On the hardness of solving edge matching puzzles as SAT or CSP problems

Abstract

Edge matching puzzles have been amongst us for a long time now and traditionally they have been considered, both, a children’s game and an interesting mathematical divertimento. Their main characteristics have already been studied, and their worst-case complexity has been properly classified as a NP-complete problem. It is in recent times, specially after being used as the problem behind a money-prized contest, with a prize of 2US$ million for the first solver, that edge matching puzzles have attracted mainstream attention from wider audiences, including, of course, computer science people working on solving hard problems. We consider these competitions as an interesting opportunity to showcase SAT/CSP solving techniques when confronted to a real world problem to a broad audience, a part of the intrinsic, i.e. monetary, interest of such a contest. This article studies the NP-complete problem known as edge matching puzzle using SAT and CSP approaches for solving it. We will focus on providing, first and foremost, a theoretical framework, including a generalized definition of the problem. We will design and show algorithms for easy and fast problem instances generation, generators with easily tunable hardness. Afterwards we will provide with SAT and CSP models for the problems and we will study problem complexity, both typical case and worst-case complexity. We will also provide some specially crafted heuristics that result in a boost in solving time and study which is the effect of such heuristics.