Has There Been Progress on the P vs. NP Question?

Has There Been Progress on the P vs. NP Question?

Scott Aaronson (MIT)

P vs. NP: I Assume You’ve Heard of It

Frank Wilczek (Physics Nobel 2004) was recently asked: “If you could ask a superintelligent alien one yes-or-no question, what would it be?”

His response: “P vs. NP. That basically contains all the other questions, doesn’t it?”

A Depressing Possibility…From the standpoint of P vs. NP, the last 50 years of complexity theory have taken us around in circles and been a complete waste of time.

This talk:

We might be nowhere close to a proof, but at least the depressing possibility doesn’t hold!

We’ve found (and continue to find) nontrivial insights that will play a role in the solution, assuming there is one.

The end is not in sight, but we’re not at the beginning either.

Achievement 1: Increased Confidence That P vs. NP Was The

Right Question To Ask

NP-completeness

Goal: Lift this box

SAT

PCPWhat NP-completeness accomplished…

The Unreasonable Robustness of PA half-century of speculation about alternative computational models has taken us only slightly beyond P

Would-be PNP provers: don’t get discouraged!

“But can’t soap bubbles solve the Minimum Steiner Tree problem in an instant, rendering P vs. NP irrelevant?”

Likewise for spin glasses, folding proteins, DNA computers, analog computers…

More serious challenges to the Polynomial-Time Church-Turing Thesis have also been addressed…Randomness: P = BPP under plausible assumptions(indeed, assumptions that will probably have to be proved before PNP) [NW94], [IW97], …

Nonuniform Algorithms: P/poly is “almost the same as P,” for P vs. NP purposes [Karp-Lipton 82]

Quantum Computing: BQP probably is larger than P. But even NPBQP doesn’t look like a “radically” different conjecture from PNP

Quantum Gravity? What little we know is consistent with BQP being the “end of the line”E.g., topological quantum field theories can be simulated in BQP [FKLW’02]

Achievement 2: Half a Century of Experience with Efficient

Computation, Increasing One’s Confidence That PNP

PDynamic Programming

Linear Programming

Semidefinite/Convex Programming

#P Problems with Miraculous CancellationDeterminant, counting planar perfect matchings,

3-regular-planar-mod-7-SAT…

#P Problems with Miraculous Positivity

TestMatching, Littlewood-

Richardson coefficients…

Polynomial Identity Testing (assuming P=BPP)

Matrix Group Membership

(modulo discrete log)

Polynomial Factoring

Trivial Problems

Experimental Complexity Theory

We now have a fairly impressive “statistical physics understanding” of the hardness of NP-complete problems

[Achlioptas, Ricci-Tersenghi 2006] Known heuristic CSP algorithms fail when a large connected cluster of solutions “melts” into exponentially many disconnected pieces

Claim: Had we been physicists, we would’ve long ago declared PNP a law of

nature

When people say: “What if P=NP? What if there’s an n10000 algorithm for SAT? Or an nlogloglog(n) algorithm?”

Feynman apparently had trouble accepting that P vs. NP was an open problem at all!

Response: What if the aliens killed JFK to keep him from discovering that algorithm?

“But couldn’t you have said the same about Linear Programming before Khachiyan, or primality before AKS?”No. In those cases we had plenty of hints about what was coming, from both theory and practice.“But haven’t there been lots of surprises in complexity?”

Achievement 3: Knowing What A Nontrivial Lower Bound Looks Like

Can P vs. NP Be Solved By A “Fool’s Mate?”

Fact [A.-Wigderson ’08]: Given a 3SAT formula , suppose a randomized verifier needs (polylog n) queries to to decide if is satisfiable, even given polylog(n) communication with a competing yes-prover and no-prover (both of whom can exchange private messages not seen by the other prover). Then PNP.

(A 5-line observation that everyone somehow missed?)

Suppose P=NP. Then clearly PA=NPA for all oracles A. But this is known to be false; hence PNP.

Proof: If P=NP, then NEXP=EXP=RG (where RG = Refereed Games), and indeed NEXPA=EXPA=RGA for all oracles A.

So What Does A Real Chess Match Look Like?

Time-space tradeoffs for SAT

Monotone lower bound for CLIQUE [Razborov]

Lower bounds for constant-depth circuits [FSS, Ajtai, RS]

Lower bounds on proof complexity

nlog(n) lower bound on multilinear formula size [Raz]

Lower bounds for specific algorithms (DPLL, GSAT…)

Bounds on spectral gaps for NP-complete problems [DMV, FGG]

Circuit lower bounds for PP, MAEXP, etc. [BFT, Vinodchandran, Santhanam]

Circuit lower bounds based on algebraic degree

[Strassen, Mulmuley…]

Metaquestion: Given how short these results fall of proving PNP, can we infer anything from them about what a proof of PNP would look like?

