Deolalikar P vs NP paper

From Polymath Wiki
Revision as of 07:53, 11 August 2010 by Teorth (talk | contribs) (→‎Media and aggregators: New Slashdot article)
Jump to navigationJump to search

Note: This is an UNOFFICIAL page on Deolalikar's P!=NP paper; it is not affiliated with a Polymath project.

This is a clearinghouse wiki page for the analysis of Vinay Deolalikar's recent preprint claiming to prove that P != NP, and to aggregate various pieces of news and information about this paper. Corrections and new contributions to this page are definitely welcome. Of course, any new material should be sourced whenever possible, and remain constructive and objectively neutral; in particular, personal subjective opinions or speculations are to be avoided. This page is derived from an earlier collaborative document created by Suresh Venkatasubramanian.

For the latest discussion on the technical points of the paper, see this thread of Dick Lipton and Ken Regan. For meta-discussion of this wiki (and other non-mathematical or meta-mathematical issues), see this thread of Suresh Venkatasubramanian.

The paper

These links are taken from Vinay Deolalikar's web page.

  1. First draft, Aug 6, 2010
  2. Second draft Aug 9, 2010. File removed, Aug 10 2010.
  3. draft 2 + ε, Aug 9 2010.

Here is the list of updates between the different versions.

Typos and minor errors

  • (Second draft, page 31, Definition 2.16): "Perfect man" should be "Perfect map". (via Blake Stacey)
  • (Second draft) Some (but not all) of the instances of the [math]\displaystyle{ O() }[/math] notation should probably be [math]\displaystyle{ \Theta() }[/math] or [math]\displaystyle{ \Omega() }[/math] instead, e.g. on pages 4, 9, 16, 28, 33, 57, 68, etc. (via András Salamon)
  • (Second draft, page 27) [math]\displaystyle{ n 2^n }[/math] independent parameters → [math]\displaystyle{ n 2^k }[/math] independent parameters
  • (draft 2 + e, p.34, Def. 3.8): [math]\displaystyle{ n }[/math] → [math]\displaystyle{ k }[/math]
  • (Second draft, page 52) [math]\displaystyle{ \sum C_{li}S_i-k\gt 0 }[/math] → [math]\displaystyle{ \sum C_{li}S_i+k\gt 0 }[/math]

Proof strategy

(Excerpted from this comment of Ken Regan)

Deolalikar has constructed a vocabulary V which apparently obeys the following properties:

  1. Satisfiability of a k-CNF formula can be expressed by NP-queries over V—in particular, by an NP-query Q over V that ties in to algorithmic properties.
  2. All P-queries over V can be expressed by FO(LFP) formulas over V.
  3. NP = P implies Q is expressible by an FO(LFP) formula over V.
  4. If Q is expressible by an LFP formula over V, then by the algorithmic tie-in, we get a certain kind of polynomial-time LFP-based algorithm.
  5. Such an algorithm, however, contradicts known statistical properties of randomized k-SAT when k >= 9.

An alternate perspective

Leonid Gurvits:

...the discrete probabilistic distributions in the paper can be viewed as tensors, or very special multilinear polynomials. The assumptions “P=NP” somehow gives a (polynomial?) upper bound on the tensor rank. And finally, using known probabilistic results, he gets nonmatching (exponential?) lower bound on the same rank.

If I am right, then this approach is a very clever, in a good sense elementary, way to push the previous algebraic-geometric approaches.

Possible issues

Issues with LFP

There appear to be 4 issues related to the use of the characterization of P in terms of first order logic, an ordering and a least fixed point operator. All of these are discussed in the Lipton/Regan post, with contributions from David Barrington, Paul Christiano, Lance Fortnow, James Gate, Arthur Milchior, Charanjit Jutla and Julian Bradfield.

  • Is the lack of ordering in the logical structures used to define the LFP structure a problem (since parity can not be expressed without an ordering even with LFP, hence P is not captured without order).
In chapter 7 this issue seems to disappear since he introduces a successor relation over the variables [math]\displaystyle{ x_1\lt \dots\lt x_n\lt \neg x_1\lt \dots\lt \neg x_n }[/math].
If it was possible to express k-SAT in FO(NLFP,without succ) (NLFP=non deterministic LFP) or in relational-NP, as introduced in [AVV1997] then by an extension of the Abiteboul-Vianu theorem it would be enough to prove that k-SAT is not in FO(LFP,without succ). This would avoid the problem of the order
  • The paper requires that a certain predicate in the FO(LFP) formula be unary, and forces this by expanding neighborhoods and constructing k-tuples of parameters to act as single parameters. It is not clear how this affects the arguments about the propagation of local neighborhoods.
  • Does the logical vocabulary created to express the LFP operation suffice to capture all P-time operations ?
  • Charanjit Jutla has pointed out that the argument in section 4.3 (with which several other people have also had issues) depends on the absence of a greatest fixed point. "This is a usual mistake most people new to fixed-point logics fall prey to. For example, now he has to deal with formulas of the kind [math]\displaystyle{ \nu x (f(y, x) \and g(y, x)). }[/math] Section 4.2 deals with just one least fixed point operator…where his idea is correct. But, in the next section 4.3, where he deals with complex fixed point iterations, he is just hand waving, and possibly way off."
A few comments later, he appears to revise this objection, while bringing up a new issue about the boundedness of the universe relating to the LFP operator.

