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\(P\mathop{ =}\limits^{?}NP\)

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Open Problems in Mathematics

Abstract

In 1950, John Nash sent a remarkable letter to the National Security Agency, in which—seeking to build theoretical foundations for cryptography—he all but formulated what today we call the \(\mathsf{P}\mathop{ =}\limits^{?}\mathsf{NP}\) problem, and consider one of the great open problems of science. Here I survey the status of this problem in 2016, for a broad audience of mathematicians, scientists, and engineers. I offer a personal perspective on what it’s about, why it’s important, why it’s reasonable to conjecture that P ≠ NP is both true and provable, why proving it is so hard, the landscape of related problems, and crucially, what progress has been made in the last half-century toward solving those problems. The discussion of progress includes diagonalization and circuit lower bounds; the relativization, algebrization, and natural proofs barriers; and the recent works of Ryan Williams and Ketan Mulmuley, which (in different ways) hint at a duality between impossibility proofs and algorithms.

For a more recent version of this survey, with numerous corrections and revisions as well as new material, the reader is invited to visit http://www.scottaaronson.com/papers/pvsnp.pdf

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Notes

  1. 1.

    Here we’re using the observation that, once we fix a formal system (say, first-order logic plus the axioms of ZF set theory), deciding whether a given statement has a proof at most n symbols long in that system is an NP problem, which can therefore be solved in time polynomial in n assuming P = NP. We’re also assuming that the other six Clay conjectures have ZF proofs that are not too enormous: say, 1012 symbols or fewer, depending on the exact running time of the assumed algorithm. In the case of the Poincaré Conjecture, this can almost be taken to be a fact, modulo the translation of Perelman’s proof [179] into the language of ZF.

  2. 2.

    The most celebrated examples of nonconstructive proofs that algorithms exist all come from the Robertson-Seymour graph minors theory, one of the great achievements of twentieth-century combinatorics (for an accessible introduction, see for example Fellows [75]). The Robertson-Seymour theory typically deals with parameterized problems: for example, “given a graph G, decide whether G can be embedded on a sphere with k handles.” In those cases, typically a fast algorithm A k can be abstractly shown to exist for every value of k. The central problem is that each A k requires hard-coded data—in the above example, a finite list of obstructions to the desired embedding—that no one knows how to find given k, and whose size might also grow astronomically as a function of k. On the other hand, once the finite obstruction set for a given k was known, one could then use it to solve the problem for any graph G in time \(O\left (\left \vert G\right \vert ^{3}\right )\), where the constant hidden by the big-O depended on k.

    Robertson-Seymour theory also provides a few examples of non-parameterized problems that are abstractly proved to be in P but with no bound on the exponent, or abstractly proved to be \(O\left (n^{3}\right )\) or \(O\left (n^{2}\right )\) but with no bound on the constant. Thus, one cannot rule out the possibility that the same would happen with an NP-complete problem, and Donald Knuth [131] has explicitly speculated that P = NP will be proven in that way. To me, however, it is unclear whether he speculates this because there is a positive reason for thinking it true, or just because it would be cool and interesting if it was true.

  3. 3.

    As an amusing side note, there is a trick called Levin’s universal search [141], in which one “dovetails” over all Turing machines M 1, M 2, … (that is, for all t, runs M 1, …, M t for t steps each), halting when and if any M i has outputs a valid solution to one’s NP search problem. If we know P = NP, then we know this particular algorithm will find a valid solution, whenever one exists, in polynomial time—because clearly some M i does so, and all the machines other than M i increase the total running time by “only” a polynomial factor! With more work, one can even decrease this to a constant factor. Admittedly, however, the polynomial or constant factor will be so enormous as to negate this algorithm’s practical use.

  4. 4.

    The reason for this caveat is that, if a programming language were inherently limited to (say) 64K of memory, there would be only finitely many possible program behaviors, so in principle we could just cache everything in a giant lookup table. Many programming languages do impose a finite upper bound on the addressable memory, but they could easily be generalized to remove this restriction (or one could consider programs that store information on external I/O devices).

  5. 5.

    I should stress that, once we specify which computational models we have in mind—Turing machines, Intel machine code, etc.—the polynomial-time equivalence of those models is typically a theorem, though a rather tedious one. The “thesis” of the Extended Church-Turing Thesis, the part not susceptible to proof, is that all other “reasonable” models of digital computation will also be equivalent to those models.

  6. 6.

    In practice, often one only needs a special kind of Turing reduction called a many-one reduction or Karp reduction, which is a polynomial-time algorithm that maps every yes-instance of \(L^{{\prime}}\) to a yes-instance of L, and every no-instance of \(L^{{\prime}}\) to a no-instance of L. The additional power of Turing reductions—to make multiple queries to the L-oracle (with later queries depending on the outcomes of earlier ones), post-process the results of those queries, etc.—is needed only in a minority of cases. Nevertheless, for conceptual simplicity, throughout this survey I’ll talk in terms of Turing reductions.

