Euclidea is APX-hard: Complexity of Optimizing Euclidean Constructions

Abstract

We prove that it is APX-hard to find the shortest straightedge-and-compass construction of a given geometric object, under either measure of “moves” in the puzzle video game Euclidea. In particular, the problem is NP-hard, even to approximate within a factor of √2 (or a factor of 2 under the Unique Games Conjecture).