Subquadratic Approximation Algorithms for Separating Two Points with Objects in the Plane

TL;DR

This paper introduces subquadratic algorithms for the point-separation problem in the plane, achieving near-optimal solutions faster than previous quadratic-time methods by using novel approximation techniques.

Contribution

It presents the first subquadratic algorithms for approximate point-separation, surpassing the quadratic lower bounds of prior APSP-based approaches.

Findings

Monte Carlo algorithms run in O(n^{3/2}) for certain objects.

Deterministic algorithms achieve (1+psilon) approximation in O(n/psilon).

Algorithms outperform previous methods in runtime for specific geometric objects.

Abstract

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.