffold Filling is FPT wrt. a parameter combining the number of missing genes, the number of gene repetitions and the target distance.

Laurent Bulteau, Guillaume Fertin, and Christian Komusiewicz. Beyond Adjacency Maximization: Scaffold Filling for New String Distances. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2017.27, author = {Bulteau, Laurent and Fertin, Guillaume and Komusiewicz, Christian}, title = {{Beyond Adjacency Maximization: Scaffold Filling for New String Distances}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {27:1--27:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.27}, URN = {urn:nbn:de:0030-drops-73364}, doi = {10.4230/LIPIcs.CPM.2017.27}, annote = {Keywords: computational biology, strings, FPT algorithms, kernelization} }

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

In this article, the steering committee of the Parameterized Algorithms and Computational Experiments challenge (PACE) reports on the first iteration of the challenge. Where did PACE come from, how did it go, who won, and what's next?

Holger Dell, Thore Husfeldt, Bart M. P. Jansen, Petteri Kaski, Christian Komusiewicz, and Frances A. Rosamond. The First Parameterized Algorithms and Computational Experiments Challenge. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 30:1-30:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dell_et_al:LIPIcs.IPEC.2016.30, author = {Dell, Holger and Husfeldt, Thore and Jansen, Bart M. P. and Kaski, Petteri and Komusiewicz, Christian and Rosamond, Frances A.}, title = {{The First Parameterized Algorithms and Computational Experiments Challenge}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {30:1--30:9}, 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.2016.30}, URN = {urn:nbn:de:0030-drops-69310}, doi = {10.4230/LIPIcs.IPEC.2016.30}, annote = {Keywords: treewidth, feedback vertex set, contest, implementation challenge, FPT} }

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**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

Let G=(V,E) be a vertex-colored graph, where C is the set of colors used to color V. The Graph Motif (or GM) problem takes as input G, a multiset M of colors built from C, and asks whether there is a subset S subseteq V such that (i) G[S] is connected and (ii) the multiset of colors obtained from S equals M. The Colorful Graph Motif problem (or CGM) is a constrained version of GM in which M=C, and the List-Colored Graph Motif problem (or LGM) is the extension of GM in which each vertex v of V may choose its color from a list L(v) of colors.
We study the three problems GM, CGM and LGM, parameterized by l:=|V|-|M|. In particular, for general graphs, we show that, assuming the strong exponential-time hypothesis, CGM has no (2-epsilon)^l * |V|^{O(1)}-time algorithm, which implies that a previous algorithm, running in O(2^l\cdot |E|) time is optimal. We also prove that LGM is W[1]-hard even if we restrict ourselves to lists of at most two colors. If we constrain the input graph to be a tree, then we show that, in contrast to CGM, GM can be solved in O(4^l *|V|) time but admits no polynomial kernel, while CGM can be solved in O(sqrt{2}^l + |V|) time and admits a polynomial kernel.

Guillaume Fertin and Christian Komusiewicz. Graph Motif Problems Parameterized by Dual. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 7:1-7:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fertin_et_al:LIPIcs.CPM.2016.7, author = {Fertin, Guillaume and Komusiewicz, Christian}, title = {{Graph Motif Problems Parameterized by Dual}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {7:1--7:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.7}, URN = {urn:nbn:de:0030-drops-60837}, doi = {10.4230/LIPIcs.CPM.2016.7}, annote = {Keywords: NP-hard problem, subgraph problem, fixed-parameter algorithm, lowerbounds, kernelization} }

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

We consider the recognition problem for two graph classes that generalize split and unipolar graphs, respectively.
First, we consider the recognizability of graphs that admit a monopolar partition: a partition of the vertex set into sets A,B such that G[A] is a disjoint union of cliques and G[B] an independent set. If in such a partition G[A] is a single clique, then G is a split graph. We show that in
O(2^k * k^3 * (|V(G)| + |E(G)|)) time we can decide whether G admits a monopolar partition
(A,B) where G[A] has at most k cliques. This generalizes the linear-time algorithm for recognizing split graphs corresponding to the case when k=1.
Second, we consider the recognizability of graphs that admit a 2-subcoloring: a partition of the vertex set into sets A,B such that each of G[A] and G[B] is a disjoint union of cliques. If in such a partition G[A] is a single clique, then G is a unipolar graph. We show that in
O(k^(2k+2) * (|V(G)|^2+|V(G)| * |E(G)|)) time we can decide whether G admits a
2-subcoloring (A,B) where G[A] has at most k cliques. This generalizes the polynomial-time algorithm for recognizing unipolar graphs corresponding to the case when k=1.
We also show that in O(4^k) time we can decide whether G admits a 2-subcoloring (A,B) where G[A] and G[B] have at most k cliques in total.
To obtain the first two results above, we formalize a technique, which we dub inductive recognition, that can
be viewed as an adaptation of iterative compression to recognition problems. We believe that the formalization
of this technique will prove useful in general for designing parameterized algorithms for recognition problems.
Finally, we show that, unless the Exponential Time Hypothesis fails, no subexponential-time algorithms for the
above recognition problems exist, and that, unless P=NP, no generic fixed-parameter algorithm exists for the
recognizability of graphs whose vertex set can be bipartitioned such that one part is a disjoint union of k
cliques.

