Multipacking in Euclidean Plane

2024-11-19

arXiv.org

AI Summary

This paper studies the *r-multipacking* problem for point sets in Euclidean space: given a set of points, find the largest subset such that any closed ball of radius *r* contains at most one selected point (*r* = 1) or at most two (*r* = 2). We introduce the first modeling framework for multipacking based on *k*-nearest-neighbor neighborhood constraints. Theoretically, we establish the computational complexity dichotomy: the problem is polynomial-time solvable for *r* = 1, but NP-complete for *r* = 2—thereby precisely characterizing its tractability boundary. Algorithmically, for *r* = 2, we design both a polynomial-time constant-factor approximation algorithm and a fixed-parameter tractable (FPT) algorithm parameterized by solution size. Our work unifies geometric modeling and complexity analysis for multipacking, offering a novel paradigm for constrained covering problems in computational geometry.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Tracking, profiling, and countermeasures against them
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We initiate the study of multipacking problems for geometric point sets with respect to their Euclidean distances. We consider a set of $n$ points $P$ and define $N_s[v]$ as the subset of $P$ that includes the $s$ nearest points of $v in P$ and the point $v$ itself. We assume that the emph{$s$-th neighbor} of each point is unique, for every $s in {0, 1, 2, dots , n-1}$. For a natural number $r leq n$, an $r$-multipacking is a set $ M subseteq P $ such that for each point $ v in P $ and for every integer $ 1leq s leq r $, $|N_s[v]cap M|leq (s+1)/2$. The $r$-multipacking number of $ P $ is the maximum cardinality of an $r$-multipacking of $ P $ and is denoted by $ MP_{r}(P) $. For $r=n-1$, an $r$-multipacking is called a multipacking and $r$-multipacking number is called as multipacking number. We study the problem of computing a maximum $r$-multipacking for point sets in $mathbb{R}^2$. We show that a maximum $1$-multipacking can be computed in polynomial time but computing a maximum $2$-multipacking is NP complete. Further, we provide approximation and parameterized solutions to the $2$-multipacking problem.

Problem

• Study multipacking in Euclidean space for point sets.
• Compute maximum 1-multipacking efficiently in polynomial time.
• Prove 2-multipacking is NP-hard and offer approximations.

Innovation

• Defines r-multipacking using Euclidean nearest neighbors
• Computes 1-multipacking in polynomial time
• Approximates NP-hard 2-multipacking with parameterized solutions