MILP and Max-Clique based heuristics for the Eternity II puzzle

TL;DR

This paper introduces MILP and Max-Clique formulations for the Eternity II puzzle, providing new heuristics and hard instances, and demonstrates that local search improves initial solutions effectively.

Contribution

It presents novel MILP and Max-Clique formulations for the Eternity II puzzle, along with heuristic methods and new challenging instances for research.

Findings

Local search improves initial heuristic solutions.

Formulations enable development of heuristic decompositions.

Experimental results show competitiveness with state-of-the-art methods.

Abstract

The present paper considers a hybrid local search approach to the Eternity II puzzle and to unsigned, rectangular, edge matching puzzles in general. Both an original mixed-integer linear programming (MILP) formulation and a novel Max-Clique formulation are presented for this NP-hard problem. Although the presented formulations remain computationally intractable for medium and large sized instances, they can serve as the basis for developing heuristic decompositions and very large scale neighbourhoods. As a side product of the Max-Clique formulation, new hard-to-solve instances are published for the academic research community. Two reasonably well performing MILP-based constructive methods are presented and used for determining the initial solution of a multi-neighbourhood local search approach. Experimental results confirm that this local search can further improve the results obtained by the constructive heuristics and is quite competitive with the state of the art procedures.