What if P equals NP?

“You don’t have to believe in God, but you should believe in The Book.”, Paul Erdős, 1985.¹

“No more half measures, Walter”, Mike Ehrmantraut in “Breaking Bad”, 2010.

“The evidence in favor of [\(\mathbf{P}\neq \mathbf{NP}\)] and [ its algebraic counterpart ] is so overwhelming, and the consequences of their failure are so grotesque, that their status may perhaps be compared to that of physical laws rather than that of ordinary mathematical conjectures.”, Volker Strassen, laudation for Leslie Valiant, 1986.

“Suppose aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the [fifth Ramsey number]. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the [sixth Ramsey number], however, we would have no choice but to launch a preemptive attack.”, Paul Erdős, as quoted by Graham and Spencer, 1990.²

We have mentioned that the question of whether \(\mathbf{P}=\mathbf{NP}\), which is equivalent to whether there is a polynomial-time algorithm for \(3\ensuremath{\mathit{SAT}}\), is the great open question of Computer Science. But why is it so important? In this chapter, we will try to figure out the implications of such an algorithm.

First, let us get one qualm out of the way. Sometimes people say, “What if \(\mathbf{P}=\mathbf{NP}\) but the best algorithm for 3SAT takes \(n^{1000}\) time?” Well, \(n^{1000}\) is much larger than, say, \(2^{0.001\sqrt{n}}\) for any input smaller than \(2^{50}\), as large as a harddrive as you will encounter, and so another way to phrase this question is to say “what if the complexity of 3SAT is exponential for all inputs that we will ever encounter, but then grows much smaller than that?” To me this sounds like the computer science equivalent of asking, “what if the laws of physics change completely once they are out of the range of our telescopes?”. Sure, this is a valid possibility, but wondering about it does not sound like the most productive use of our time.

So, as the saying goes, we’ll keep an open mind, but not so open that our brains fall out, and assume from now on that:

• There is a mathematical god,

and

• She does not “beat around the bush” or take “half measures”.

What we mean by this is that we will consider two extreme scenarios:

• 3SAT is very easy: \(3\ensuremath{\mathit{SAT}}\) has an \(O(n)\) or \(O(n^2)\) time algorithm with a not too huge constant (say smaller than \(10^6\).)

• 3SAT is very hard: \(3\ensuremath{\mathit{SAT}}\) is exponentially hard and cannot be solved faster than \(2^{\epsilon n}\) for some not too tiny \(\epsilon>0\) (say at least \(10^{-6}\)). We can even make the stronger assumption that for every sufficiently large \(n\), the restriction of \(3\ensuremath{\mathit{SAT}}\) to inputs of length \(n\) cannot be computed by a circuit of fewer than \(2^{\epsilon n}\) gates.

At the time of writing, the fastest known algorithm for \(3\ensuremath{\mathit{SAT}}\) requires more than \(2^{0.35 n}\) to solve \(n\) variable formulas, while we do not even know how to rule out the possibility that we can compute \(3\ensuremath{\mathit{SAT}}\) using \(10n\) gates. To put it in perspective, for the case \(n=1000\) our lower and upper bounds for the computational costs are apart by a factor of about \(10^{100}\). As far as we know, it could be the case that \(1000\)-variable \(3\ensuremath{\mathit{SAT}}\) can be solved in a millisecond on a first-generation iPhone, and it can also be the case that such instances require more than the age of the universe to solve on the world’s fastest supercomputer.

So far, most of our evidence points to the latter possibility of 3SAT being exponentially hard, but we have not ruled out the former possibility either. In this chapter we will explore some of the consequences of the “\(3\ensuremath{\mathit{SAT}}\) easy” scenario.

Search-to-decision reduction

A priori, having a fast algorithm for 3SAT might not seem so impressive. Sure, such an algorithm allows us to decide the satisfiability of not just 3CNF formulas but also of quadratic equations, as well as find out whether there is a long path in a graph, and solve many other decision problems. But this is not typically what we want to do. It’s not enough to know if a formula is satisfiable: we want to discover the actual satisfying assignment. Similarly, it’s not enough to find out if a graph has a long path: we want to actually find the path.

