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

Jayson Lynch, Jack Spalding-Jamieson

TL;DR

This work studies the unweighted point-separation problem in the plane, where the goal is to select a minimum set of geometric objects whose union separates two points $s$ and $t$. Building on the homology-cover framework, the authors bypass the quadratic barrier of all-pairs shortest paths by developing subquadratic additive and multiplicative-additive approximations, using a combination of well-approximated paths, Monte Carlo sampling, and a divide-and-conquer strategy. The approach yields near-optimal results across several object classes (disks, line segments, and rectilinear polylines), with running times tied to fast single-source shortest-path subroutines in the homology-cover intersection graph. They also establish tight connections to $k$-cycle detection via a fine-grained hypothesis, deriving both upper and lower bounds for multiplicative approximations and highlighting the conditional hardness of faster exact or multiplicative schemes. Overall, the paper delivers subquadratic algorithms for unweighted point-separation and clarifies the trade-offs between approximation quality, object class, and underlying shortest-path computations, offering practical sublinear improvements over previous APSP-based methods.

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.

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

TL;DR

This work studies the unweighted point-separation problem in the plane, where the goal is to select a minimum set of geometric objects whose union separates two points and . Building on the homology-cover framework, the authors bypass the quadratic barrier of all-pairs shortest paths by developing subquadratic additive and multiplicative-additive approximations, using a combination of well-approximated paths, Monte Carlo sampling, and a divide-and-conquer strategy. The approach yields near-optimal results across several object classes (disks, line segments, and rectilinear polylines), with running times tied to fast single-source shortest-path subroutines in the homology-cover intersection graph. They also establish tight connections to -cycle detection via a fine-grained hypothesis, deriving both upper and lower bounds for multiplicative approximations and highlighting the conditional hardness of faster exact or multiplicative schemes. Overall, the paper delivers subquadratic algorithms for unweighted point-separation and clarifies the trade-offs between approximation quality, object class, and underlying shortest-path computations, offering practical sublinear improvements over previous APSP-based methods.

Abstract

The (unweighted) point-separation problem asks, given a pair of points and in the plane, and a set of candidate geometric objects, for the minimum-size subset of objects whose union blocks all paths from to . 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 or . Our algorithms are fundamentally different from the APSP-based approach. In particular, we give Monte Carlo randomized additive approximation algorithms running in time for disks, axis-aligned line segments and constant-complexity rectilinear polylines, and time for line segments and constant-complexity polylines. We will also give deterministic multiplicative-additive approximation algorithms that, for any value , guarantee a solution of size while running in time for disks, axis-aligned line segments and constant-complexity rectilinear polylines, and time for line segments and constant-complexity polylines.

Paper Structure

This paper contains 12 sections, 29 theorems, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Reduction to Shortest-Paths
  4. Shortest-Path Algorithms, Notation and Time Complexity
  5. Well-Approximated Paths
  6. A Monte Carlo Algorithm
  7. Divide and Conquer
  8. Main Theorems
  9. A Formal Description of the Homology Cover Space
  10. A Simple Deterministic Additive $+k$ Approximation
  11. Fast Shortest-Paths in the Homology Cover
  12. Multiplicative Approximations and $k$-Cycle Detection

Key Result

Proposition 2

For a set of objects $\mathcal{C}$, the graph $\overline G$, and an object $c\in\mathcal{C}$, let $D_c$ denote the elements along an arbitrary shortest-path in $\overline{G}$ from $c^{-1}$ to $c^1$, and let $F_c\subset\mathcal{C}$ denote the corresponding planar objects that separate $s$ and $t$. If

Figures (12)

  • Figure 1: An instance of the point-separation problem with a set of line segment objects (left), and a minimum subset of segments that separate $s$ and $t$ (right). The region of the plane reachable from $s$ without crossing the blocking objects (which does not include $t$) is shaded.
  • Figure 2: Visualization of two curves in the homology cover, and their projections onto the plane. One of the resulting projected curves separates $s$ and $t$, and the other does not.
  • Figure 3: Visualization of objects in the homology cover (left), a path $D_c$ through the intersection graph in the homology cover, the corresponding set of planar objects $F_c$ (middle, top and bottom, respectively), and an induced path whose projection into the plane separates $s$ and $t$ (right).
  • Figure 4: A path through the intersection graph in the homology cover that contains the path in \ref{['fig:homology-cover-objects-projection']} (middle) as a sub-path, and the corresponding set of planar objects. There are $2$ additional objects in the path in the homology cover, but only $1$ additional object in the planar set.
  • Figure 5: The intersection graph $\overline{G}$ in the homology cover, using disk centres as canonical points.
  • ...and 7 more figures

Theorems & Definitions (29)

  • Proposition 2: spalding2025separating
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Proposition 9
  • Theorem 10
  • Proposition 11
  • ...and 19 more