Why Pigeons at Rest Are at the Center of Complexity Theory

By January 2020, Papadimitriou had been reasoning astir the pigeonhole rule for 30 years. So helium was amazed erstwhile a playful speech with a predominant collaborator led them to a elemental twist connected the rule that they’d ne'er considered: What if determination are less pigeons than holes? In that case, immoderate statement of pigeons indispensable permission immoderate bare holes. Again, it seems obvious. But does inverting the pigeonhole rule person immoderate absorbing mathematical consequences?

It whitethorn dependable arsenic though this “empty-pigeonhole” rule is conscionable the archetypal 1 by different name. But it’s not, and its subtly antithetic quality has made it a caller and fruitful instrumentality for classifying computational problems.

To recognize the empty-pigeonhole principle, let’s spell backmost to the bank-card example, transposed from a shot stadium to a performance hallway with 3,000 seats—a smaller fig than the full imaginable four-digit PINs. The empty-pigeonhole rule dictates that immoderate imaginable PINs aren’t represented astatine all. If you privation to find 1 of these missing PINs, though, determination doesn’t look to beryllium immoderate amended mode than simply asking each idiosyncratic their PIN. So far, the empty-pigeonhole rule is conscionable similar its much celebrated counterpart.

The quality lies successful the trouble of checking solutions. Imagine that idiosyncratic says they’ve recovered 2 circumstantial radical successful the shot stadium who person the aforesaid PIN. In this case, corresponding to the archetypal pigeonhole scenario, there’s a elemental mode to verify that claim: Just cheque with the 2 radical successful question. But successful the performance hallway case, ideate that idiosyncratic asserts that nary idiosyncratic has a PIN of 5926. Here, it’s intolerable to verify without asking everyone successful the assemblage what their PIN is. That makes the empty-pigeonhole rule overmuch much vexing for complexity theorists.

Two months aft Papadimitriou began reasoning astir the empty-pigeonhole principle, helium brought it up successful a speech with a prospective postgraduate student. He remembers it vividly, due to the fact that it turned retired to beryllium his past in-person speech with anyone earlier the Covid-19 lockdowns. Cooped up astatine location implicit the pursuing months, helium wrestled with the problem’s implications for complexity theory. Eventually helium and his colleagues published a insubstantial astir hunt problems that are guaranteed to person solutions due to the fact that of the empty-pigeonhole principle. They were particularly funny successful problems wherever pigeonholes are abundant—that is, wherever they acold outnumber pigeons. In keeping with a contented of unwieldy acronyms successful complexity theory, they dubbed this people of problems APEPP, for “abundant polynomial empty-pigeonhole principle.”

One of the problems successful this people was inspired by a celebrated 70-year-old impervious by the pioneering machine idiosyncratic Claude Shannon. Shannon proved that astir computational problems indispensable beryllium inherently hard to solve, utilizing an statement that relied connected the empty-pigeonhole rule (though helium didn’t telephone it that). Yet for decades, machine scientists person tried and failed to beryllium that circumstantial problems are genuinely hard. Like missing bank-card PINs, hard problems indispensable beryllium retired there, adjacent if we can’t place them.

Historically, researchers haven’t thought astir the process of looking for hard problems arsenic a hunt occupation that could itself beryllium analyzed mathematically. Papadimitriou’s approach, which grouped that process with different hunt problems connected to the empty-pigeonhole principle, had a self-referential spirit diagnostic of overmuch caller enactment successful complexity theory—it offered a caller mode to crushed astir the trouble of proving computational difficulty.