Representation Changes in CombinatorialProblems :

The Pigeonhole Principle (PHP) has been one of the most appealing methods of solving combinatorial optimization problems. Variations of the Pigeonhole Principle, sometimes called the \Hidden" Pigeonhole Principle (HPHP), are even more powerful and often produce the most elegant solutions to nontrivial problems. However, some Operations Research approaches, such as the Linear Programming Relaxation (LPR), are strong competitors to PHP and HPHP. They can also be applied to combinatorial optimization problems to derive upper bounds. It has been an open question whether PHP or LPR establish tighter upper bounds and how eeciently, when applied to the same problem. Challenged by this open question, we identify that the main reason for the lack of ability to compare the eeciency of PHP and LPR is the fact that diierent problem representations are required by the two methods. We introduce a problem representation change into an Integer Programming form which allows for an alternative way of solving combinatorial problems. We also introduce several combinatorial optimization problems, and show how to perform representation changes to convert the original problems into the Integer Programming form. Using the new problem model, we re-deene the Pigeonhole Principle as a method of solving Integer Programming problems, determine the diierence between PHP and HPHP, prove that PHP has the same bounding power as LPR, and demonstrate that HPHP and Integer cuts are actually similar representation changes of the problem domains. The choice of representation is well-known to be one of the most diicult parameters to adjust and evaluate in machine learning algorithms. This work is therefore of particular relevance to machine learning researchers interested in understanding the impact of diierent representations. By analyzing representations used in the Operational Research area, we study and demonstrate the impact of representation choices in problem solving eeciency.