eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-08-30
48:1
48:16
10.4230/LIPIcs.ESA.2023.48
article
Kernelization for Spreading Points
Fomin, Fedor V.
1
https://orcid.org/0000-0003-1955-4612
Golovach, Petr A.
1
https://orcid.org/0000-0002-2619-2990
Inamdar, Tanmay
1
https://orcid.org/0000-0002-0184-5932
Saurabh, Saket
2
1
Zehavi, Meirav
3
https://orcid.org/0000-0002-3636-5322
University of Bergen, Norway
Institute of Mathematical Sciences, Chennai, India
Ben-Gurion University, Beer-Sheva, Israel
We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of points is "close" to each other. More precisely, for a family of n points, an integer k, and a real number d > 0, we ask whether at most k points could be relocated, each point at distance at most d from its original location, such that the distance between each pair of points is at least a fixed constant, say 1. A number of approximation algorithms for variants of this problem, under different names like distant representatives, disk dispersing, or point spreading, are known in the literature. However, to the best of our knowledge, the parameterized complexity of this problem remains widely unexplored. We make the first step in this direction by providing a kernelization algorithm that, in polynomial time, produces an equivalent instance with 𝒪(d²k³) points. As a byproduct of this result, we also design a non-trivial fixed-parameter tractable (FPT) algorithm for the problem, parameterized by k and d. Finally, we complement the result about polynomial kernelization by showing a lower bound that rules out the existence of a kernel whose size is polynomial in k alone, unless NP ⊆ coNP/poly.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol274-esa2023/LIPIcs.ESA.2023.48/LIPIcs.ESA.2023.48.pdf
parameterized algorithms
kernelization
spreading points
distant representatives
unit disk packing