# Hitting Meets Packing: How Hard Can It Be?

conference contribution

posted on 2024-10-08, 08:24 authored by Jacob FockeJacob Focke, Fabian FreiFabian Frei, Shaohua Li, DΓ‘niel MarxDΓ‘niel Marx, Philipp SchepperPhilipp Schepper, Roohani Sharma, Karol W\kegrzyckiWe 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" }## Usage metrics

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