Computational complexity of art gallery problems
IEEE Transactions on Information Theory
Art gallery theorems and algorithms
Art gallery theorems and algorithms
Probabilistic checking of proofs and hardness of approximation problems
Probabilistic checking of proofs and hardness of approximation problems
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
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In the Art Gallery problem, given is a polygonal gallery and the goal is to guard the gallery's interior or walls with a number of guards that must be placed strategically in the interior, on walls or on corners of the gallery. Here we consider a more realistic version: exhibits now have size and may have different costs. Moreover the meaning of guarding is relaxed: we use a new concept, that of watching an expensive art item, i.e. overseeing a part of the item. The main result of the paper is that the problem of maximizing the total value of a guarded weighted boundary is APX-complete. This is shown by an appropriate gap-preserving reduction from the Max-5-occurrence-3-Sat problem. We also show that this technique can be applied to a number of maximization variations of the art gallery problem. In particular we consider the following problems: given a polygon with or without holes and k available guards, maximize a) the length of walls guarded and b) the total cost of paintings watched or overseen. We prove that all the above problems are APX-complete.