A deterministic algorithm finds O(OPT log(h+2) log(OPT log(h+2))) points that see at least (1-δ) area of a polygon with h holes in time polynomial in input size and log(1/δ).
O’Rourke.Art Gallery Theorems and Algorithms
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A Deterministic Bicriteria Approximation Algorithm for the Art Gallery Problem
A deterministic algorithm finds O(OPT log(h+2) log(OPT log(h+2))) points that see at least (1-δ) area of a polygon with h holes in time polynomial in input size and log(1/δ).