Kernelization for Spreading Points

by   Fedor V. Fomin, et al.

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 β€œclose" 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(d^2k^3) 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 π–­π–―βŠ†π–Όπ—ˆπ–­π–―/poly.


page 1

page 2

page 3

page 4

βˆ™ 08/17/2021

Distant Representatives for Rectangles in the Plane

The input to the distant representatives problem is a set of n objects i...
βˆ™ 05/19/2021

Approximation Algorithms For The Dispersion Problems in a Metric Space

In this article, we consider the c-dispersion problem in a metric space ...
βˆ™ 03/17/2023

Connectivity with uncertainty regions given as line segments

For a set Q of points in the plane and a real number Ξ΄β‰₯ 0, let 𝔾_Ξ΄(Q) be...
βˆ™ 07/01/2023

Detecting Points in Integer Cones of Polytopes is Double-Exponentially Hard

Let d be a positive integer. For a finite set X βŠ†β„^d, we define its inte...
βˆ™ 03/26/2018

Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH

The k-Even Set problem is a parameterized variant of the Minimum Distanc...
βˆ™ 03/05/2020

Optimal Discretization is Fixed-parameter Tractable

Given two disjoint sets W_1 and W_2 of points in the plane, the Optimal ...
βˆ™ 05/16/2019

Parameterized Inapproximability of Exact Cover and Nearest Codeword

The k-ExactCover problem is a parameterized version of the ExactCover pr...

Please sign up or login with your details

Forgot password? Click here to reset