Kernelization for Spreading Points

Authors Fedor V. Fomin , Petr A. Golovach , Tanmay Inamdar , Saket Saurabh, Meirav Zehavi

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Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Tanmay Inamdar
  • University of Bergen, Norway
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
  • University of Bergen, Norway
Meirav Zehavi
  • Ben-Gurion University, Beer-Sheva, Israel


We thank the anonymous reviewers for their valuable comments.

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Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi. Kernelization for Spreading Points. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Parameterized complexity and exact algorithms
  • parameterized algorithms
  • kernelization
  • spreading points
  • distant representatives
  • unit disk packing


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