Polynomial Algorithm for Solving Cross-matching Puzzles

Abstract

The aim of this paper it to analyze the cross-matching puzzle and to propose a fast and deterministic algorithm that can solve it. Nevertheless, there is a bigger goal than designing an algorithm for a particular problem. We want to show that while AI researchers constantly look for new constraint-satisfaction problems that could be utilized for testing various problem-solving techniques it is possible to come up with the problem that can be solved by much simpler algorithms. We would like to stress that there is an important misconception related to NP class that a huge number of potential solutions to the specific problem almost automatically implies that the relevant problem belongs to the class of NP. Such a misunderstanding and misclassification of the particular problem leads to false impression that there is no chance to design a simple and fast algorithm for the problem. Therefore, various heuristics or general problem-solving techniques are unnecessarily employed in order to solve it. And moreover, the wrong impression that the problem is difficult is further supported. We believe that our paper can help to raise the awareness that not all the problems with immense search spaces are hard to be solved and the polynomial algorithm to tackle the cross-matching puzzle that is described here is a good example of such an approach.