Abstract
Computing small kernels for the hitting set problem is a wellstudied computational problem where we are given a hypergraph with n vertices and m hyperedges, each of size d for some small constant d, and a parameter k. The task is to compute a new hypergraph, called a kernel, whose size is polynomial with respect to the parameter k and which has a sizek hitting set if, and only if, the original hypergraph has one. Stateoftheart algorithms compute kernels of size k^d (which is a polynomial kernel size as d is a constant), and they do so in time m⋅ 2^d poly(d) for a small polynomial poly(d) (which is a linear runtime as d is again a constant).
We generalize this task to the dynamic setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a sizek^d hitting set kernel in memory (including moments when no sizek hitting set exists). This paper presents a deterministic solution with worstcase time 3^d poly(d) for updating the kernel upon hyperedge inserts and time 5^d poly(d) for updates upon deletions. These bounds nearly match the time 2^d poly(d) needed by the best static algorithm per hyperedge. Let us stress that for constant d our algorithm maintains a dynamic hitting set kernel with constant, deterministic, worstcase update time that is independent of n, m, and the parameter k. As a consequence, we also get a deterministic dynamic algorithm for keeping track of sizek hitting sets in dhypergraphs with update times O(1) and query times O(c^k) where c = d  1 + O(1/d) equals the best base known for the static setting.
BibTeX  Entry
@InProceedings{bannach_et_al:LIPIcs.IPEC.2021.7,
author = {Bannach, Max and Heinrich, Zacharias and Reischuk, R\"{u}diger and Tantau, Till},
title = {{Dynamic Kernels for Hitting Sets and Set Packing}},
booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
pages = {7:17:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772167},
ISSN = {18688969},
year = {2021},
volume = {214},
editor = {Golovach, Petr A. and Zehavi, Meirav},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15390},
URN = {urn:nbn:de:0030drops153900},
doi = {10.4230/LIPIcs.IPEC.2021.7},
annote = {Keywords: Kernelization, Dynamic Algorithms, Hitting Set, Set Packings}
}
Keywords: 

Kernelization, Dynamic Algorithms, Hitting Set, Set Packings 
Collection: 

16th International Symposium on Parameterized and Exact Computation (IPEC 2021) 
Issue Date: 

2021 
Date of publication: 

22.11.2021 