It turns out that if we can solve these decision problems, we can solve the corresponding search problems as well:

Optimization

Theorem 16.1 allows us to find solutions for \(\mathbf{NP}\) problems if \(\mathbf{P}=\mathbf{NP}\), but it is not immediately clear that we can find the optimal solution. For example, suppose that \(\mathbf{P}=\mathbf{NP}\), and you are given a graph \(G\). Can you find the longest simple path in \(G\) in polynomial time?

The answer is Yes. The idea is simple: if \(\mathbf{P}=\mathbf{NP}\) then we can find out in polynomial time if an \(n\)-vertex graph \(G\) contains a simple path of length \(n\), and moreover, by Theorem 16.1, if \(G\) does contain such a path, then we can find it. (Can you see why?) If \(G\) does not contain a simple path of length \(n\), then we will check if it contains a simple path of length \(n-1\), and continue in this way to find the largest \(k\) such that \(G\) contains a simple path of length \(k\).

The above reasoning was not specifically tailored to finding paths in graphs. In fact, it can be vastly generalized to proving the following result:

Example: Supervised learning

One classical optimization task is supervised learning. In supervised learning we are given a list of examples \(x_0,x_1,\ldots,x_{m-1}\) (where we can think of each \(x_i\) as a string in \(\{0,1\}^n\) for some \(n\)) and the labels for them \(y_0,\ldots,y_{m-1}\) (which we will think of simply bits, i.e., \(y_i\in \{0,1\}\)). For example, we can think of the \(x_i\)’s as images of either dogs or cats, for which \(y_i=1\) in the former case and \(y_i=0\) in the latter case. Our goal is to come up with a hypothesis or predictor \(h:\{0,1\}^n \rightarrow \{0,1\}\) such that if we are given a new example \(x\) that has an (unknown to us) label \(y\), then with high probability \(h\) will predict the label. That is, with high probability it will hold that \(h(x)=y\). The idea in supervised learning is to use the Occam’s Razor principle: the simplest hypothesis that explains the data is likely to be correct. There are several ways to model this, but one popular approach is to pick some fairly simple function \(H:\{0,1\}^{k+n} \rightarrow \{0,1\}\). We think of the first \(k\) inputs as the parameters and the last \(n\) inputs as the example data. (For example, we can think of the first \(k\) inputs of \(H\) as specifying the weights and connections for some neural network that will then be applied on the latter \(n\) inputs.) We can then phrase the supervised learning problem as finding, given a set of labeled examples \(S=\{ (x_0,y_0),\ldots,(x_{m-1},y_{m-1}) \}\), the set of parameters \(\theta_0,\ldots,\theta_{k-1} \in \{0,1\}\) that minimizes the number of errors made by the predictor \(x \mapsto H(\theta,x)\) (This is often known as Empirical Risk Minimization.)

In other words, we can define for every set \(S\) as above the function \(F_S:\{0,1\}^k \rightarrow [m]\) such that \(F_S(\theta) = \sum_{(x,y)\in S} |H(\theta,x)-y|\). Now, finding the value \(\theta\) that minimizes \(F_S(\theta)\) is equivalent to solving the supervised learning problem with respect to \(H\). For every polynomial-time computable \(H:\{0,1\}^{k+n} \rightarrow \{0,1\}\), the task of minimizing \(F_S(\theta)\) can be “massaged” to fit the form of Theorem 16.3 and hence if \(\mathbf{P}=\mathbf{NP}\), then we can solve the supervised learning problem in great generality. In fact, this observation extends to essentially any learning model, and allows for finding the optimal predictors given the minimum number of examples. (This is in contrast to many current learning algorithms, which often rely on having access to an extremely large number of examples\(-\) far beyond the minimum needed, and in particular far beyond the number of examples humans use for the same tasks.)