  7. 7.

    Technically, Daskalakis et al. showed that the search problem of finding a Nash equilibrium is complete for a complexity class called PPAD. This could be loosely interpreted as saying that the problem is “as close to NP-hard as it could possibly be, subject to Nash’s theorem showing why the decision version is trivial.”

  8. 8.

    In defining the kth level of the hierarchy, we could also have given oracles for Π k−1 P rather than \(\Sigma _{k-1}^{\mathsf{P}}\): it doesn’t matter. Note also that “an oracle for complexity class \(\mathcal{C}\)” should be read as “an oracle for any \(\mathcal{C}\)-complete language L.”

  9. 9.

    This requires one nontrivial result, that every prime number has a succinct certificate—or in other words, that primality testing is in NP [180]. Since 2002, it is even known that primality testing is in P [14].

  10. 10.

    A further surprising result from 1987, called the Immerman-Szelepcsényi Theorem [110, 218], says that \(\mathsf{NSPACE}\left (f\left (n\right )\right ) = \mathsf{coNSPACE}\left (f\left (n\right )\right )\) for every “reasonable” memory bound \(f\left (n\right )\). (By contrast, Savitch’s Theorem produces a quadratic blowup when simulating nondeterministic space by deterministic space, and it remains open whether that blowup can be removed.) This further illustrates how space complexity behaves differently than we expect time complexity to behave.

  11. 11.

    Unlike P or PSPACE, classes like \(\mathsf{TIME}\left (n^{2}\right )\), \(\mathsf{SPACE}\left (n^{3}\right )\), etc. can be sensitive to whether we are talking about Turing machines, RAM machines, or some other model of computation. But in any case, one can simply fix one of those models any time the classes are mentioned in this survey, and nothing will go wrong.

  12. 12.

    I like to joke that, if computer scientists had been physicists, we’d simply have declared P ≠ NP to be an observed law of Nature, analogous to the laws of thermodynamics. A Nobel Prize would even be given for the discovery of that law. (And in the unlikely event that someone later proved P = NP, a second Nobel Prize would be awarded for the law’s overthrow.)

  13. 13.

    Strictly speaking, this is for the variant of 3Sat in which every clause must have exactly three literals, rather than at most three.

    Also note that, if we allow the use of randomness, then we can satisfy a 7∕8 fraction of the clauses in expectation by just setting each of the n variables uniformly at random! This is because a clause with three literals has 23 − 1 = 7 ways to be satisfied, and only one way to be unsatisfied. A deterministic polynomial-time algorithm that’s guaranteed to satisfy at least 7∕8 of the clauses requires only a little more work.

  14. 14.

    Indeed, a natural conjecture would be that the problem is NP-hard under randomized reductions for all k ≠ 7, but this remains open (Valiant, personal communication).

  15. 15.

    Note also that, by the Shoenfield absoluteness theorem [202], one’s beliefs about the Axiom of Choice, the Continuum Hypothesis, or other statements proven independent of ZF via forcing can have no effect on the provability of arithmetical statements such as P ≠ NP.

  16. 16.

    If a \(\Pi _{1}\)-sentence like the Goldbach Conjecture or the Riemann Hypothesis were known to be independent of ZF, then it would also be known to be true, since any counterexample would have a trivial finite proof! On the other hand, we could also imagine, say, the Goldbach Conjecture being proven equivalent to the consistency of ZF, in which case we could say only that either ZF is consistent and Goldbach is true but ZF doesn’t prove either, or else ZF proves anything. In any case, none of this directly applies to P ≠ NP, which is a \(\Pi _{2}\)-sentence.

  17. 17.

    I won’t have much to say about linear or semidefinite programming in this survey, so perhaps this is as good a place as any to mention that today, we know a great deal about the impossibility of solving NP-complete problems in polynomial time by formulating them as “natural” linear programs. This story starts in 1987, with a preprint by Swart [217] that claimed to prove P = NP by reducing the Traveling Salesperson Problem to a linear program with n 8 variables and constraints. Swart’s preprint inspired a landmark paper by Yannakakis [244] (making it possibly the most productive failed P = NP proof in history!), in which Yannakakis showed that there is no “symmetric” linear program with \(n^{o\left (n\right )}\) variables and constraints that has the “Traveling Salesperson Polytope” as its projection onto a subset of the variables. This ruled out Swart’s approach. Yannakakis also showed that the polytope corresponding to the maximum matching problem has no symmetric LP of subexponential size, but the polytope for the minimum spanning tree problem does have a polynomial-size LP. In general, expressibility by such an LP is sufficient for a problem to be in P, but not necessary.

    Later, in 2012, Fiorini et al. [76] substantially improved Yannakakis’s result, getting rid of the symmetry requirement. There have since been other major results in this direction: in 2014, Rothvoß[194] showed that the perfect matching polytope requires exponentially-large LPs (again with no symmetry requirement), while in 2015, Lee, Raghavendra, and Steurer [140] extended many of these lower bounds from linear to semidefinite programs.

