TY - UNPB

T1 - The complexity of high-dimensional cuts

AU - Bauer, Ulrich

AU - Rathod, Abhishek

AU - Zehavi, Meirav

N1 - DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.
abs/2108.10195

PY - 2021/8/23

Y1 - 2021/8/23

N2 - Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. For instance, the minimum cut, the minimum s-t cut, the minimum multiway cut, and the minimum k-way cut are some of the commonly encountered cut problems. Many of these problems have been extensively studied over several decades. In this paper, we initiate the algorithmic study of some cut problems in high dimensions. The first problem we study, namely, Topological Hitting Set (THS), is defined as follows: Given a nontrivial r-cycle ζ in a simplicial complex K, find a set S of r-dimensional simplices of minimum cardinality so that S meets every cycle homologous to ζ. Our main result is that this problem admits a polynomial time solution on triangulations of closed surfaces. Interestingly, the optimal solution is given in terms of the cocycles of the surface. For general complexes, we show that THS is W[1]-hard with respect to the solution size k. On the positive side, we show that THS admits an FPT algorithm with respect to k + d, where d is the maximum degree of the Hasse graph of the complex K. We also define a problem called Boundary Nontrivialization (BNT): Given a bounding rcycle ζ in a simplicial complex K, find a set S of (r+ 1)-dimensional simplices of minimum cardinality so that the removal of S from K makes ζ non-bounding. We show that BNT is W[1]-hard with respect to the solution size as the parameter, and has an O(log n)-approximation FPT algorithm for (r + 1)-dimensional complexes with the (r + 1)-th Betti number βr+1 as the parameter. Finally, we provide randomized (approximation) FPT algorithms for the global variants of THS and BNT.

AB - Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. For instance, the minimum cut, the minimum s-t cut, the minimum multiway cut, and the minimum k-way cut are some of the commonly encountered cut problems. Many of these problems have been extensively studied over several decades. In this paper, we initiate the algorithmic study of some cut problems in high dimensions. The first problem we study, namely, Topological Hitting Set (THS), is defined as follows: Given a nontrivial r-cycle ζ in a simplicial complex K, find a set S of r-dimensional simplices of minimum cardinality so that S meets every cycle homologous to ζ. Our main result is that this problem admits a polynomial time solution on triangulations of closed surfaces. Interestingly, the optimal solution is given in terms of the cocycles of the surface. For general complexes, we show that THS is W[1]-hard with respect to the solution size k. On the positive side, we show that THS admits an FPT algorithm with respect to k + d, where d is the maximum degree of the Hasse graph of the complex K. We also define a problem called Boundary Nontrivialization (BNT): Given a bounding rcycle ζ in a simplicial complex K, find a set S of (r+ 1)-dimensional simplices of minimum cardinality so that the removal of S from K makes ζ non-bounding. We show that BNT is W[1]-hard with respect to the solution size as the parameter, and has an O(log n)-approximation FPT algorithm for (r + 1)-dimensional complexes with the (r + 1)-th Betti number βr+1 as the parameter. Finally, we provide randomized (approximation) FPT algorithms for the global variants of THS and BNT.

U2 - 10.48550/arXiv.2108.10195

DO - 10.48550/arXiv.2108.10195

M3 - Preprint

BT - The complexity of high-dimensional cuts

ER -