TY - GEN

T1 - Kernelization for Spreading Points

AU - Fomin, Fedor V.

AU - Golovach, Petr A.

AU - Inamdar, Tanmay

AU - Saurabh, Saket

AU - Zehavi, Meirav

N1 - Publisher Copyright:
© Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi.

PY - 2023/9/1

Y1 - 2023/9/1

N2 - 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.

AB - 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.

KW - distant representatives

KW - kernelization

KW - parameterized algorithms

KW - spreading points

KW - unit disk packing

UR - http://www.scopus.com/inward/record.url?scp=85173536965&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2023.48

DO - 10.4230/LIPIcs.ESA.2023.48

M3 - Conference contribution

AN - SCOPUS:85173536965

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 31st Annual European Symposium on Algorithms, ESA 2023

A2 - Li Gortz, Inge

A2 - Farach-Colton, Martin

A2 - Puglisi, Simon J.

A2 - Herman, Grzegorz

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 31st Annual European Symposium on Algorithms, ESA 2023

Y2 - 4 September 2023 through 6 September 2023

ER -