    Collectively, these results rule out one “natural” approach to proving P = NP: namely, to start from famous NP-hard optimization problems like TSP, and then find a polynomial-size linear or semidefinite program that projects onto the polytope whose extreme points are the valid solutions. Of course, we can’t yet rule out the possibility that linear or semidefinite programs could help prove P = NP in some more indirect way (or via some NP-hard problem other than the specific ones that were studied); ruling that out seems essentially tantamount to proving P ≠ NP itself.

  18. 18.

    Despite the term “circuit,” which comes from electrical engineering, circuits in theoretical computer science are always free of cycles; they proceed from the inputs to the output via layers of logic gates.

  19. 19.

    This is a rare instance where the non-containment can actually be proved: for example, any unary language (i.e., language of the form \(\left \{0^{n}: n \in S\right \}\)) is clearly in P∕poly, but there is an uncountable infinity of such languages, so almost all of them cannot be in P.

  20. 20.

    Note that, if each logic gate depends on at most 2 inputs, then \(\log _{2}n\) is the smallest depth that allows the output to depend on all n input bits.

  21. 21.

    Bshouty, Cleve, and Eberly [53] showed that the size of the depth-reduced formula can even be taken to be \(O\left (S^{1+\varepsilon }\right )\) , for any constant \(\varepsilon > 0\).

  22. 22.

    But making matters more complicated still, survey propagation fails badly on random 4Sat.

  23. 23.

    We could also allow sampling from some distribution close to \(\mathcal{D}_{n}\), but we will ignore that complication here.

  24. 24.

    There are closely-related objects, such as “lossy” trapdoor OWFs (see [178]), that also suffice for building public-key cryptosystems.

  25. 25.

    At least, not for arbitrary polynomials computed by small formulas or circuits. A great deal of progress has been made derandomizing PIT for restricted classes of polynomials. In fact, the deterministic primality test of Agrawal et al. [14] was based on a derandomization of one extremely special case of PIT.

  26. 26.

    One can also consider the QMA-complete problems, which are a quantum generalization of the NP-complete problems themselves (see [48]), but we will not pursue that here.

  27. 27.

    One can artificially design an NP-complete problem with a superpolynomial quantum speedup over the best known classical algorithm by, for example, taking the language

    $$\displaystyle\begin{array}{rcl} L& =& \left \{0\varphi 0\cdots 0\ \vert \ \varphi \ \text{ is a satisfiable 3SAT instance of size }\ n^{0.01}\right \} \cup {}\\ & &\left \{1x\ \vert \ x\ \text{ is a binary encoding of a positive integer with an odd number of distinct prime factors}\right \}.{}\\ \end{array}$$

    Clearly L is NP-complete, and a quantum algorithm can decide L in \(O\left (c^{n^{0.01} }\right )\) time for some c, whereas the best known classical algorithm will take \(\exp \left (n\right )\) time.

    Conversely, there are also NP-complete problems with no significant quantum speedup known—say, because the best known classical algorithm is based on dynamic programming, and it’s unknown how to combine that with Grover’s algorithm. A candidate example is the Traveling Salesman Problem, which is solvable in \(O\left (2^{n}\mathop{\mathrm{poly}}\nolimits \left (n\right )\right )\) time using the Held-Karp dynamic programming algorithm [107], whereas Grover’s algorithm seems to yield only the worse bound \(O(\sqrt{n!})\).

  28. 28.

    Going even further, Raz [184] proved in 2010 that, if we manage to show that any explicit d-dimensional tensor \(A: \left [n\right ]^{d} \rightarrow \mathbb{F}\) has rank at least \(n^{d\left (1-o\left (1\right )\right )}\), then we’ve also shown that the n × n permanent function has no polynomial-size arithmetic formulas. It’s easy to construct explicit d-dimensional tensors with rank \(n^{\left \lfloor d/2\right \rfloor }\), but the current record is an explicit d-dimensional tensor with rank at least \(2n^{\left \lfloor d/2\right \rfloor } + n - O\left (d\log n\right )\) [18].

    Note that, if we could show that the permanent had no \(n^{O\left (\log n\right )}\)-size arithmetic formulas, that would imply Valiant’s famous Conjecture 66: that the permanent has no polynomial-size arithmetic circuits. However, Raz’s technique seems incapable of proving formula-size lower bounds better than \(n^{\Omega \left (\log \log n\right )}\).

  29. 29.

    With some effort, Shannon’s lower bound can be shown to be tight: that is, every n-variable Boolean function can be represented by a circuit of size \(O\left (2^{n}/n\right )\) . (The obvious upper bound is \(O\left (n2^{n}\right )\).)

  30. 30.

    Crucially, this will be a different language for each k; otherwise we would get \(\mathsf{PSPACE}\not\subset \mathsf{P/poly}\), which is far beyond our current ability to prove.

