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

In this paper, we study the computational complexity of the Maximum Cut problem parameterized above guarantee. Our main result provides a linear kernel for the Maximum Cut problem in connected graphs parameterized above the spanning tree. This kernel significantly improves the previous O(k⁵) kernel given by Madathil, Saurabh, and Zehavi [ToCS 2020]. We also provide subexponential running time algorithms for this problem in special classes of graphs: chordal, split, and co-bipartite. We complete the picture by lower bounds under the assumption of the ETH. Moreover, we initiate a study of the Maximum Cut problem above 2/3|E| lower bound in tripartite graphs.

Ivan Bliznets and Vladislav Epifanov. MaxCut Above Guarantee. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bliznets_et_al:LIPIcs.MFCS.2023.22, author = {Bliznets, Ivan and Epifanov, Vladislav}, title = {{MaxCut Above Guarantee}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {22:1--22:14}, 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.2023.22}, URN = {urn:nbn:de:0030-drops-185560}, doi = {10.4230/LIPIcs.MFCS.2023.22}, annote = {Keywords: Tripartite, 3-colorable, chordal, maximum cut, FPT-algorithm, linear kernel} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

For a fixed graph H, the H-free Edge Deletion/Completion/Editing problem asks to delete/add/edit the minimum possible number of edges in G to get a graph that does not contain an induced subgraph isomorphic to the graph H. In this work, we investigate H-free Edge Deletion/Completion/Editing problems in terms of the hardness of their approximation. We formulate a conjecture according to which all the graphs with at least five vertices can be classified into several groups of graphs with specific structural properties depending on the hardness of approximation for the corresponding H-free Edge Deletion/Completion/Editing problems. Also, we make significant progress in proving that conjecture by showing that it is sufficient to resolve it only for a finite set of graphs.

Tatiana Belova and Ivan Bliznets. Hardness of Approximation for H-Free Edge Modification Problems: Towards a Dichotomy. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{belova_et_al:LIPIcs.ISAAC.2022.24, author = {Belova, Tatiana and Bliznets, Ivan}, title = {{Hardness of Approximation for H-Free Edge Modification Problems: Towards a Dichotomy}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.24}, URN = {urn:nbn:de:0030-drops-173097}, doi = {10.4230/LIPIcs.ISAAC.2022.24}, annote = {Keywords: Parameterized complexity, Hardness of approximation, Edge modification problems} }

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

In this paper we consider the Target Set Selection problem. The problem naturally arises in many fields like economy, sociology, medicine. In the Target Set Selection problem one is given a graph G with a function thr: V(G) -> N cup {0} and integers k, l. The goal of the problem is to activate at most k vertices initially so that at the end of the activation process there is at least l activated vertices. The activation process occurs in the following way: (i) once activated, a vertex stays activated forever; (ii) vertex v becomes activated if at least thr(v) of its neighbours are activated. The problem and its different special cases were extensively studied from approximation and parameterized points of view. For example, parameterizations by the following parameters were studied: treewidth, feedback vertex set, diameter, size of target set, vertex cover, cluster editing number and others.
Despite the extensive study of the problem it is still unknown whether the problem can be solved in O^*((2-epsilon)^n) time for some epsilon >0. We partially answer this question by presenting several faster-than-trivial algorithms that work in cases of constant thresholds, constant dual thresholds or when the threshold value of each vertex is bounded by one-third of its degree. Also, we show that the problem parameterized by l is W[1]-hard even when all thresholds are constant.

Ivan Bliznets and Danil Sagunov. Solving Target Set Selection with Bounded Thresholds Faster than 2^n. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bliznets_et_al:LIPIcs.IPEC.2018.22, author = {Bliznets, Ivan and Sagunov, Danil}, title = {{Solving Target Set Selection with Bounded Thresholds Faster than 2^n}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {22:1--22:14}, 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.2018.22}, URN = {urn:nbn:de:0030-drops-102235}, doi = {10.4230/LIPIcs.IPEC.2018.22}, annote = {Keywords: exact exponential algorithms, target set, vertex thresholds, social influence, irreversible conversions of graphs, bootstrap percolation} }

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

Clustering is a well-known and important problem with numerous applications. The graph-based model is one of the typical cluster models. In the graph model generally clusters are defined as cliques. However, such approach might be too restrictive as in some applications, not all objects from the same cluster must be connected. That is why different types of cliques relaxations often considered as clusters.
In our work, we consider a problem of partitioning graph into clusters and a problem of isolating cluster of a special type where by cluster we mean highly connected subgraph. Initially, such clusterization was proposed by Hartuv and Shamir. And their HCS clustering algorithm was extensively applied in practice. It was used to cluster cDNA fingerprints, to find complexes in protein-protein interaction data, to group protein sequences hierarchically into superfamily and family clusters, to find families of regulatory RNA structures. The HCS algorithm partitions graph in highly connected subgraphs. However, it is achieved by deletion of not necessarily the minimum number of edges. In our work, we try to minimize the number of edge deletions. We consider problems from the parameterized point of view where the main parameter is a number of allowed edge deletions. The presented algorithms significantly improve previous known running times for the Highly Connected Deletion (improved from \cOs\left(81^k\right) to \cOs\left(3^k\right)), Isolated Highly Connected Subgraph (from \cOs(4^k) to \cOs\left(k^{\cO\left(k^{\sfrac{2}{3}}\right)}\right) ), Seeded Highly Connected Edge Deletion (from \cOs\left(16^{k^{\sfrac{3}{4}}}\right) to \cOs\left(k^{\sqrt{k}}\right)) problems. Furthermore, we present a subexponential algorithm for Highly Connected Deletion problem if the number of clusters is bounded. Overall our work contains three subexponential algorithms which is unusual as very recently there were known very few problems admitting subexponential algorithms.

Ivan Bliznets and Nikolai Karpov. Parameterized Algorithms for Partitioning Graphs into Highly Connected Clusters. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bliznets_et_al:LIPIcs.MFCS.2017.6, author = {Bliznets, Ivan and Karpov, Nikolai}, title = {{Parameterized Algorithms for Partitioning Graphs into Highly Connected Clusters}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {6:1--6:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.6}, URN = {urn:nbn:de:0030-drops-80728}, doi = {10.4230/LIPIcs.MFCS.2017.6}, annote = {Keywords: clustering, parameterized complexity, highly connected} }

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

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

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

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