Why the bathroom in comfort in the center of the theory of complexity

By January 2020, Babadmeterio was considering the principle of a 30 -year dove. So he was surprised when a fun conversation led them with a repeated collaborator to a slight development about the principle in which they never thought: What if there was a bath less than holes? In this case, any arrangement of the bathroom should leave some empty holes. Again, it looks clear. But does the mind of the principle of a pigeon have any interesting sporting consequences?

It might seem as if the “empty Pigonol” principle is just the original name with another. But it is not, and it has made her character skillfully with a new and fruitful tool to classify mathematical problems.

To understand the principle of empty pigeon, let’s go back to the example of the bank card, transported from the football field to a 3000-seat concert hall-which is smaller than the total number of four possible numbers. The principle of empty vacuum dictates that some possible nails do not represent at all. If you want to find one of these lost nails, though, it does not seem to be a better way than just asking everyone. To date, the empty pigeon principle is similar to its most famous counterpart.

The difference lies in the difficulty of checking solutions. Imagine someone who says he found two specific people in the football field and have the same pin. In this case, the corresponding to the original bathroom scenario, there is a simple way to check this claim: just check with the two persons concerned. But in the concert hall, imagine that someone confirms that no one has a pin of 5926. Here, it is impossible to check without asking everyone in the audience. This makes the empty pigeon principle more harassed by the theory of complexity.

Two months after Papadimitrio began thinking about the principle of empty pigeon, he presented it in a conversation with a potential graduate student. He clearly remembers it, because it turned out to be his last personal conversation with anyone before Covid-19s. You have at home during the following months, struggling with the consequences of the problem on the theory of complexity. In the end, he and his colleagues published a paper About search problems that ensure that they have solutions due to the principle of empty pigeon. They were particularly interested in the problems where the bathroom holes were abundant – that is, the number of pigeons exceeds. Watching with a tradition Unbreakable shortcuts In the theory of complexity, they called this category of Apepp, for “the principle of abundance empty borders.”

One of the problems in this chapter was inspired by a famous A 70 -year -old guide By the world of the pioneering computer Claude Shannon. Shannon has proven that most of the calculations should be difficult to solve by nature, using an empty pigeon argument (although he did not call it). However, for decades, computer scientists have tried to prove that the specific problems are really difficult. Like the bank card is missing, there should be difficult problems, even if we cannot identify them.

Historically, researchers did not think about the search for difficult problems as a research problem that can be analyzed mathematically. The PapadimitRIou approach, which assembled this process with other research problems related to the Bigonol Empty principle, had a self -reference flavor A lot of modern work In the theory of complexity – a new way to cause difficulty proving the arithmetic difficulty was offered.