We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of π³-HitPack asks: Can removing k (deletable) vertices of a graph G prevent us from packing π vertex-disjoint objects of type π³? This problem captures a spectrum of problems with standard hitting and packing on opposite ends. Our main motivating question is whether the combination π³-HitPack can be significantly harder than these two base problems. Already for one particular choice of π³, this question can be posed for many different complexity notions, leading to a large, so-far unexplored domain at the intersection of the areas of hitting and packing problems. At a high level, we present two case studies: (1) π³ being all cycles, and (2) π³ being all copies of a fixed graph H. In each, we explore the classical complexity as well as the parameterized complexity with the natural parameters k+π and treewidth. We observe that the combined problem can be drastically harder than the base problems: for cycles or for H being a connected graph on at least 3 vertices, the problem is Ξ£β^π―-complete and requires double-exponential dependence on the treewidth of the graph (assuming the Exponential-Time Hypothesis). In contrast, the combined problem admits qualitatively similar running times as the base problems in some cases, although significant novel ideas are required. For π³ being all cycles, we establish a 2^{poly(k+π)}β n^{πͺ(1)} algorithm using an involved branching method, for example. Also, for π³ being all edges (i.e., H = Kβ; this combines Vertex Cover and Maximum Matching) the problem can be solved in time 2^{poly(tw)}β n^{πͺ(1)} on graphs of treewidth tw. The key step enabling this running time relies on a combinatorial bound obtained from an algebraic (linear delta-matroid) representation of possible matchings.
History
Editor
Chan T ; Fischer J ; Iacono J ; Herman G
Primary Research Area
Algorithmic Foundations and Cryptography
Name of Conference
European Symposium on Algorithms (ESA)
Journal
32nd Annual European Symposium on Algorithms (ESA 2024)
Volume
308
Page Range
55:1-55:21
Publisher
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
BibTeX
@inproceedings{Focke:Frei:Li:Marx:Schepper:Sharma:W\kegrzycki:2024,
title = "Hitting Meets Packing: How Hard Can It Be?",
author = "Focke, Jacob" AND "Frei, Fabian" AND "Li, Shaohua" AND "Marx, DΓ‘niel" AND "Schepper, Philipp" AND "Sharma, Roohani" AND "W\kegrzycki, Karol",
editor = "Chan, Timothy" AND "Fischer, Johannes" AND "Iacono, John" AND "Herman, Grzegorz",
year = 2024,
month = 9,
journal = "32nd Annual European Symposium on Algorithms (ESA 2024)",
pages = "55:1--55:21",
publisher = "Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik",
issn = "1868-8969",
doi = "10.4230/LIPIcs.ESA.2024.55"
}