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On Computing Homological Hitting Sets

Authors Ulrich Bauer , Abhishek Rathod , Meirav Zehavi



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Author Details

Ulrich Bauer
  • Department of Mathematics, TUM School of CIT, Technische Universität München, Germany
Abhishek Rathod
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Meirav Zehavi
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Acknowledgements

The authors would like to thank Izhar Oppenheim for an insightful discussion on high-dimensional expansion, Vijay Natarajan for pointing out a potential application of high-dimensional cuts to the study of biomolecules and anonymous reviewers for several valuable suggestions.

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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)
https://doi.org/10.4230/LIPIcs.ITCS.2023.13

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
  • Mathematics of computing
Keywords
  • Algorithmic topology
  • Cut problems
  • Surfaces
  • Parameterized complexity

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