Example: Breaking cryptosystems

We will discuss cryptography later in this course, but it turns out that if \(\mathbf{P}=\mathbf{NP}\) then almost every cryptosystem can be efficiently broken. One approach is to treat finding an encryption key as an instance of a supervised learning problem. If there is an encryption scheme that maps a “plaintext” message \(p\) and a key \(\theta\) to a “ciphertext” \(c\), then given examples of ciphertext/plaintext pairs of the form \((c_0,p_0),\ldots,(c_{m-1},p_{m-1})\), our goal is to find the key \(\theta\) such that \(E(\theta,p_i)=c_i\) where \(E\) is the encryption algorithm. While you might think getting such “labeled examples” is unrealistic, it turns out (as many amateur home-brew crypto designers learn the hard way) that this is actually quite common in real-life scenarios, and that it is also possible to relax the assumption to having more minimal prior information about the plaintext (e.g., that it is English text). We defer a more formal treatment to Chapter 21.

Finding mathematical proofs

In the context of Gödel’s Theorem, we discussed the notion of a proof system (see Section 11.1). Generally speaking, a proof system can be thought of as an algorithm \(V:\{0,1\}^* \rightarrow \{0,1\}\) (known as the verifier) such that given a statement \(x\in \{0,1\}^*\) and a candidate proof \(w\in \{0,1\}^*\), \(V(x,w)=1\) if and only if \(w\) encodes a valid proof for the statement \(x\). Any type of proof system that is used in mathematics for geometry, number theory, analysis, etc., is an instance of this form. In fact, standard mathematical proof systems have an even simpler form where the proof \(w\) encodes a sequence of lines \(w^0,\ldots,w^m\) (each of which is itself a binary string) such that each line \(w^i\) is either an axiom or follows from some prior lines through an application of some inference rule. For example, Peano’s axioms encode a set of axioms and rules for the natural numbers, and one can use them to formalize proofs in number theory. Also, there are some even stronger axiomatic systems, the most popular one being Zermelo–Fraenkel with the Axiom of Choice or ZFC for short. Thus, although mathematicians typically write their papers in natural language, proofs of number theorists can typically be translated to ZFC or similar systems, and so in particular the existence of an \(n\)-page proof for a statement \(x\) implies that there exists a string \(w\) of length \(poly(n)\) (in fact often \(O(n)\) or \(O(n^2)\)) that encodes the proof in such a system. Moreover, because verifying a proof simply involves going over each line and checking that it does indeed follow from the prior lines, it is fairly easy to do that in \(O(|w|)\) or \(O(|w|^2)\) (where as usual \(|w|\) denotes the length of the proof \(w\)). This means that for every reasonable proof system \(V\), the following function \(\ensuremath{\mathit{SHORTPROOF}}_V:\{0,1\}^* \rightarrow \{0,1\}\) is in \(\mathbf{NP}\), where for every input of the form \(x1^m\), \(\ensuremath{\mathit{SHORTPROOF}}_V(x,1^m)=1\) if and only if there exists \(w\in \{0,1\}^*\) with \(|w|\leq m\) s.t. \(V(xw)=1\). That is, \(\ensuremath{\mathit{SHORTPROOF}}_V(x,1^m)=1\) if there is a proof (in the system \(V\)) of length at most \(m\) bits that \(x\) is true. Thus, if \(\mathbf{P}=\mathbf{NP}\), then despite Gödel’s Incompleteness Theorems, we can still automate mathematics in the sense of finding proofs that are not too long for every statement that has one. (Frankly speaking, if the shortest proof for some statement requires a terabyte, then human mathematicians won’t ever find this proof either.) For this reason, Gödel himself felt that the question of whether \(\ensuremath{\mathit{SHORTPROOF}}_V\) has a polynomial time algorithm is of great interest. As Gödel wrote in a letter to John von Neumann in 1956 (before the concept of \(\mathbf{NP}\) or even “polynomial time” was formally defined):

