Hardness and Approximation Schemes for Discrete Packing and Domination

TL;DR

This work develops PTASes based on local search for the discrete geometric problems of Maximum Discrete Independent Set ($\mathsf{IS}$) and Minimum Discrete Dominating Set ($\mathsf{DS}$) when objects are disks of arbitrary radii or axis-parallel squares of arbitrary side length, extending Chan-Har-Peled’s approach to broader geometries. It proves that both $\mathsf{IS}$ and $\mathsf{DS}$ admit $(1-\varepsilon)$-approximation schemes via $t$-level local search with $t=O(1/\varepsilon^2)$, leveraging Additive Weighted Voronoi Diagrams and planar separator arguments to obtain tight competitive guarantees. The paper also establishes APX-hardness for the $\mathsf{DS}$ problem across a wide range of object classes and proves NP-hardness for restricted instances, illustrating a nuanced complexity landscape between discrete and continuous variants. These results advance understanding of geometric packing and domination, delineating the boundary between tractable PTAS regimes and hardness, with implications for related geometric optimization problems and algorithmic design in planar settings.

Abstract

We present polynomial-time approximation schemes based on local search} technique for both geometric (discrete) independent set (\mdis) and geometric (discrete) dominating set (\mdds) problems, where the objects are arbitrary radii disks and arbitrary side length axis-parallel squares. Further, we show that the \mdds~problem is \apx-hard for various shapes in the plane. Finally, we prove that both \mdis~and \mdds~problems are \np-hard for unit disks intersecting a horizontal line and axis-parallel unit squares intersecting a straight line with slope $-1$.

Figures (9)

• Figure 1: An illustration of an existence of a point $q$ on the segment $\mathsf{seg}(p, \mathsf{cen}(L))$.
• Figure 2: Here $L_1 \in \mathcal{L}$ and $O_1, O_2, O_3 \in \mathcal{O}$. Two edges $\mathscr{C}(\mathsf{cen}(L_1), q_1, \mathsf{cen}(O_1))$ (in green) and $\mathscr{C}(\mathsf{cen}(L_1), q_3, \mathsf{cen}(O_3))$ (in purple) added to set $E_1$.
• Figure 3: Illustration of segment $\mathsf{seg}(x, x^\prime)$ (in green). The edge $\mathscr{C}(\mathsf{cen}(L), x, x^\prime, \mathsf{cen}(O))$ is the chain of $\mathsf{seg}(\mathsf{cen}(L), x)$ (in blue), $\mathsf{seg}(x, x^\prime)$ (in green), and $\mathsf{seg}(x^\prime, \mathsf{cen}(O))$ (in red).
• Figure 4: A possible placement of disks and points such that edges $e_1 = \mathscr{C}(\mathsf{cen}(L_1), x_1, x_1^\prime, \mathsf{cen}(O_1))$ and $e_2 = \mathscr{C}(\mathsf{cen}(L_2), x_2, x_2^\prime, \mathsf{cen}(O_2))$ in $E = E_1 \cup E_2$ intersect. \ref{['edge_intersect_a']} Both edges $e_1, e_2 \in E_2$\ref{['edge_intersect_b']}$e_1 \in E_1$ and $e_2 \in E_2$.
• Figure 5: Edge perturbation step illustration. The shaded region defines $\mathsf{cell}(L_1)$. The segment $\mathsf{seg}(x_2, x_2^\prime)$ in the edge is replaced with the curve $\mathscr{C}_1(x_2, x_2^\prime)$ (the chain of segment segment $\mathsf{seg}(x_2, t_2)$, curve $\mathcal{C}(t_2, t_2^\prime)$, and segment $\mathsf{seg}(t_2^\prime, x_2^\prime)$) which connects $x_2$ and $x_2^\prime$.

Theorems & Definitions (40)

• Definition 1
• Lemma 1: Gibson2010
• proof
• Lemma 2: Wan2011
• proof
• Lemma 3: Frederickson1987
• Theorem 1
• proof
• Theorem 2
• proof