Given a collection S of subsets of some set Í, and Å ⊂ Í, the set cover problem is to find the smallest subcollection C ⊂ S such that Å is a subset of the union of the sets in C. While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually Í = d . Combining previously known techniques [3, 4], we show that polynomial time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset R ⊂ S and function f (), there is a decomposition of the complement Í \ ∪Y ∈RY into an expected f (|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f (|C|)) can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f (c) that are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudodisks, by a family of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects, and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in 3 , and for guarding an xmonotone polygonal chain.
Of course the problem can be generalized further: see REF for a discussion of geometric set cover problems.
Improved approximation algorithms for geometric set cover
SCG '05
Mathematics Computer Science