One can obviously easily construct a Turing machine, which for every formula \(F\) in first order predicate logic and every natural number \(n\), allows one to decide if there is a proof of \(F\) of length \(n\) (length = number of symbols). Let \(\psi(F,n)\) be the number of steps the machine requires for this and let \(\varphi(n) = \max_F \psi(F,n)\). The question is how fast \(\varphi(n)\) grows for an optimal machine. One can show that \(\varphi \geq k \cdot n\) [for some constant \(k>0\)]. If there really were a machine with \(\varphi(n) \sim k \cdot n\) (or even \(\sim k\cdot n^2\)), this would have consequences of the greatest importance. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem,³ the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine. After all, one would simply have to choose the natural number \(n\) so large that when the machine does not deliver a result, it makes no sense to think more about the problem.

For many reasonable proof systems (including the one that Gödel referred to), \(\ensuremath{\mathit{SHORTPROOF}}_V\) is in fact \(\mathbf{NP}\)-complete, and so Gödel can be thought of as the first person to formulate the \(\mathbf{P}\) vs \(\mathbf{NP}\) question. Unfortunately, the letter was only discovered in 1988.

Quantifier elimination (advanced)

If \(\mathbf{P}=\mathbf{NP}\) then we can solve all \(\mathbf{NP}\) search and optimization problems in polynomial time. But can we do more? It turns out that the answer is that Yes we can!

An \(\mathbf{NP}\) decision problem can be thought of as the task of deciding, given some string \(x\in \{0,1\}^*\) the truth of a statement of the form

for some polynomial-time algorithm \(V\) and polynomial \(p:\N \rightarrow \N\). That is, we are trying to determine, given some string \(x\), whether there exists a string \(y\) such that \(x\) and \(y\) satisfy some polynomial-time checkable condition \(V\). For example, in the independent set problem, the string \(x\) represents a graph \(G\) and a number \(k\), the string \(y\) represents some subset \(S\) of \(G\)’s vertices, and the condition that we check is whether \(|S| \geq k\) and there is no edge \(\{u,v\}\) in \(G\) such that both \(u\in S\) and \(v\in S\).

We can consider more general statements such as checking, given a string \(x\in \{0,1\}^*\), the truth of a statement of the form

which in words corresponds to checking, given some string \(x\), whether there exists a string \(y\) such that for every string \(z\), the triple \((x,y,z)\) satisfy some polynomial-time checkable condition. We can also consider more levels of quantifiers such as checking the truth of the statement

\[ \exists_{y \in \{0,1\}^{p_0(|x|)}} \forall_{z\in \{0,1\}^{p_1(|x|)}}\exists_{w\in \{0,1\}^{p_2(|x|)}} V(xyzw)=1 \;\;(16.2) \]

and so on and so forth.

For example, given an \(n\)-input NAND-CIRC program \(P\), we might want to find the smallest NAND-CIRC program \(P'\) that computes the same function as \(P\). The question of whether there is such a \(P'\) that can be described by a string of at most \(s\) bits can be phrased as

which has the form Equation 16.1. (Since NAND-CIRC programs are equivalent to Boolean circuits, the search problem corresponding to Equation 16.3 known as the circuit minimization problem and is widely studied in Engineering. You can skip ahead to Section 16.4.1 to see a particularly compelling application of this.)

Another example of a statement involving \(a\) levels of quantifiers would be to check, given a chess position \(x\), whether there is a strategy that guarantees that White wins within \(a\) steps. For example is \(a=3\) we would want to check if given the board position \(x\), there exists a move \(y\) for White such that for every move \(z\) for Black there exists a move \(w\) for White that ends in a a checkmate.

