Geometric set cover problem

The geometric set cover problem is the special case of the set cover problem in geometric settings. The input is a range space ${\displaystyle \Sigma =(X,{\mathcal {R}})}$ where ${\displaystyle X}$ is a universe of points in ${\displaystyle \mathbb {R} ^{d}}$ and ${\displaystyle {\mathcal {R}}}$ is a family of subsets of ${\displaystyle X}$ called ranges, defined by the intersection of ${\displaystyle X}$ and geometric shapes such as disks and axis-parallel rectangles. The goal is to select a minimum-size subset ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {R}}}$ of ranges such that every point in the universe ${\displaystyle X}$ is covered by some range in ${\displaystyle {\mathcal {C}}}$.

Given the same range space ${\displaystyle \Sigma }$, a closely related problem is the geometric hitting set problem, where the goal is to select a minimum-size subset ${\displaystyle H\subseteq X}$ of points such that every range of ${\displaystyle {\mathcal {R}}}$ has nonempty intersection with ${\displaystyle H}$, i.e., is hit by ${\displaystyle H}$.

In the one-dimensional case, where ${\displaystyle X}$ contains points on the real line and ${\displaystyle {\mathcal {R}}}$ is defined by intervals, both the geometric set cover and hitting set problems can be solved in polynomial time using a simple greedy algorithm. However, in higher dimensions, they are known to be NP-complete even for simple shapes, i.e., when ${\displaystyle {\mathcal {R}}}$ is induced by unit disks or unit squares.^[1] The discrete unit disc cover problem is a geometric version of the general set cover problem which is NP-hard.^[2]

Many approximation algorithms have been devised for these problems. Due to the geometric nature, the approximation ratios for these problems can be much better than the general set cover/hitting set problems. Moreover, these approximate solutions can even be computed in near-linear time.^[3]

Approximation algorithms

The greedy algorithm for the general set cover problem gives ${\displaystyle O(\log n)}$ approximation, where ${\displaystyle n=\max\{|X|,|{\mathcal {R}}|\}}$. This approximation is known to be tight up to constant factor.^[4] However, in geometric settings, better approximations can be obtained. Using a multiplicative weight algorithm,^[5] Brönnimann and Goodrich^[6] showed that an ${\displaystyle O(\log {\mathsf {OPT}})}$-approximate set cover/hitting set for a range space ${\displaystyle \Sigma }$ with constant VC-dimension can be computed in polynomial time, where ${\displaystyle {\mathsf {OPT}}\leq n}$ denotes the size of the optimal solution. The approximation ratio can be further improved to ${\displaystyle O(\log \log {\mathsf {OPT}})}$ or ${\displaystyle O(1)}$ when ${\displaystyle {\mathcal {R}}}$ is induced by axis-parallel rectangles or disks in ${\displaystyle \mathbb {R} ^{2}}$, respectively.

Near-linear-time algorithms

Based on the iterative-reweighting technique of Clarkson^[7] and Brönnimann and Goodrich,^[6] Agarwal and Pan^[3] gave algorithms that computes an approximate set cover/hitting set of a geometric range space in ${\displaystyle O(n~\mathrm {polylog} (n))}$ time. For example, their algorithms computes an ${\displaystyle O(\log \log {\mathsf {OPT}})}$-approximate hitting set in ${\displaystyle O(n\log ^{3}n\log \log \log {\mathsf {OPT}})}$ time for range spaces induced by 2D axis-parallel rectangles; and it computes an ${\displaystyle O(1)}$-approximate set cover in ${\displaystyle O(n\log ^{4}n)}$ time for range spaces induced by 2D disks.

See also

• Set cover problem
• Vertex cover
• Lebesgue covering dimension
• Carathéodory's extension theorem