TY - GEN

T1 - On Computing Homological Hitting Sets

AU - Bauer, Ulrich

AU - Rathod, Abhishek

AU - Zehavi, Meirav

N1 - Funding Information:
Funding Ulrich Bauer: Supported by DFG Collaborative Research Center SFB/TRR 109 “Dis-cretization in Geometry and Dynamics”. Abhishek Rathod: Supported by DFG Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics”. Meirav Zehavi: Supported by European Research Council (ERC) grant titled PARAPATH.
Publisher Copyright:
© Ulrich Bauer, Abhishek Rathod, and Meirav Zehavi; licensed under Creative Commons License CC-BY 4.0.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - 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 + ∆, where ∆ is the maximum degree of the Hasse graph of the complex K.

AB - 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 + ∆, where ∆ is the maximum degree of the Hasse graph of the complex K.

KW - Algorithmic topology

KW - Cut problems

KW - Parameterized complexity

KW - Surfaces

UR - http://www.scopus.com/inward/record.url?scp=85147539612&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2023.13

DO - 10.4230/LIPIcs.ITCS.2023.13

M3 - Conference contribution

AN - SCOPUS:85147539612

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 14th Innovations in Theoretical Computer Science Conference, ITCS 2023

A2 - Kalai, Yael Tauman

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 14th Innovations in Theoretical Computer Science Conference, ITCS 2023

Y2 - 10 January 2023 through 13 January 2023

ER -