  31. 31.

    Note that, if we encode the input string using the so-called dual-rail representation—in which every 0 is represented by the 2-bit string 01, and every 1 by 10—then the monotone circuit complexities of Clique, Matching, and so on do become essentially equivalent to their non-monotone circuit complexities, since we can push all the NOT gates to the bottom layer of the circuit using de Morgan’s laws. Unfortunately, Razborov’s lower bound techniques also break down under dual-rail encoding.

  32. 32.

    Assume for simplicity that n is a power of 2. Then x 1 ⊕⋯ ⊕ x n can be written as y ⊕ z, where y: = x 1 ⊕⋯ ⊕ x n∕2 and z: = x n∕2+1 ⊕⋯ ⊕ x n . This in turn can be written as \(\left (y \wedge \overline{z}\right ) \vee \left (\overline{y} \wedge z\right )\). Expanding recursively now yields a size-n 2 formula for Parity, made of AND, OR, and NOT gates.

  33. 33.

    In theoretical computer science, the term non-negligible means lower-bounded by \(1/n^{O\left (1\right )}\).

  34. 34.

    Technically, the problem of distinguishing random from pseudorandom functions is equivalent to the problem of inverting one-way functions, which is not quite as strong as solving NP-complete problems in polynomial time—only solving average-case NP-complete problems with planted solutions. For more see Sect. 5.3.

  35. 35.

    The problem, given as input the truth table of a Boolean function \(f: \left \{0, 1\right \}^{n} \rightarrow \left \{0, 1\right \}\), of computing or approximating the circuit complexity of f is called the Minimum Circuit Size Problem (MCSP). It is a longstanding open problem whether or not MCSP is NP-hard; at any rate, there are major obstructions to proving it NP-hard with existing techniques (see Kabanets and Cai [117] and Murray and Williams [169]). On the other hand, MCSP cannot be in P (or BPP) unless there are no cryptographically-secure pseudorandom generators. At any rate, what is relevant to natural proofs is just whether there is an efficient algorithm to certify a large fraction of Boolean functions as being hard, which is a weaker requirement than solving MCSP.

  36. 36.

    Allender and Koucký’s paper partly builds on 2003 work by Srinivasan [211], who showed that, to prove P ≠ NP, one would “merely” need to show that any algorithm to compute weak approximations for the MaxClique problem takes \(\Omega \left (n^{1+\varepsilon }\right )\) time, for some constant \(\varepsilon > 0\). The way Srinivasan proved this striking statement was, again, by using a sort of self-reduciblity: he showed that, if there’s a polynomial-time algorithm for MaxClique, then by running that algorithm on smaller graphs sampled from the original graph, one can solve approximate versions of MaxClique in \(n^{1+o\left (1\right )}\) time.

  37. 37.

    We can assume without loss of generality that Merlin’s strategy is deterministic, since Merlin is computationally unbounded, and any convex combination of strategies must contain a deterministic strategy that causes Arthur to accept with at least as great a probability as the convex combination does.

  38. 38.

    This is so because a polynomial-space Turing machine can treat the entire interaction between Merlin and Arthur as a game, in which Merlin is trying to get Arthur to accept with the largest possible probability. The machine can then evaluate the exponentially large game tree using depth-first recursion.

  39. 39.

    Actually, for this proof one does not really need either Toda’s Theorem, or the slightly-nontrivial result that \(\mathsf{\varSigma }_{2}^{\mathsf{P}}\) does not have circuits of size n k. Instead, one can just argue directly that at any rate, P #P does not have circuits of size n k, using a slightly more careful version of the argument of Theorem 37. For details see Aaronson [4].

  40. 40.

    In complexity theory, a promise problem is a pair of subsets \(\Pi _{\mathop{\mathrm{YES}}\nolimits },\Pi _{\mathop{\mathrm{NO}}\nolimits } \subseteq \left \{0, 1\right \}^{{\ast}}\) with \(\Pi _{\mathop{\mathrm{YES}}\nolimits } \cap \Pi _{\mathop{\mathrm{NO}}\nolimits } = \varnothing \). An algorithm solves the problem if it accepts all inputs in \(\Pi _{\mathop{\mathrm{YES}}\nolimits }\) and rejects all inputs in \(\Pi _{\mathop{\mathrm{NO}}\nolimits }\). Its behavior on inputs neither in \(\Pi _{\mathop{\mathrm{YES}}\nolimits }\) nor \(\Pi _{\mathop{\mathrm{NO}}\nolimits }\) (i.e., inputs that “violate the promise”) can be arbitrary. A typical example of a promise problem is: given a Boolean circuit C, decide whether C accepts at least 2∕3 of all inputs \(x \in \left \{0, 1\right \}^{n}\) or at most 1∕3 of them, promised that one of those is true. This problem is in BPP (or technically, PromiseBPP). The role of the promise here is to get rid of those inputs for which random sampling would accept with probability between 1∕3 and 2∕3, violating the definition of BPP.

