In 2015, Haviv introduced the Remote set problem (RSP) and studied the complexity of the covering radius problem (CRP), which is a classical problem in lattices. The RSP aims to identify a set containing a point that is sufficiently distant from a given lattice \(\pmb {\mathcal {L}}\). It introduced a new method for analyzing the complexity of CRP. An open question in RSP is whether we can obtain the approximation factor \(\gamma =1/2\). This paper investigates this question and proposes a probabilistic polynomial-time algorithm for RSP with an approximation factor of \(1/2-1/(c\lambda ^{(p)}_n)\), where \(c\in \mathbb {Z}^{+}\) and \(\lambda ^{(p)}_n\) is the n-th successive minima in lattice under \(l_p\)-norm. For a given lattice \(\pmb {\mathcal {L}}\) with rank n and positive integer d, our algorithm outputs a set S of size d in polynomial time. This set S includes a point at least \((\frac{1}{2}-\frac{1}{c\lambda ^{(p)}_n}){{\rho }^{(p)}}(\pmb {\mathcal {L}})\) from lattice \(\pmb {\mathcal {L}}\) with a probability greater than \(1-1/2^d\). Here, c is a positive integer and \(\rho ^{(p)}(\pmb {\mathcal {L}})\) denotes the covering radius of \(\pmb {\mathcal {L}}\) in \(l_p\)-norm(\(1\le p\le \infty \)). Based on this, we obtain that \(\text {GAPCRP}_{2+1/2^{O(n)}}\) belongs to the complexity class coRP, and we provide new reductions from GAPCRP to GAPCVP.

2015年，Haviv引入了远程集问题（RSP），并研究了格中经典问题覆盖半径问题（CRP）的复杂性。RSP 旨在识别包含距离给定格\(\pmb {\mathcal {L}}\)足够远的点的集合。它引入了一种分析 CRP 复杂性的新方法。RSP 中的一个悬而未决的问题是我们是否可以获得近似因子\(\gamma =1/2\)。本文研究了这个问题，并提出了一种 RSP 概率多项式时间算法，其近似因子为\(1/2-1/(c\lambda ^{(p)}_n)\)，其中\(c\in \ mathbb {Z}^{+}\)和\(\lambda ^{(p)}_n\)是在\(l_p\) -范数下格子中的第n个连续最小值。对于给定的具有秩n和正整数d的格\(\pmb {\mathcal {L}}\) ，我们的算法在多项式时间内输出大小为d的集合S。该集合S至少包含一个点\((\frac{1}{2}-\frac{1}{c\lambda ^{(p)}_n}){{\rho }^{(p)}} (\pmb {\mathcal {L}})\)来自格子\(\pmb {\mathcal {L}}\)，概率大于\(1-1/2^d\)。这里，c是一个正整数，\(\rho ^{(p)}(\pmb {\mathcal {L}})\)表示\(\pmb {\mathcal {L}}\)的覆盖半径\(l_p\) -norm( \(1\le p\le \infty \) )。基于此，我们得到\(\text {GAPCRP}_{2+1/2^{O(n)}}\)属于复杂度类 coRP，并且我们提供了从 GAPCRP 到 GAPCVP 的新约简。