Yes! Any proof of PNP (or at least of NPP/poly, NPcoNP, etc.) will have to contain most of the known lower bounds as special cases

Analogy: We don’t have a quantum theory of gravity, but the fact that it has to contain the existing theories (QM and GR) as limiting cases constrains it pretty severely

This provides another explanation for why PNP is so hard, as well as a criterion for evaluating proposed approaches

Achievement 4: Formal Barriers

Relativization [BGS’75]: Any proof of PNP (or even NEXPP/poly, etc.) will need to use something specific about NP-complete problems—something that wouldn’t be true in a fantasy universe where P and NP machines could both solve PSPACE-complete problems for free

Algebrization [AW’08]: Any proof of PNP (or even NEXPP/poly, etc.) will need to use something specific about NP-complete problems, besides the extendibility to low-degree polynomials used in IP=PSPACE and other famous non-relativizing results

Natural Proofs [RR’97]: Any proof of NPP/poly (or even NPTC0, etc.) will need to use something specific about NP-complete problems—some property that can’t be exploited to efficiently certify a random Boolean function as hard (thereby breaking pseudorandom generators, and doing many of very things we were trying to prove intractable)

The known barriers, in one sentence each

But don’t serious mathematicians ignore all these barriers, and just plunge ahead and tackle hard problems—their minds

unpolluted by pessimism?

If you like to be unpolluted by pessimism, why are you thinking about P vs. NP?

P EXP [Hartmanis-Stearns]

PARITY AC0 [FSS, Ajtai]

MAEXP P/poly [BFT]

RELATIVIZATION

NATURAL PROOFS

ALGEBRIZATION

P NP

Achievement 5: Connections to “Real” Math

The Blum-Cucker-Shub-Smale ModelOne can define analogues of P and NP over an arbitrary field F

When F is finite (e.g., F=F2), we recover the usual P vs. NP question

When F=R or F=C, we get an interesting new question with a “mathier” feel

All three cases (F=F2, F=R, and F=C) are open, and no implications are known among them

But the continuous versions (while ridiculously hard themselves) seem likely to be “easier” than the discrete version

Even Simpler: PERMANENT vs. DETERMINANT

[Valiant 70’s]: Given an nn matrix A, suppose you can’t write per(A) as det(B), where B is a poly(n)poly(n) matrix of linear combinations of the entries of A. Then AlgNCAlg#P.

This is important! It reduces a barrier problem in circuit lower bounds to algebraic geometry—a subject about which there are yellow books.

nS

n

iiiaA

1,sgndet

nS

n

iiiaA

1,per

Mulmuley’s GCT Program:The String Theory of Computer Science

To each (real) complexity class C, one can associate a (real) algebraic variety XC

X#P(n) = “Orbit closure” of the nn Permanent function, under invertible linear transformations

of the entries

XNC(m) = “Orbit closure” of the mm Determinant

function, for some m=poly(n)

Dream: Show that X#P(n) has “too little

symmetry” to be embedded into

XNC(m). This would imply

AlgNCAlg#P.

Mulmuley’s GCT Program:The String Theory of Computer Science

But where do we get any new leverage?

Proposal: Exploit the “exceptional” nature of the Permanent and Determinant functions—the fact that these functions can be uniquely characterized by their symmetries—to reduce the embeddability problem to a problem in representation theory

(Which merely requires a generalization of a generalization of a generalization of the Riemann Hypothesis over finite fields)

Indeed, we already knew from Relativization / Algebrization / Natural Proofs that we’d have to exploit some special properties of the Permanent and Determinant, besides their being low-degree polynomials

Metaquestion: Why should P NP be provable at all?Indeed, people have speculated since the 70s about its possible independence from set theory—see [A.’03]

If P NP is a “universal mathematical statement”, why shouldn’t the proof require an infinite number of mathematical ideas?

More concretely: if the proof needs to “know” that MATCHING is in P, LINEAR PROGRAMMING is in P, etc., what doesn’t it need to know is in P?

GCT suggests one possible answer: the proof would only need to know about “exceptional” problems in P (e.g., problems characterized by their symmetries)

ConclusionsA proof of PNP might have to be the greatest synthesis of mathematical ideas ever

But don’t let that discourage you

“Obvious” starting point is PERMANENT vs. DETERMINANT

My falsifiable prediction: Progress will come not by ignoring the last half-century of complexity theory and starting afresh, but by subsuming the many disparate facts we already know into something terrifyingly bigger

If nothing else, this provides a criterion for evaluating proposed P vs. NP attempts

Open Problems• P vs. NP

• Use GCT (or pieces of it) to prove something new about computation

One natural place to look: Polynomial Identity Testing

• Evade the algebrization and natural proofs barriers, by exploiting additional “magic properties” of NP- and #P-complete problems

Beyond the locality of 3SAT and the low degree / self-correctibility of the permanent

• “Experimental complexity theory”: What else can we do?