Kernelization for Spreading Points

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 $O(d^2k^3)$ 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 $\mathsf{NP} \subseteq \mathsf{coNP}/\text{poly}$

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Dispersion in Unit Disks

We present two new approximation algorithms with (improved) constant ratios for selecting $n$ points in $n$ unit disks such that the minimum pairwise distance among the points is maximized. (I) A very simple $O(n log{n})$-time algorithm with ratio $0.5110$ for disjoint unit disks. In combination with an algorithm of Cabello~cite{Ca07}, it yields a $O(n^2)$-time algorithm with ratio of $0.4487$ for dispersion in $n$ not necessarily disjoint unit disks. (II) A more sophisticated LP-based algorithm with ratio $0.6495$ for disjoint unit disks that uses a linear number of variables and constraints, and runs in polynomial time. The algorithm introduces a novel technique which combines linear programming and projections for approximating distances. The previous best approximation ratio for disjoint unit disks was $frac{1}{2}$. Our results give a partial answer to an open question raised by Cabello~cite{Ca07}, who asked whether $frac{1}{2}$ could be improved