It turns out that if \(\mathbf{P}=\mathbf{NP}\) then we can solve these kinds of problems as well:

The algorithm of Theorem 16.6 can also solve the search problem as well: find the value \(y_0\) that certifies the truth of Equation 16.4. We note that while this algorithm is in polynomial time, the exponent of this polynomial blows up quite fast. If the original NANDSAT algorithm required \(\Omega(n^2)\) time, solving \(a\) levels of quantifiers would require time \(\Omega(n^{2^a})\).⁴

Application: self improving algorithm for \(3\ensuremath{\mathit{SAT}}\)

Suppose that we found a polynomial-time algorithm \(A\) for \(3\ensuremath{\mathit{SAT}}\) that is “good but not great”. For example, maybe our algorithm runs in time \(cn^2\) for some not too small constant \(c\). However, it’s possible that the best possible SAT algorithm is actually much more efficient than that. Perhaps, as we guessed before, there is a circuit \(C^*\) of at most \(10^6 n\) gates that computes 3SAT on \(n\) variables, and we simply haven’t discovered it yet. We can use Theorem 16.6 to “bootstrap” our original “good but not great” 3SAT algorithm to discover the optimal one. The idea is that we can phrase the question of whether there exists a size \(s\) circuit that computes 3SAT for all length \(n\) inputs as follows: there exists a size \(\leq s\) circuit \(C\) such that for every formula \(\varphi\) described by a string of length at most \(n\), if \(C(\varphi)=1\) then there exists an assignment \(x\) to the variables of \(\varphi\) that satisfies it. One can see that this is a statement of the form Equation 16.2 and hence if \(\mathbf{P}=\mathbf{NP}\) we can solve it in polynomial time as well. We can therefore imagine investing huge computational resources in running \(A\) one time to discover the circuit \(C^*\) and then using \(C^*\) for all further computation.

Approximating counting problems and posterior sampling (advanced, optional)

Given a Boolean circuit \(C\), if \(\mathbf{P}=\mathbf{NP}\) then we can find an input \(x\) (if one exists) such that \(C(x)=1\). But what if there is more than one \(x\) like that? Clearly we can’t efficiently output all such \(x\)’s; there might be exponentially many. But we can get an arbitrarily good multiplicative approximation (i.e., a \(1\pm \epsilon\) factor for arbitrarily small \(\epsilon>0\)) for the number of such \(x\)’s, as well as output a (nearly) uniform member of this set. The details are beyond the scope of this book, but this result is formally stated in the following theorem (whose proof is omitted).

In other words, the algorithm \(\ensuremath{\mathit{COUNT}}_V\) gives an approximation up to a factor of \(1 \pm \epsilon\) for the number of witnesses for \(x\) with respect to the verifying algorithm \(V\). Once again, to understand this theorem it can be useful to see how it implies that if \(\mathbf{P}=\mathbf{NP}\) then there is a polynomial-time algorithm that given a graph \(G\) and a number \(k\), can compute a number \(K\) that is within a \(1 \pm 0.01\) factor equal to the number of simple paths in \(G\) of length \(k\). (That is, \(K\) is between \(0.99\) to \(1.01\) times the number of such paths.)

Posterior sampling and probabilistic programming. The algorithm for counting can also be extended to sampling from a given posterior distribution. That is, if \(C:\{0,1\}^n \rightarrow \{0,1\}^m\) is a Boolean circuit and \(y\in \{0,1\}^m\), then if \(\mathbf{P}=\mathbf{NP}\) we can sample from (a close approximation of) the distribution of uniform \(x\in \{0,1\}^n\) conditioned on \(C(x)=y\). This task is known as posterior sampling and is crucial for Bayesian data analysis. These days it is known how to achieve posterior sampling only for circuits \(C\) of very special form, and even in these cases more often than not we do have guarantees on the quality of the sampling algorithm. The field of making inferences by sampling from posterior distribution specified by circuits or programs is known as probabilistic programming.

What does all of this imply?

So, what will happen if we have a \(10^6n\) algorithm for \(3\ensuremath{\mathit{SAT}}\)? We have mentioned that \(\mathbf{NP}\)-hard problems arise in many contexts, and indeed scientists, engineers, programmers and others routinely encounter such problems in their daily work. A better \(3\ensuremath{\mathit{SAT}}\) algorithm will probably make their lives easier, but that is the wrong place to look for the most foundational consequences. Indeed, while the invention of electronic computers did of course make it easier to do calculations that people were already doing with mechanical devices and pen and paper, the main applications computers are used for today were not even imagined before their invention.

