New Results on the Remote Set Problem and Its Applications in Complexity Study

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 \(\pmb {\mathcal {L}}\) $\mathcal{L}$ . It introduced a new method for analyzing the complexity of CRP. An open question in RSP is whether we can obtain the approximation factor \(\gamma =1/2\) $\gamma =1/2$ . This paper investigates this question and proposes a probabilistic polynomial-time algorithm for RSP with an approximation factor of \(1/2-1/(c\lambda ^{(p)}_n)\) $1/2-1/(c{\lambda }_n^{(p)})$ , where \(c\in \mathbb {Z}^{+}\) $c\in {\mathbb{Z}}^{+}$ and \(\lambda ^{(p)}_n\) ${\lambda }_n^{(p)}$ is the n-th successive minima in lattice under \(l_p\) $l_p$ -norm. For a given lattice \(\pmb {\mathcal {L}}\) $\mathcal{L}$ 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 \((\frac{1}{2}-\frac{1}{c\lambda ^{(p)}_n}){{\rho }^{(p)}}(\pmb {\mathcal {L}})\) $(\frac{1}{2}-\frac{1}{c{\lambda }_n^{(p)}}){\rho }^{(p)}(\mathcal{L})$ from lattice \(\pmb {\mathcal {L}}\) $\mathcal{L}$ with a probability greater than \(1-1/2^d\) $1-1/2^d$ . Here, c is a positive integer and \(\rho ^{(p)}(\pmb {\mathcal {L}})\) ${\rho }^{(p)}(\mathcal{L})$ denotes the covering radius of \(\pmb {\mathcal {L}}\) $\mathcal{L}$ in \(l_p\) $l_p$ -norm( \(1\le p\le \infty \) $1\le p\le \infty$ ). Based on this, we obtain that \(\text {GAPCRP}_{2+1/2^{O(n)}}\) ${\text{GAPCRP}}_{2+1/2^{O(n)}}$ belongs to the complexity class coRP, and we provide new reductions from GAPCRP to GAPCVP.