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In this clip, Donald Knuth tells Lex Fridman that he suspects P equals NP, but that a proof would not hand us useful algorithms. He points to the game of Hex and the Robertson–Seymour graph minor theorem as cases where algorithms provably exist yet remain unknown. His intuition rests on the sheer number of possible algorithms, though he stresses it is no proof.
- 1An algorithm can exist without anyone knowing itKnuth argues that the total number of algorithms is mind-boggling, and most are far beyond human comprehension, so a proof of P = NP would not directly yield a practical method.
- 2Hex shows existence without constructionThe game cannot end in a draw, so one of the two players must have a winning strategy, yet nobody knows an algorithm for finding it.
- 3The Robertson–Seymour theorem guarantees polynomial-time algorithms we cannot writeAny class of graphs closed under minors is characterized by a finite list of forbidden minors, but we may not know that list.
- 4The intuition for P = NP comes from the size of the search spaceWith so many possible algorithms, Knuth says, surprising coincidences are to be expected, much like two people sharing the same number of hairs.
- 5Failure to find an algorithm is weak evidenceKnuth notes that the same logic could be flipped, since many smart people have also failed to prove P is unequal to NP.
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A proof that an algorithm exists is not an algorithm
Fridman opens by recalling that Donald Knuth has said he believes P may equal NP, while also arguing that such a proof would not directly lead to algorithms for hard problems. Knuth explains that the popular assumption is that if an algorithm exists, somebody will eventually find it. He says that is wrong: many more algorithms exist than anyone can ever understand or discover, because they are way beyond human comprehension.
To make the point concrete, Knuth turns to the game of Hex, played by placing pebbles on a hexagonal board. One player tries to connect left to right and the other bottom to top, there is no capture, and, crucially, there are no draws. Because it is a finite game of perfect information, one of the two players must have a winning strategy.
Knuth says you can even show that it has to be one or the other, using an argument about the second player mimicking the first. Yet this tells us nothing about which player wins or how to play. In his words, we know the algorithm exists but have no idea what it is.
Graph minors give polynomial algorithms nobody can write down
Knuth's second example is the Robertson–Seymour theorem from graph theory. It says that every class of graphs closed under taking minors has a polynomial-time membership test. Planar graphs are the standard example: shrinking an edge to a point or deleting an edge leaves a planar graph planar. Knuth says the proof spans a sequence of some twenty papers and is deep work.
The key result is that any such family has a finite number of minimal forbidden graphs. For planar graphs there are two, the five-pointed star in which everything connects to everything and the three-utilities-to-three-houses graph. Any non-planar graph can be shrunk down until it contains one of these two bad graphs. Knuth notes that the theorem applies to any arbitrary minor-closed class, and that many important classes arising in applications have this property.
Knuth then walks through why this gives a polynomial algorithm. You test whether the input graph contains each bad minor, each test takes polynomial time, and there are only finitely many tests. The catch is that for an arbitrary class we only know the list is finite. We might know one or two of the forbidden minors while there could be 125 more, so we hold a polynomial-time algorithm that we cannot actually write.
Why Knuth still leans toward P = NP, though not by proof
Fridman asks why Knuth holds the intuition that P equals NP at all. Knuth answers that to believe otherwise you would have to rule out an enormous number of possible algorithms. He gives an example of an odd but perfectly valid polynomial-time procedure, such as encoding a graph with prime numbers and combining them with bitwise operations. There are so many algorithms of this kind that some random-looking one may well happen to solve a problem.
He compares this to coincidences in everyday counting. It seems unlikely that two people have the same number of hairs on their heads, but given how many people there are, such pairs must exist. Likewise, when the space of possibilities is huge, a particular combination of operations might just happen to prove that a graph has a Hamiltonian path. Knuth emphasizes that this is by no means a proof.
He also rejects the opposite argument that P cannot equal NP because brilliant people and million-dollar prizes have not produced an algorithm. By the same logic, he says, one could argue P must equal NP because so many smart people have tried and failed to prove otherwise. Fridman likens it to the search for aliens, and Knuth agrees that the number of planets makes their existence plausible, even if they have not been found.
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