Why pigeons are at rest in the center of complexity theory
By January 2020, Papadimitriv had been thinking about the pigeon’s principle for 30 years. So he was amazed when he was a playful conversation with a repeated colleague to a simple twist on the principles he never considered: What if there is less pigeon than the hole? In this case, any arrangement of pigeons must leave empty holes. Again, it looks clear. But does the reversal of the pigeon’s principle have some interesting mathematical consequences?
It may seem that the “empty hole” principle is only with another original name. But that’s not the case, and its completely different personality has made it a new and fruitful tool to classify computational problems.
To understand the principle of the blank page, let’s go back to the bank card sample, which has been moved from a football stadium to a concert hall with 3,000 seats-fewer than total four-digit pins. The principle of empty Peugeot indicates that some possible pins are not shown at all. If you want to find one of these missing pins, it doesn’t seem to be a better way to ask each person your pin. So far, the principle of empty Peugeot is exactly the same as its famous counterpart.
The difference lies in the difficulty of examining the solutions. Imagine that someone says they have found two special people in the football stadium who have the same pin. In this case, according to the original pigeon scenario, there is a simple way to confirm this claim: Check with only two people in question. But in the case of the concert hall, imagine that someone claims that no one has 5926 pins. Here, it is impossible to confirm all of the audience. This makes it very uncomfortable for complexity theorists.
Two months after Papadimitrio began to think about the principle of empty Peugeot, he discussed it in a conversation with a forward -looking graduate. He recalls it clearly, because it turned out to be his latest in-person conversation with anyone before Coveid-19 locks. In the following months, he collaborated at home, with the consequences for the theory of complexity of the ship. Eventually, he and his colleagues published an article on search problems that are guaranteed because of the principle of empty Peugeot. They were particularly interested in the problems that the pigeons are abundant – where they were away from the pigeons. In accordance with the tradition of unpleasant abbreviation in the theory of complexity, they called this class of APEPPP problems for the “polynomial principle devoid of empty haunted”.
One of the problems in this class was inspired by a famous 70 -year -old proof by the pioneer computer scientist Claude Shannon. Shannon proved that most computational problems must be solved inherently, using an argument that relies on the principle of empty Peugeot (though he did not call it). However, for decades, computer scientists have tried and have failed to prove that specific problems are really difficult. Like the disappearance of bank card pins, there should be difficult problems there, even if we cannot identify them.
Historically, the researchers have not thought about the searches of difficult problems as a search problem that can be mathematically analyzed. The Papadimitriv approach, which has grown this process with other search problems related to the principle of empty hole, has a self-reference taste from a very recent work in complexity theory-provides a new way to argue about difficulty proving computational difficulties.