An exponentially faster algorithm for all \(\mathbf{NP}\) problems would be no less radical an improvement (and indeed, in some sense would be more) than the computer itself, and it is as hard for us to imagine what it would imply as it was for Babbage to envision today’s world. For starters, such an algorithm would completely change the way we program computers. Since we could automatically find the “best” (in any measure we chose) program that achieves a certain task, we would not need to define how to achieve a task, but only specify tests as to what would be a good solution, and could also ensure that a program satisfies an exponential number of tests without actually running them.

The possibility that \(\mathbf{P}=\mathbf{NP}\) is often described as “automating creativity”. There is something to that analogy, as we often think of a creative solution as one that is hard to discover but that, once the “spark” hits, is easy to verify. But there is also an element of hubris to that statement, implying that the most impressive consequence of such an algorithmic breakthrough will be that computers would succeed in doing something that humans already do today. Nevertheless, artificial intelligence, like many other fields, will clearly be greatly impacted by an efficient 3SAT algorithm. For example, it is clearly much easier to find a better Chess-playing algorithm when, given any algorithm \(P\), you can find the smallest algorithm \(P'\) that plays Chess better than \(P\). Moreover, as we mentioned above, much of machine learning (and statistical reasoning in general) is about finding “simple” concepts that explain the observed data, and if \(\mathbf{NP}=\mathbf{P}\), we could search for such concepts automatically for any notion of “simplicity” we see fit. In fact, we could even “skip the middle man” and do an automatic search for the learning algorithm with smallest generalization error. Ultimately the field of Artificial Intelligence is about trying to “shortcut” billions of years of evolution to obtain artificial programs that match (or beat) the performance of natural ones, and a fast algorithm for \(\mathbf{NP}\) would provide the ultimate shortcut. (One interesting theory is that \(\mathbf{P}=\mathbf{NP}\) and evolution has already discovered this algorithm, which we are already using without realizing it. At the moment, there seems to be very little evidence for such a scenario. In fact, we have some partial results in the other direction showing that, regardless of whether \(\mathbf{P}=\mathbf{NP}\), many types of “local search” or “evolutionary” algorithms require exponential time to solve 3SAT and other \(\mathbf{NP}\)-hard problems.)

More generally, a faster algorithm for \(\mathbf{NP}\) problems would be immensely useful in any field where one is faced with computational or quantitative problems\(-\) which is basically all fields of science, math, and engineering. This will not only help with concrete problems such as designing a better bridge, or finding a better drug, but also with addressing basic mysteries such as trying to find scientific theories or “laws of nature”. In a fascinating talk, physicist Nima Arkani-Hamed discusses the effort of finding scientific theories in much the same language as one would describe solving an \(\mathbf{NP}\) problem, for which the solution is easy to verify or seems “inevitable”, once found, but that requires searching through a huge landscape of possibilities to reach, and that often can get “stuck” at local optima:

“the laws of nature have this amazing feeling of inevitability… which is associated with local perfection.”

“The classical picture of the world is the top of a local mountain in the space of ideas. And you go up to the top and it looks amazing up there and absolutely incredible. And you learn that there is a taller mountain out there. Find it, Mount Quantum…. they’re not smoothly connected … you’ve got to make a jump to go from classical to quantum … This also tells you why we have such major challenges in trying to extend our understanding of physics. We don’t have these knobs, and little wheels, and twiddles that we can turn. We have to learn how to make these jumps. And it is a tall order. And that’s why things are difficult.”

Finding an efficient algorithm for \(\mathbf{NP}\) amounts to always being able to search through an exponential space and find not just the “local” mountain, but the tallest peak.

