Until January 2020, Papadimitrioou thought for 30 years about the principle of a drawer. So he was surprised when a comical conversation with a recurrent collaborator led them to a uncomplicated return of the principle that they never considered: what if there is less pigeons than holes? In this case, each pigeon system must leave empty holes. It seems obvious again. But does the reversal of the drawer principle have any fascinating mathematical consequences?
It may sound as if this “empty pigenhole” principle was simply original after the second name. But this is not the case, and his subtly different character has made him a recent and fruitful tool for classification of computing problems.
To understand the principle of empty pigenhole, let’s return to the example of banking cards, transposed from the football stadium to the concert hall with 3000 places-a smaller number than the total possible four-digit high heels. The principle of empty pigenhole decides that some possible heels are not represented at all. However, if you want to find one of these missing pins, it seems that there is no better way than the usual question of each person. So far, the principle of empty Pigeonhole is like his more known counterpart.
The difference is the difficulties of checking solutions. Imagine that someone says that he found two specific people at the football stadium. In this case, when corresponding to the original script of the drawer, there is a uncomplicated way to verify this claim: just contact two people. But in the case of the concert hall, imagine that someone claims that no person has a pin of 5926. Here you can not verify without asking everyone in the audience what their pins are. This makes the principle of empty Pigeonhole much more annoying for theoretics of complexity.
Two months after Papadimitriou began to think about the principle of empty Pigenhole, he moved her in an interview with a potential graduate. He remembers it vividly, because it turned out that it was his last personal conversation with anyone in front of Covid-19 curls. Over the next months he stopped at home, struggled with implications of a problem for the theory of complexity. Finally he and his colleagues published paper About search problems that guarantee solutions due to the principle of empty pigenomole. They were particularly interested in problems in which the drawers are bountiful – that is, where they significantly exceed the number of pigeons. According to tradition Incorrect acronym In the theory of complexity, they called this class of APEPS problems, for the “abundant multi -core principle of empty Pigeonhole.”
One of the problems in this class was inspired by the celebrated 70-year evidence by pioneering computer science Claude Shannon. Shannon proved that most of the computing problems must be challenging to solve by nature, using an argument that was based on an empty pigenomol (although he did not call it). However, for decades, IT specialists have tried and not proved that specific problems are really challenging. Like the missing pins with banking cards, challenging problems must be there, even if we cannot identify them.
Historically, researchers did not think about the process of searching for challenging problems as a search problem that could be examined mathematically. Papadimitrou’s approach, which grouped this process with other problems related to searching related to the principle of empty Pigeonhole, had a characteristic characteristic taste Many of the latest works In the theory of complexity – he offered a recent way of reasoning about difficulties in proving computing difficulties.