  41. 41.

    Indeed, the hybrid circuit lower bounds of Sect. 6.3.2 could already be considered examples of ironic complexity theory. In this section, we discuss other examples.

  42. 42.

    On unrealistic models such as one-tape Turing machines, one can prove up to \(\Omega \left (n^{2}\right )\) lower bounds for 3Sat and many other problems (even recognizing palindromes), but only by exploiting the fact that the tape head needs to waste a lot of time moving back and forth across the input.

  43. 43.

    On the other hand, by proving size-depth tradeoffs for so-called branching programs, researchers have been able to obtain time-space tradeoffs for certain special problems in P. Unlike the 3Sat tradeoffs, the branching program tradeoffs involve only slightly superlinear time bounds; on the other hand, they really do represent a fundamentally different way to prove time-space tradeoffs, one that makes no appeal to NP-completeness, diagonalization, or hierarchy theorems. As one example, in 2000 Beame et al. [37], building on earlier work by Ajtai [17], used branching programs to prove the following: there exists a problem in P, based on binary quadratic forms, for which any RAM algorithm (even a nonuniform one) that uses \(n^{1-\Omega \left (1\right )}\) space must also use \(\Omega \left (n \cdot \sqrt{\log n/\log \log n}\right )\) time.

  44. 44.

    Interestingly, both the polynomial method and the proof of Lemma 64 are also closely related to the proof of Toda’s Theorem (Theorem 13), that \(\mathsf{PH} \subseteq \mathsf{P}^{\mathsf{\#P}}\).

  45. 45.

    Curiously, this step can only be applied to the ACC circuits themselves, which of course allow OR gates. It cannot be applied to the Boolean functions of low-degree polynomials that one derives from the ACC circuits.

  46. 46.

    Very recently, Kane and Williams [119] managed to give an explicit Boolean function that requires depth-2 threshold circuits with \(\Omega \left (n^{3/2}/\log ^{3}n\right )\) gates. However, their argument does not proceed via a better-than-brute-force algorithm for depth-2 TC 0 Sat; the latter problem appears to remain open.

  47. 47.

    I’ll restrict to fields here for simplicity, but one can also consider (e.g.) rings.

  48. 48.

    To clarify, 0 and 2x are equal as functions over the finite field \(\mathbb{F}_{2}\), but not equal as formal polynomials.

  49. 49.

    The central difference between the \(\mathsf{P}_{\mathbb{R}}\mathop{ =}\limits^{?}\mathsf{NP}_{\mathbb{R}}\) and \(\mathsf{P}_{\mathbb{C}}\mathop{ =}\limits^{?}\mathsf{NP}_{\mathbb{C}}\) questions is simply that, because \(\mathbb{R}\) is an ordered field, one defines Turing machines over \(\mathbb{R}\) to allow comparisons ( < , ≤ ) and branching on their results.

  50. 50.

    In later work, Bürgisser [59] showed that the same conjecture about \(\tau ^{{\ast}}\left (n!\right )\) (called the τ-conjecture) would also imply Valiant’s Conjecture  66, that the permanent has no polynomial-size arithmetic circuits.

  51. 51.

    Fields of characteristic 2, such as \(\mathbb{F}_{2}\) , are a special case: there, the permanent and determinant are equivalent, so in particular \(\mathop{\mathrm{Per}}\nolimits \left (X\right )\) has polynomial-size arithmetic circuits.

  52. 52.

    In the literature, Conjecture 66 is often called the VP ≠ VNP conjecture, with VP and VNP being arithmetic analogues of P and NP respectively. I won’t use that terminology in this survey, for several reasons: (1) VP is arguably more analogous to NC than to P, (2) VNP is arguably more analogous to #P than to NP, and (3) Conjecture 66 is almost always studied as a nonuniform conjecture, more analogous to \(\mathsf{NP}\not\subset \mathsf{P/poly}\) than to P ≠ NP.

  53. 53.

    Indeed, #P would even have polynomial-size circuits of depth \(\log ^{O\left (1\right )}n\).

  54. 54.

    An immediate corollary is that any multilinear circuit for \(\mathop{\mathrm{Per}}\nolimits \left (X\right )\) or \(\mathop{\mathrm{Det}}\nolimits \left (X\right )\) requires depth \(\Omega \left (\log ^{2}n\right )\).

  55. 55.

    For simplicity, here I’ll assume that we mean an “ordinary” (Boolean) polynomial-time algorithm, though one could also require polynomial-time algorithms in the arithmetic model.

  56. 56.

    Although so far, I haven’t seen a conjectured role for topology or logic in GCT.

  57. 57.