But perhaps more than any computational speedups, a fast algorithm for \(\mathbf{NP}\) problems would bring about a new type of understanding. In many of the areas where \(\mathbf{NP}\)-completeness arises, it is not as much a barrier for solving computational problems as it is a barrier for obtaining “closed-form formulas” or other types of more constructive descriptions of the behavior of natural, biological, social and other systems. A better algorithm for \(\mathbf{NP}\), even if it is “merely” \(2^{\sqrt{n}}\)-time, seems to require obtaining a new way to understand these types of systems, whether it is characterizing Nash equilibria, spin-glass configurations, entangled quantum states, or any of the other questions where \(\mathbf{NP}\) is currently a barrier for analytical understanding. Such new insights would be very fruitful regardless of their computational utility.

Can \(\mathbf{P} \neq \mathbf{NP}\) be neither true nor false?

The Continuum Hypothesis is a conjecture made by Georg Cantor in 1878, positing the non-existence of a certain type of infinite cardinality. (One way to phrase it is that for every infinite subset \(S\) of the real numbers \(\R\), either there is a one-to-one and onto function \(f:S \rightarrow \R\) or there is a one-to-one and onto function \(f:S \rightarrow \N\).) This was considered one of the most important open problems in set theory, and settling its truth or falseness was the first problem put forward by Hilbert in the 1900 address we mentioned before. However, using the theories developed by Gödel and Turing, in 1963 Paul Cohen proved that both the Continuum Hypothesis and its negation are consistent with the standard axioms of set theory (i.e., the Zermelo-Fraenkel axioms + the Axiom of choice, or “ZFC” for short). Formally, what he proved is that if ZFC is consistent, then so is ZFC when we assume either the continuum hypothesis or its negation.

Today, many (though not all) mathematicians interpret this result as saying that the Continuum Hypothesis is neither true nor false, but rather is an axiomatic choice that we are free to make one way or the other. Could the same hold for \(\mathbf{P} \neq \mathbf{NP}\)?

In short, the answer is No. For example, suppose that we are trying to decide between the “3SAT is easy” conjecture (there is an \(10^6n\) time algorithm for 3SAT) and the “3SAT is hard” conjecture (for every \(n\), any NAND-CIRC program that solves \(n\) variable 3SAT takes \(2^{10^{-6}n}\) lines). Then, since for \(n = 10^8\), \(2^{10^{-6}n} > 10^6 n\), this boils down to the finite question of deciding whether or not there is a \(10^{13}\)-line NAND-CIRC program deciding 3SAT on formulas with \(10^8\) variables. If there is such a program then there is a finite proof of its existence, namely the approximately 1TB file describing the program, and for which the verification is the (finite in principle though infeasible in practice) process of checking that it succeeds on all inputs.⁵ If there isn’t such a program, then there is also a finite proof of that, though any such proof would take longer since we would need to enumerate over all programs as well. Ultimately, since it boils down to a finite statement about bits and numbers; either the statement or its negation must follow from the standard axioms of arithmetic in a finite number of arithmetic steps. Thus, we cannot justify our ignorance in distinguishing between the “3SAT easy” and “3SAT hard” cases by claiming that this might be an inherently ill-defined question. Similar reasoning (with different numbers) applies to other variants of the \(\mathbf{P}\) vs \(\mathbf{NP}\) question. We note that in the case that 3SAT is hard, it may well be that there is no short proof of this fact using the standard axioms, and this is a question that people have been studying in various restricted forms of proof systems.

Is \(\mathbf{P}=\mathbf{NP}\) “in practice”?

The fact that a problem is \(\mathbf{NP}\)-hard means that we believe there is no efficient algorithm that solves it in the worst case. It does not, however, mean that every single instance of the problem is hard. For example, if all the clauses in a 3SAT instance \(\varphi\) contain the same variable \(x_i\) (possibly in negated form), then by guessing a value to \(x_i\) we can reduce \(\varphi\) to a 2SAT instance which can then be efficiently solved. Generalizations of this simple idea are used in “SAT solvers”, which are algorithms that have solved certain specific interesting SAT formulas with thousands of variables, despite the fact that we believe SAT to be exponentially hard in the worst case. Similarly, a lot of problems arising in economics and machine learning are \(\mathbf{NP}\)-hard.⁶ And yet vendors and customers manage to figure out market-clearing prices (as economists like to point out, there is milk on the shelves) and mice succeed in distinguishing cats from dogs. Hence people (and machines) seem to regularly succeed in solving interesting instances of \(\mathbf{NP}\)-hard problems, typically by using some combination of guessing while making local improvements.