Iyad Kanj, Christian Komusiewicz, Manuel Sorge, and Erik Jan van Leeuwen. Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kanj_et_al:LIPIcs.SWAT.2016.14, author = {Kanj, Iyad and Komusiewicz, Christian and Sorge, Manuel and Jan van Leeuwen, Erik}, title = {{Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {14:1--14:14}, 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.2016.14}, URN = {urn:nbn:de:0030-drops-60360}, doi = {10.4230/LIPIcs.SWAT.2016.14}, annote = {Keywords: vertex-partition problems, monopolar graphs, subcolorings, split graphs, unipolar graphs, fixed-parameter algorithms} }

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

We consider the following graph cut problem called Critical Node Cut (CNC): Given a graph G on n vertices, and two positive integers k and x, determine whether G has a set of k vertices whose removal leaves G with at most x connected pairs of vertices. We analyze this problem in the framework of parameterized complexity. That is, we are interested in whether or not this problem is solvable in f(kappa) * n^{O(1)} time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters kappa. We consider four such parameters:
- The size k of the required cut.
- The upper bound x on the number of remaining connected pairs.
- The lower bound y on the number of connected pairs to be removed.
- The treewidth w of G.
We determine whether or not CNC is fixed-parameter tractable for each of these parameters. We determine this also for all possible aggregations of these four parameters, apart from w+k. Moreover, we also determine whether or not CNC admits a polynomial kernel for all these parameterizations. That is, whether or not there is an algorithm that reduces each instance of CNC in polynomial time to an equivalent instance of size kappa^{O(1)}, where kappa is the given parameter.

Danny Hermelin, Moshe Kaspi, Christian Komusiewicz, and Barak Navon. Parameterized Complexity of Critical Node Cuts. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 343-354, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hermelin_et_al:LIPIcs.IPEC.2015.343, author = {Hermelin, Danny and Kaspi, Moshe and Komusiewicz, Christian and Navon, Barak}, title = {{Parameterized Complexity of Critical Node Cuts}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {343--354}, 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.2015.343}, URN = {urn:nbn:de:0030-drops-55950}, doi = {10.4230/LIPIcs.IPEC.2015.343}, annote = {Keywords: graph cut problem, NP-hard problem, treewidth} }

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**Published in:** OASIcs, Volume 48, 15th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2015)

We show that any alpha(n)-approximation algorithm for the n-vertex metric asymmetric Traveling Salesperson problem yields O(alpha(C))-approximation algorithms for various mixed, windy, and capacitated arc routing problems. Herein, C is the number of weakly-connected components in the subgraph induced by the positive-demand arcs, a number that can be expected to be small in applications. In conjunction with known results, we derive constant-factor approximations if C is in O(log n) and O(log(C)/log(log(C)))-approximations in general.

René van Bevern, Christian Komusiewicz, and Manuel Sorge. Approximation Algorithms for Mixed, Windy, and Capacitated Arc Routing Problems. In 15th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2015). Open Access Series in Informatics (OASIcs), Volume 48, pp. 130-143, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{vanbevern_et_al:OASIcs.ATMOS.2015.130, author = {van Bevern, Ren\'{e} and Komusiewicz, Christian and Sorge, Manuel}, title = {{Approximation Algorithms for Mixed, Windy, and Capacitated Arc Routing Problems}}, booktitle = {15th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2015)}, pages = {130--143}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-99-6}, ISSN = {2190-6807}, year = {2015}, volume = {48}, editor = {Italiano, Giuseppe F. and Schmidt, Marie}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2015.130}, URN = {urn:nbn:de:0030-drops-54575}, doi = {10.4230/OASIcs.ATMOS.2015.130}, annote = {Keywords: vehicle routing, transportation, Rural Postman, Chinese Postman, NP- hard problem, parameterized algorithm, combinatorial optimization} }

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**Published in:** LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

Given a bipartite graph G=(V_c,V_t,E) and a non-negative integer k, the NP-complete Minimum-Flip Consensus Tree problem asks whether G can be transformed, using up to k edge insertions and deletions, into a graph that does not contain an induced P_5 with its first vertex in V_t (a so-called M-graph or Sigma-graph). This problem plays an important role in computational phylogenetics, V_c standing for the characters and V_t standing for taxa. Chen et al. [IEEE/ACM TCBB 2006] showed that Minimum-Flip Consensus Tree is NP-complete and presented a parameterized algorithm with running time O(6^k\cdot |V_t|\cdot |V_c|).
Recently, Boecker et al. [IWPEC'08] presented a refined search tree algorithm
with running time O(4.83^k(|V_t|+|V_c|) + |V_t|\cdot |V_c|).
We complement these results by polynomial-time executable data reduction rules yielding a problem kernel with O(k^3) vertices.

Christian Komusiewicz and Johannes Uhlmann. A Cubic-Vertex Kernel for Flip Consensus Tree. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 280-291, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{komusiewicz_et_al:LIPIcs.FSTTCS.2008.1760, author = {Komusiewicz, Christian and Uhlmann, Johannes}, title = {{A Cubic-Vertex Kernel for Flip Consensus Tree}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {280--291}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-08-8}, ISSN = {1868-8969}, year = {2008}, volume = {2}, editor = {Hariharan, Ramesh and Mukund, Madhavan and Vinay, V}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1760}, URN = {urn:nbn:de:0030-drops-17600}, doi = {10.4230/LIPIcs.FSTTCS.2008.1760}, annote = {Keywords: Fixed-parameter algorithm, problem kernel, NP-hard problem, graph modification problem, computational phylogenetics} }