    Furthermore, just as string theory didn’t predict what new data there has been in fundamental physics in recent decades (e.g., the dark energy), so GCT played no role in, e.g., the proof of \(\mathsf{NEXP}\not\subset \mathsf{ACC}\) (Sect. 6.4.2), or the breakthrough lower bounds for small-depth circuits computing the permanent (Sect. 6.5.2). In both cases, to point this out is to hold the theory to a high and probably unfair standard, but also, ipso facto, to pay the theory a compliment.

  58. 58.

    MaxFlow can easily be reduced to linear programming, but Ford and Fulkerson [78] also gave a much faster and direct way to solve it, which can be found in any undergraduate algorithms textbook. There have been further improvements since.

  59. 59.

    We can assume, if we like, that G, s, and t are hardwired into the circuit. We can also allow constants such as 0 and 1 as inputs, but this is not necessary, as we can also generate constants ourselves using comparison gates and arithmetic.

  60. 60.

    More broadly, I’ve found, there are many confusing points about GCT whose resolutions require reminding ourselves that we’re talking about orbit closures, and not only about orbits. For example, the plan of GCT is ultimately to show, roughly speaking, that the n × n permanent has “too little symmetry” to be embedded into the m × m determinant, unless m is much larger than n. But in that case, what about (say) a random arithmetic formula f of size n, which has no nontrivial symmetries, but which clearly can be embedded into the \(\left (n + 1\right ) \times \left (n + 1\right )\) determinant, by Theorem 68? Even though this f clearly isn’t characterized by its symmetries, mustn’t the embedding obstructions for f be a strict superset of the embedding obstructions for the permanent—since f’s symmetries are a strict subset of the permanent’s symmetries—and doesn’t that give rise to a contradiction? According to Mulmuley (personal communication), the solution to the apparent paradox is that this argument would be valid if we were talking only about orbits, but it’s not valid for orbit closures. With orbit closures, the set of obstructions doesn’t depend in a simple way on the symmetries of the original function, so that it’s possible that an obstruction for the permanent would fail to be an obstruction for f.

  61. 61.

    Note that we don’t even need to assume the symmetry \(p\left (X\right ) = p\left (X^{T}\right )\) ; that comes as a free byproduct. Also, it might seem like “cheating” that we use the permanent to state the symmetries that characterize the permanent, and likewise for the determinant. But we’re just using the permanent and determinant as convenient ways to specify which matrices A,B we want, and could give slightly more awkward symmetry conditions that avoided them. (This is especially clear for the permanent, since if A is diagonal, then \(\mathop{\mathrm{Per}}\nolimits \left (A\right )\) is just the product of the diagonal entries.)

  62. 62.

    See Grochow [95, Proposition 3.4.9] for a simple proof of this, via a dimension argument.

  63. 63.

    In principle, \(\rho _{\mathop{\mathrm{Det}}\nolimits }\) and \(\rho _{\mathop{\mathrm{Per}}\nolimits }\) are infinite-dimensional representations, so an algorithm could search them forever for obstructions without halting. On the other hand, if we impose some upper bound on the degrees of the polynomials in the coordinate ring, we get an algorithm that takes “merely” doubly- or triply-exponential time.

  64. 64.

    A very loose analogy: well before Andrew Wiles proved Fermat’s Last Theorem [233], that x n + y n = z n has no nontrivial integer solutions for any n ≥ 3, number theorists knew a reasonably efficient algorithm that took an exponent n as input, and that (in practice, in all the cases that were tried) proved FLT for that n. Using that algorithm, in 1993—just before Wiles announced his proof—Buhler et al. [55] proved FLT for all n up to 4 million.

  65. 65.

    As pointed out in Sect. 2.2.6, there are other cases in complexity theory where deciding positivity is much easier than exact counting: for example, deciding whether a graph has at least one perfect matching (counting the number of perfect matchings is #P-complete).

  66. 66.

    A language L is called P-complete if (1) L ∈ P, and (2) every \(L^{{\prime}}\in \mathsf{P}\) can be reduced to L by some form of reduction weaker than arbitrary polynomial-time ones (LOGSPACE reductions are often used for this purpose).

  67. 67.

    This result of Lipton’s provided the germ of the proof that IP = PSPACE; see Sect. 6.3.1.

    Mulmuley’s test improves over Lipton’s by, for example, requiring only nonadaptive queries to C rather than adaptive ones.

  68. 68.

    A standard example is the 2 × 2 × 2 tensor whose \(\left (2, 1, 1\right )\), \(\left (1, 2, 1\right )\), and \(\left (1, 1, 2\right )\) entries are all 1, and whose 5 remaining entries are all 0. One can check that this tensor has a rank of 3 but border rank of 2.

  69. 69.

    It would be interesting to find a function in P, more natural than the P-complete H function, that’s completely characterized by its symmetries, and then try to understand explicitly why there’s no representation-theoretic obstruction to that function’s orbit closure being contained in χ P —something we already know must be true.

  70. 70.

    In fact, the list can be found in the class ZPP NP, where ZPP stands for Zero-Error Probabilistic Polynomial-Time. This means that, whenever the randomized algorithm succeeds in constructing the list, it’s certain that it’s done so.

