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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. In this paper, we initiate the algorithmic study of a high-dimensional cut problem. The problem we study, namely, Homological Hitting Set (HHS), is defined as follows: Given a nontrivial r-cycle z in a simplicial complex, find a set 𝒮 of r-dimensional simplices of minimum cardinality so that 𝒮 meets every cycle homologous to z. Our first result is that HHS admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the minimal solution is given in terms of the cocycles of the surface. Next, we provide an example of a 2-complex for which the (unique) minimal hitting set is not a cocycle. Furthermore, for general complexes, we show that HHS is W[1]-hard with respect to the solution size p. In contrast, on the positive side, we show that HHS admits an FPT algorithm with respect to p+Δ, where Δ is the maximum degree of the Hasse graph of the complex 𝖪.

Ulrich Bauer, Abhishek Rathod, and Meirav Zehavi. On Computing Homological Hitting Sets. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bauer_et_al:LIPIcs.ITCS.2023.13, author = {Bauer, Ulrich and Rathod, Abhishek and Zehavi, Meirav}, title = {{On Computing Homological Hitting Sets}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {13:1--13:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.13}, URN = {urn:nbn:de:0030-drops-175169}, doi = {10.4230/LIPIcs.ITCS.2023.13}, annote = {Keywords: Algorithmic topology, Cut problems, Surfaces, Parameterized complexity} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We study the problem of finding a minimum homology basis, that is, a shortest set of cycles that generates the 1-dimensional homology classes with ℤ₂ coefficients in a given simplicial complex K. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al. [Dey et al., 2018], runs in O(N^ω + N² g) time, where N denotes the number of simplices in K, g denotes the rank of the 1-homology group of K, and ω denotes the exponent of matrix multiplication. In this paper, we present two conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex K. The first algorithm runs in Õ(m^ω) time, where m denotes the number of edges in K, whereas the second algorithm runs in O(m^ω + N m^{ω-1}) time.
We also study the problem of finding a minimum cycle basis in an undirected graph G with n vertices and m edges. The best known algorithm for this problem runs in O(m^ω) time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in Õ(m^ω) time.

Abhishek Rathod. Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 64:1-64:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{rathod:LIPIcs.SoCG.2020.64, author = {Rathod, Abhishek}, title = {{Fast Algorithms for Minimum Cycle Basis and Minimum Homology Basis}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {64:1--64:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.64}, URN = {urn:nbn:de:0030-drops-122223}, doi = {10.4230/LIPIcs.SoCG.2020.64}, annote = {Keywords: Computational topology, Minimum homology basis, Minimum cycle basis, Simplicial complexes, Matrix computations} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two simplicial complexes are of the same simple-homotopy type if they can be transformed into each other by a sequence of two basic homotopy equivalences, an elementary collapse and its inverse, an elementary expansion. In this article we consider the following related problem: given a 2-dimensional simplicial complex, is there a simple-homotopy equivalence to a 1-dimensional simplicial complex using at most p expansions? We show that the problem, which we call the erasability expansion height, is W[P]-complete in the natural parameter p.

Ulrich Bauer, Abhishek Rathod, and Jonathan Spreer. Parametrized Complexity of Expansion Height. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bauer_et_al:LIPIcs.ESA.2019.13, author = {Bauer, Ulrich and Rathod, Abhishek and Spreer, Jonathan}, title = {{Parametrized Complexity of Expansion Height}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.13}, URN = {urn:nbn:de:0030-drops-111346}, doi = {10.4230/LIPIcs.ESA.2019.13}, annote = {Keywords: Simple-homotopy theory, simple-homotopy type, parametrized complexity theory, simplicial complexes, (modified) dunce hat} }

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