Max 2SAT-3, Net, Euclidea: Techniques and Results in Computational Inapproximability

Abstract

This Master’s thesis investigates three diverse problem domains through the lens of computational inapproximability: Max 2SAT-3, the Net tile-rotating puzzle family, and the mobile game Euclidea. Max 2SAT-3 is a problem long known to be APX-complete, but finding a clear proof is harder than one might expect. We examine the history of Max 2SAT-3, addressing past misconceptions and clarifying where the reduction chain has been opaque, and present a novel proof of its APX-completeness. Net variants form a wide class of puzzles with lots of potential for future research. We introduce a natural optimization variant of Net and demonstrate its inapproximability, as well as consolidate existing findings and present other new results. Euclidea is a mobile game based on Euclidean straightedge-and-compass constructions. We define the game as an optimization problem and establish its APX-hardness, as well as discuss challenges in upper-bounding its complexity, relating to current knowledge gaps regarding the constructible and algebraic numbers.