  71. 71.

    The polynomial identity testing problem was defined in Sect. 5.4. Also, by a “black-box derandomization,” we mean a deterministic polynomial-time algorithm that outputs a hitting set: that is, a list of points x1 ,…,x ℓ such that, for all small arithmetic circuits C that don’t compute the identically-zero polynomial, there exists an i such that \(C\left (x_{i}\right )\neq 0\) . What makes the derandomization “black-box” is that the choice of x 1 ,…,x ℓ doesn’t depend on C.

  72. 72.

    In 1999, Allender [19] showed that the permanent, and various other natural #P-complete problems, can’t be solved by LOGTIME -uniform TC 0 circuits: in other words, constant-depth threshold circuits for which there’s an \(O\left (\log n\right )\)-time algorithm to output the ith bit of their description, for any i. Indeed, these problems can’t be solved by LOGTIME-uniform TC 0 circuits of size \(f\left (n\right )\), where f is any function that can yield an exponential when iterated a constant number of times. The proof uses a hierarchy theorem, it would be interesting to know whether it relativizes.

  73. 73.

    Examples are deleting two successive NOT gates, or applying de Morgan’s laws. By the completeness of Boolean algebra, one can give local transformation rules that suffice to convert any n-input Boolean circuit into any equivalent circuit using at most \(\exp \left (n\right )\) moves.

    From talking to experts, this problem seems closely related to the problem of proving superpolynomial lower bounds for so-called Frege proofs, but is possibly easier.

  74. 74.

    There are even what Richard Lipton has termed “galactic algorithms” [147], which beat their asymptotically-worse competitors, but only for values of n that likely exceed the information storage capacity of the galaxy or the observable universe. The currently-fastest matrix multiplication algorithms might fall into this category, although the constants don’t seem to be known in enough detail to say for sure.

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Acknowledgements

I thank Andy Drucker, Adam Klivans, Ashley Montanaro, Leslie Valiant, and Ryan Williams for helpful observations and for answering questions. I especially thank Michail Rassias for his exponential patience with my delays completing this article. Supported by an NSF Waterman Award, under grant no. 1249349.

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    Scott Aaronson

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    Michael Th. Rassias

Appendix: Glossary of Complexity Classes

Appendix: Glossary of Complexity Classes

To help you remember all the supporting characters in the ongoing soap opera of which P and NP are the stars, this appendix contains short definitions of the complexity classes that appear in this survey, with references to the sections where the classes are discussed in more detail. For a fuller list, containing over 500 classes, see for example my Complexity Zoo [8]. All the classes below are classes of decision problems—that is, languages \(L \subseteq \left \{0,1\right \}^{{\ast}}\). The known inclusion relations among many of the classes are also depicted in Fig. 3.

Fig. 3
figure 3

Known inclusion relations among 21 of the complexity classes that appear in this survey

Full size image
Π 2 P::

coNP NP, the second level of the polynomial hierarchy (with universal quantifier in front). See Sect. 2.2.3.

\(\mathsf{\varSigma }_{2}^{\mathsf{P}}\)::

NP NP, the second level of the polynomial hierarchy (with existential quantifier in front). See Sect. 2.2.3.

AC 0::

The class decidable by a nonuniform family of polynomial-size, constant-depth, unbounded-fanin circuits of AND, OR, and NOT gates. See Sect. 6.2.3.

\(\mathsf{AC}^{0}\left [m\right ]\)::

AC 0 enhanced by MOD m gates, for some specific value of m (the case of m a prime power versus a non-prime-power are dramatically different). See Sect. 6.2.4.

ACC::

AC 0 enhanced by MOD m gates, for every m simultaneously. See Sect. 6.4.2.

BPP::

Bounded-Error Probabilistic Polynomial-Time. The class decidable by a polynomial-time randomized algorithm that errs with probability at most 1∕3 on each input. See Sect. 5.4.1.

BQP::

Bounded-Error Quantum Polynomial-Time. The same as BPP except that we now allow quantum algorithms. See Sect. 5.5.

coNP::

The class consisting of the complements of all languages in NP. Complete problems include unsatisfiability, graph non-3-colorability, etc. See Sect. 2.2.3.

\(\mathsf{DTISP}\left (f\left (n\right ),g\left (n\right )\right )\)::

See Sect. 6.4.1.

EXP::

Exponential-Time, or \(\bigcup _{k}\mathsf{TIME}\left (2^{n^{k} }\right )\). Note the permissive definition of “exponential,” which allows any polynomial in the exponent. See Sect. 2.2.7.

EXPSPACE::

Exponential-Space, or \(\bigcup _{k}\mathsf{SPACE}\left (2^{n^{k} }\right )\). See Sect. 2.2.7.

