The geometric set cover problem is the special case of the set cover problem in geometric settings. The input is a range space Σ = ( X , R ) {\displaystyle \Sigma =(X,{\mathcal {R}})} where X {\displaystyle X} is a universe of points in R d {\displaystyle \mathbb {R} ^{d}} and R {\displaystyle {\mathcal {R}}} is a family of subsets of X {\displaystyle X} called ranges, defined by the intersection of X {\displaystyle X} and geometric shapes such as disks and axis-parallel rectangles. The goal is to select a minimum-size subset C ⊆ R {\displaystyle {\mathcal {C}}\subseteq {\mathcal {R}}} of ranges such that every point in the universe X {\displaystyle X} is covered by some range in C {\displaystyle {\mathcal {C}}} . Given the same range space Σ {\displaystyle \Sigma } , a closely related problem is the geometric hitting set problem, where the goal is to select a minimum-size subset H ⊆ X {\displaystyle H\subseteq X} of points such that every range of R {\displaystyle {\mathcal {R}}} has nonempty intersection with H {\displaystyle H} , i.e., is hit by H {\displaystyle H} . In the one-dimensional case, where X {\displaystyle X} contains points on the real line and R {\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 R {\displaystyle {\mathcal {R}}} is induced by unit disks or unit squares. The discrete unit disc cover problem is a geometric version of the general set cover problem which is NP-hard. 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.
Approximation algorithms The greedy algorithm for the general set cover problem gives O ( log n ) {\displaystyle O(\log n)} approximation, where n = max { | X | , | R | } {\displaystyle n=\max\{|X|,|{\mathcal {R}}|\}} . This approximation is known to be tight up to constant factor. However, in geometric settings, better approximations can be obtained. Using a multiplicative weight algorithm, Brönnimann and Goodrich showed that an O ( log O P T ) {\displaystyle O(\log {\mathsf {OPT}})} -approximate set cover/hitting set for a range space Σ {\displaystyle \Sigma } with constant VC-dimension can be computed in polynomial time, where O P T ≤ n {\displaystyle {\mathsf {OPT}}\leq n} denotes the size of the optimal solution. The approximation ratio can be further improved to O ( log log O P T ) {\displaystyle O(\log \log {\mathsf {OPT}})} or O ( 1 ) {\displaystyle O(1)} when R {\displaystyle {\mathcal {R}}} is induced by axis-parallel rectangles or disks in R 2 {\displaystyle \mathbb {R} ^{2}} , respectively.
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