By January 2020, Papadimitriou had been fascinated by the pigeonhole precept for 30 years. So he was stunned when a playful dialog with a frequent collaborator led them to a easy twist on the precept that they’d by no means thought of: What if there are fewer pigeons than holes? In that case, any association of pigeons should depart some empty holes. Once more, it appears apparent. However does inverting the pigeonhole precept have any attention-grabbing mathematical penalties?
It might sound as if this “empty-pigeonhole” precept is simply the unique one by one other title. Nevertheless it’s not, and its subtly completely different character has made it a brand new and fruitful device for classifying computational issues.
To grasp the empty-pigeonhole precept, let’s return to the bank-card instance, transposed from a soccer stadium to a live performance corridor with 3,000 seats—a smaller quantity than the whole doable four-digit PINs. The empty-pigeonhole precept dictates that some doable PINs aren’t represented in any respect. If you wish to discover considered one of these lacking PINs, although, there doesn’t appear to be any higher means than merely asking every particular person their PIN. To date, the empty-pigeonhole precept is rather like its extra well-known counterpart.
The distinction lies within the issue of checking options. Think about that somebody says they’ve discovered two particular individuals within the soccer stadium who’ve the identical PIN. On this case, comparable to the unique pigeonhole state of affairs, there’s a easy solution to confirm that declare: Simply verify with the 2 individuals in query. However within the live performance corridor case, think about that somebody asserts that no particular person has a PIN of 5926. Right here, it’s inconceivable to confirm with out asking everybody within the viewers what their PIN is. That makes the empty-pigeonhole precept way more vexing for complexity theorists.
Two months after Papadimitriou started fascinated by the empty-pigeonhole precept, he introduced it up in a dialog with a potential graduate scholar. He remembers it vividly, as a result of it turned out to be his final in-person dialog with anybody earlier than the Covid-19 lockdowns. Cooped up at house over the next months, he wrestled with the issue’s implications for complexity concept. Ultimately he and his colleagues printed a paper about search issues which might be assured to have options due to the empty-pigeonhole precept. They have been particularly thinking about issues the place pigeonholes are plentiful—that’s, the place they far outnumber pigeons. Consistent with a practice of unwieldy acronyms in complexity concept, they dubbed this class of issues APEPP, for “plentiful polynomial empty-pigeonhole precept.”
One of many issues on this class was impressed by a well-known 70-year-old proof by the pioneering laptop scientist Claude Shannon. Shannon proved that the majority computational issues have to be inherently onerous to resolve, utilizing an argument that relied on the empty-pigeonhole precept (although he didn’t name it that). But for many years, laptop scientists have tried and did not show that particular issues are really onerous. Like lacking bank-card PINs, onerous issues have to be on the market, even when we will’t establish them.
Traditionally, researchers haven’t thought concerning the technique of on the lookout for onerous issues as a search downside that would itself be analyzed mathematically. Papadimitriou’s method, which grouped that course of with different search issues linked to the empty-pigeonhole precept, had a self-referential taste attribute of much recent work in complexity concept—it supplied a brand new solution to purpose concerning the issue of proving computational issue.
