arXiv:2611.13600 · v1

Parameterized Complexity and Geometric Approximation of Riesz Energy Subset Selection

首次提交 2026-11-10 13:26

版本更新 2026-11-10 13:26

雷达观测 2026-11-11 11:04

摘要

We choose $k$ of $n$ sites to minimize Riesz $s$-energy, in which a pair at distance $t$ contributes $t^{-s}$. For $s=2$, exact selection is polynomial on a line but NP-hard in the plane. Adapting the proxy lemma of Ceccarello et al. gives an efficient parameterized approximation scheme (EPAS) for fixed Euclidean dimension and fixed $s>0$. To sharpen the bound, we start with a geometric packing. Farthest-first traversal gives $2k$ centers separated by at least the last insertion distance $r$. If one site of an optimum is removed, each of the $k-1$ remaining sites excludes at most one center closer than $r/2$, leaving at least $k+1$ centers free. The new step is a discrete exchange bound: the removed site's interaction energy is at most the average it would contribute at those free centers. In the plane at $s=2$, this reduces the representative bound from $O(k^2\log(2k)/\varepsilon^2)$ for a packing estimate of total energy to $O(k\log(2k)/\varepsilon^2)$, independently of $n$. Rational planar examples show that the resulting separation scale $r/\sqrt{\log(2k)}$ is sharp. A $(1+\varepsilon)$ approximation on rational planar sites takes $O(nk+n\log n)+(C\log(2k)/\varepsilon^2)^k$ operations of polynomial bit cost. The observed number $M$ of representatives determines the search size $\binom{M}{k}$. With one million sites, $k=5$ and $\varepsilon=1/2$, the exchange grid left 252 subsets on clustered inputs and about $9.3\times10^{26}$ on uniform inputs. The method extends to metrics of bounded doubling dimension and to compact domains with effective geometric routines, including filled simple polygons. At $s=2$, exact decision on metrics with nonzero distances in $\{1,2\}$ is W[1]-complete in $k$. A gap that shrinks with $k$ rules out a deterministic EPAS on unrestricted metrics unless FPT = W[1].

作者备注：35 pages, 6 figures. Code, data, and instructions for reproducing the experiments: https://github.com/emmerichmtm/PCGA-Riesz