IP::

Interactive Proofs, the class for which a “yes” answer can be proven (to statistical certainty) via an interactive protocol in which a polynomial-time verifier Arthur exchanges a polynomial number of bits with a computationally-unbounded prover Merlin. Turns out to equal PSPACE [200]. See Sect. 6.3.1.

LOGSPACE::

Logarithmic-Space, or \(\mathsf{SPACE}\left (\log n\right )\). Note that only read/write memory is restricted to \(O\left (\log n\right )\) bits; the n-bit input itself is stored in a read-only memory. See Sect. 6.4.1.

MA::

Merlin-Arthur, the class for which a “yes” answer can be proven to statistical certainty via a polynomial-size message from a prover (“Merlin”), which the verifier (“Arthur”) then verifies in probabilistic polynomial time. Same as NP except that the verification can be probabilistic. See Sect. 6.3.2.

MA EXP ::

The exponential-time analogue of MA, where now Merlin’s proof can be \(2^{n^{O\left (1\right )} }\) bits long, and Arthur’s probabilistic verification can also take \(2^{n^{O\left (1\right )} }\) time. See Sect. 6.3.2.

NC 1::

The class decidable by a nonuniform family of polynomial-size Boolean formulas—or equivalently, polynomial-size Boolean circuits of fanin 2 and depth \(O\left (\log n\right )\). The subclass of P∕poly that is “highly parallelizable.” See Sect. 5.2.

NEXP::

Nondeterministic Exponential-Time, or \(\bigcup _{k}\mathsf{NTIME}\left (2^{n^{k} }\right )\). The exponential-time analogue of NP. See Sect. 2.2.7.

NP::

Nondeterministic Polynomial-Time, or \(\bigcup _{k}\mathsf{NTIME}\left (n^{k}\right )\). The class for which a “yes” answer can be proven via a polynomial-size witness, which is verified by a deterministic polynomial-time algorithm. See Sect. 2.

\(\mathsf{NTIME}\left (f\left (n\right )\right )\)::

Nondeterministic \(f\left (n\right )\)-Time. The class for which a “yes” answer can be proven via an \(O\left (f\left (n\right )\right )\)-bit witness, which is verified by a deterministic \(O\left (f\left (n\right )\right )\)-time algorithm. Equivalently, the class solvable by a nondeterministic \(O\left (f\left (n\right )\right )\)-time algorithm. See Sect. 2.2.7.

P::

Polynomial-Time, or \(\bigcup _{k}\mathsf{TIME}\left (n^{k}\right )\). The class solvable by a deterministic polynomial-time algorithm. See Sect. 2.

P #P::

P with an oracle for #P problems (i.e., for counting the exact number of accepting witnesses for any problem in NP). See Sect. 2.2.6.

P∕poly::

P enhanced by polynomial-size “advice strings” \(\left \{a_{n}\right \}_{n}\), which depend only on the input size n but can otherwise be chosen to help the algorithm as much as possible. Equivalently, the class solvable by a nonuniform family of polynomial-size Boolean circuits (i.e., a different circuit is allowed for each input size n). See Sect. 5.2.

PH::

The Polynomial Hierarchy. The class expressible via a polynomial-time predicate with a constant number of alternating universal and existential quantifiers over polynomial-size strings. Equivalently, the union of \(\mathsf{\varSigma }_{1}^{\mathsf{P}} = \mathsf{NP}\), Π 1 P = coNP, \(\mathsf{\varSigma }_{2}^{\mathsf{P}} = \mathsf{NP}^{\mathsf{NP}}\), Π 2 P = coNP NP, and so on. See Sect. 2.2.3.

PP::

Probabilistic Polynomial-Time. The class decidable by a polynomial-time randomized algorithm that, for each input x, guesses the correct answer with probability greater than 1∕2. Like BPP but without the bounded-error (1∕3 versus 2∕3) requirement, and accordingly believed to be much more powerful. See Sect. 2.2.6.

PSPACE::

Polynomial-Space, or \(\bigcup _{k}\mathsf{SPACE}\left (n^{k}\right )\). See Sect. 2.2.5.

\(\mathsf{SPACE}\left (f\left (n\right )\right )\)::

The class decidable by a serial, deterministic algorithm that uses \(O\left (f\left (n\right )\right )\) bits of memory (and possibly up to \(2^{O\left (f\left (n\right )\right )}\) time). See Sect. 2.2.7.

TC 0::

AC 0 enhanced by MAJORITY gates. Also corresponds to “neural networks” (polynomial-size, constant-depth circuits of threshold gates). See Sect. 6.2.5.

\(\mathsf{TIME}\left (f\left (n\right )\right )\)::

The class decidable by a serial, deterministic algorithm that uses \(O\left (f\left (n\right )\right )\) time steps (and therefore, \(O\left (f\left (n\right )\right )\) bits of memory). See Sect. 2.2.7.

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Aaronson, S. (2016). \(P\mathop{ =}\limits^{?}NP\) . In: Nash, Jr., J., Rassias, M. (eds) Open Problems in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32162-2_1

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