The (unweighted) point-separation problem asks, given a pair of points $s$ and $t$ in the plane, and a set of candidate geometric objects, for the minimum-size subset of objects whose union blocks all paths from $s$ to $t$. Recent work has shown that the point-separation problem can be characterized as a type of shortest-path problem in a geometric intersection graph within a special lifted space. However, all known solutions to this problem essentially reduce to some form of APSP, and hence take at least quadratic time, even for special object types. We improve the conditional quadratic lower bounds for this problem, but our main results are positive: We bypass this barrier by providing subquadratic algorithms to produce solutions of size $\text{OPT}+1$ or $(1+\varepsilon)\text{OPT}+1$. Our algorithms are fundamentally different from the APSP-based approach. In particular, we give Monte Carlo randomized additive $+1$ approximation algorithms running in $\widetilde{\mathcal{O}}(n^{\frac32})$ time for disks, axis-aligned line segments and constant-complexity rectilinear polylines, and $\widetilde{\mathcal{O}}(n^{\frac{11}6})$ time for line segments and constant-complexity polylines. We will also give deterministic multiplicative-additive approximation algorithms that, for any value $\varepsilon>0$, guarantee a solution of size $(1+\varepsilon)\text{OPT}+1$ while running in $\widetilde{\mathcal{O}}\left(n/\varepsilon\right)$ time for disks, axis-aligned line segments and constant-complexity rectilinear polylines, and $\widetilde{\mathcal{O}}\left(n^{4/3}/\varepsilon\right)$ time for line segments and constant-complexity polylines.

翻译：（无权）点分离问题是指：给定平面上一对点 $s$ 和 $t$，以及一组候选几何对象，求一个最小规模的子集，使得这些对象的并集能阻挡所有从 $s$ 到 $t$ 的路径。近期研究表明，点分离问题可被刻画为一种在特殊提升空间内的几何交图上的最短路径问题。然而，目前所有已知的解决方案本质上都归结为某种形式的全对最短路径（APSP）问题，因此即使在特殊对象类型下，也至少需要二次时间。我们改进了该问题的条件二次下界，但我们的主要结果是积极的：我们通过提供次二次算法来绕过这一障碍，这些算法可生成规模为 $\text{OPT}+1$ 或 $(1+\varepsilon)\text{OPT}+1$ 的解。我们的算法从根本上不同于基于 APSP 的方法。具体而言，我们给出了蒙特卡洛随机化加性 $+1$ 近似算法，对于圆盘、轴对齐线段和常数复杂度折线，运行时间为 $\widetilde{\mathcal{O}}(n^{\frac32})$；对于线段和常数复杂度折线，运行时间为 $\widetilde{\mathcal{O}}(n^{\frac{11}6})$。我们还将给出确定性乘加性近似算法，对于任意 $\varepsilon>0$，保证解规模为 $(1+\varepsilon)\text{OPT}+1$，对于圆盘、轴对齐线段和常数复杂度折线，运行时间为 $\widetilde{\mathcal{O}}\left(n/\varepsilon\right)$；对于线段和常数复杂度折线，运行时间为 $\widetilde{\mathcal{O}}\left(n^{4/3}/\varepsilon\right)$。