It is also true that there are many interesting instances of \(\mathbf{NP}\)-hard problems that we do not currently know how to solve. Across all application areas, whether it is scientific computing, optimization, control or more, people often encounter hard instances of \(\mathbf{NP}\) problems on which our current algorithms fail. In fact, as we will see, all of our digital security infrastructure relies on the fact that some concrete and easy-to-generate instances of, say, 3SAT (or, equivalently, any other \(\mathbf{NP}\)-hard problem) are exponentially hard to solve.

Thus it would be wrong to say that \(\mathbf{NP}\) is easy “in practice”, nor would it be correct to take \(\mathbf{NP}\)-hardness as the “final word” on the complexity of a problem, particularly when we have more information about how any given instance is generated. Understanding both the “typical complexity” of \(\mathbf{NP}\) problems, as well as the power and limitations of certain heuristics (such as various local-search based algorithms) is a very active area of research. We will see more on these topics later in this course.

What if \(\mathbf{P} \neq \mathbf{NP}\)?

So, \(\mathbf{P}=\mathbf{NP}\) would give us all kinds of fantastical outcomes. But we strongly suspect that \(\mathbf{P} \neq \mathbf{NP}\), and moreover that there is no much-better-than-brute-force algorithm for 3SAT. If indeed that is the case, is it all bad news?

One might think that impossibility results, telling you that you cannot do something, is the kind of cloud that does not have a silver lining. But in fact, as we already alluded to before, it does. A hard (in a sufficiently strong sense) problem in \(\mathbf{NP}\) can be used to create a code that cannot be broken, a task that for thousands of years has been the dream of not just spies but of many scientists and mathematicians over the generations. But the complexity viewpoint turned out to yield much more than simple codes, achieving tasks that people had previously not even dared to dream of. These include the notion of public key cryptography, allowing two people to communicate securely without ever having exchanged a secret key; electronic cash, allowing private and secure transaction without a central authority; and secure multiparty computation, enabling parties to compute a joint function on private inputs without revealing any extra information about it. Also, as we will see, computational hardness can be used to replace the role of randomness in many settings.

Furthermore, while it is often convenient to pretend that computational problems are simply handed to us, and that our job as computer scientists is to find the most efficient algorithm for them, this is not how things work in most computing applications. Typically even formulating the problem to solve is a highly non-trivial task. When we discover that the problem we want to solve is \(\mathbf{NP}\)-hard, this might be a useful sign that we used the wrong formulation for it.

Beyond all these, the quest to understand computational hardness \(-\) including the discoveries of lower bounds for restricted computational models, as well as new types of reductions (such as those arising from “probabilistically checkable proofs”) \(-\) has already had surprising positive applications to problems in algorithm design, as well as in coding for both communication and storage. This is not surprising since, as we mentioned before, from group theory to the theory of relativity, the pursuit of impossibility results has often been one of the most fruitful enterprises of mankind.

Exercises

Bibliographical notes

As mentioned before, Aaronson’s survey (Aaronson, 2016) is a great exposition of the \(\mathbf{P}\) vs \(\mathbf{NP}\) problem. Another recommended survey by Aaronson is (Aaronson, 2005) which discusses the question of whether \(\mathbf{NP}\) complete problems could be computed by any physical means.

The paper (Buchfuhrer, Umans, 2011) discusses some results about problems in the polynomial hierarchy.

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Compiled on 12/06/2023 00:07:50

Copyright 2023, Boaz Barak.
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