A Global Approach for Solving Edge-Matching Puzzles

TL;DR

This paper introduces an algebraic and polynomial system-based approach to solve edge-matching puzzles, transforming the discrete problem into a continuous one and proposing convex relaxation algorithms for approximate solutions.

Contribution

It presents a novel algebraic representation and polynomial formulation for edge-matching puzzles, enabling the use of convex relaxation techniques for solution approximation.

Findings

Algebraic representation precisely characterizes puzzle configurations.

Polynomial system formulation allows continuous optimization approaches.

Convex relaxation algorithms effectively generate approximate solutions.

Abstract

We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this problem and provide conditions under which it exactly characterizes a puzzle. Using the new representation, we recast the combinatorial, discrete problem of solving puzzles as a global, polynomial system of equations with continuous variables. We further propose new algorithms for generating approximate solutions to the continuous problem by solving a sequence of convex relaxations.