Issues with phase transitions

A brief synopsis of the terms discussed can be found here

  1. The nomenclature of phase transitions: In the statistical physics picture, there is not a single phase transition, but rather a set of different well defined transitions called clustering (equivalently d1RSB), condensation, and freezing (Florent Krzakala and Lenka Zdeborova). In the current version of the paper, properties of d1RSB (clustering), and freezing are mixed-up. Whereas following the established definitions, and contrary to some earlier conjectures, it is now agreed that some polynomial algorithms work beyond the d1RSB (clustering) or condensation thresholds. Graph coloring provides some evidence of this when one compares the performance of algorithms with the statistical physics predictions. The property of the solution space of random K-SAT the paper is actually using is called freezing. It was conjectured in the statistical physics community (Florent Krzakala, Lenka Zdeborova and Cris Moore) that really hard instances appears in the frozen phase, i.e. when all solutions have non-trivial cores. Existence of such a region was proven rigorously by Achlioptas and Ricci-Tersenghi and their theorem appears as Theorem 5.1 in the paper.
  2. The XOR-SAT objection : The conjecture that frozen variables make a problem hard is however restricted to NP-complete problems such as K-SAT and Q-COL. Indeed a linear problem such as random k-XORSAT also has a clustering transition, frozen variables, etc., and is not easy to solve with most algorithms, but is of course in P as one can use Gauss elimination and exploit the linear structure to solve it in polynomial time (Cris Moore, Alif Wahid, and Lenka Zdeborova). Similar problem might exists in other restricted CSP which are in P, but may exibit freezing stage, as pointed by several other people.

Issues with random k-SAT

  1. Complex solution spaces are uncorrelated with time complexity. (The below is a greatly expanded version of a series of twitter comments by Ryan Williams, on twitter) The author tries to use the fact that for certain distributions of random k-SAT, the solution space has a "hard structure". For certain parameterizations, the space of satisfying assignments to a random k-SAT instance has some intriguing structure. If SAT is in P, then SAT can be captured in a certain logic (equivalent to P in some sense). The author claims that anything captured in this logic can't have a solution space with this intriguing structure. There are two "meta" objections to this. One is that "intriguing structure in the solution space is not sufficient for NP hardness". The second is that "intriguing structure is not necessary for NP hardness". They don't actually point to a place where the proof is wrong. But they do appear to give an obstacle to the general proof method.
    1. Polytime solvable problems (such as perfect matching on random graphs) can also have complicated solution distributions. In fact it is not hard to design 2-SAT formulas (in this case not random, but specifically designed ones) so that they have exponentially many clusters of solutions, each cluster being "far" from the others, etc. That is, the fact that random k-SAT has a "hard" distribution of solutions does not seem to be relevant for proving a time lower bound on k-SAT. It is not sufficient to use a problem with a hard distribution of solutions, if you're separating P from NP. This is the objection which seems most germane to the current proposed proof: it opposes the claim that "anything in P can't have a solution space with this intriguing structure". It appears there must be some error in either the translation to this logic, or the analysis of solution spaces that this logic permits.
    2. Moreover, it's also worth pointing out that a hard distribution of solutions is not necessary for NP-hardness, either. A weird distribution is not what makes a problem hard, it's the representation of that solution space (e.g., a 3-CNF formula, a 2-CNF formula, etc.). The "hard" case of 3-SAT is the case where there is at most one satisfying assignment. There is a randomized reduction from 3-SAT to 3-SAT with at most ONE satisfying assignment (Valiant-Vazirani). This reduction increases the number of clauses and the number of variables, but that doesn't really matter. The point is that you can always reduce 3-SAT with a "complex" solution space to one with an "easy" solution space, so how can a proof separating P from NP rely on the former? Suppose Valiant-Vazirani can be derandomized to run in deterministic polynomial time (which is true if plausible circuit lower bounds hold up). For every LFP formula F that is to solve k-SAT, replace it with an LFP formula F' that has computationally equivalent behavior to the following algorithm: first "Valiant-Vazirani-ize" your input formula (reduce it to having at most one solution in polynomial time) then evaluate F on the result. These new formulas only have at most "one" solution to deal with. To summarize, there is essentially no correlation between the "hard structure" of the solution space for instances of some problem, and the NP-hardness of that problem.

Barriers

Any P vs NP proof must deal with the three known barriers described below. The concerns around this paper have, for the most part, not yet reached this stage yet.

Relativization

Quick overview of Relativization Barrier at Shiva Kintali's blog post

Natural proofs

See Razborov and Rudich, "Natural proofs" Proceedings of the twenty-sixth annual ACM symposium on Theory of computing (1994).

Algebrization

See Aaronson and Widgerson, "Algebrization: A New Barrier in Complexity Theory" ACM Transactions on Computation Theory (2009).

The paper is all about the local properties of a specific NP-complete problem (k-SAT), and for that reason, I don't think relativization is relevant. Personally, I'm more interested in why the argument makes essential use of uniformity (which is apparently why it's supposed to avoid Razborov-Rudich). (Scott Aaronson)

Terminology

Online reactions

Theoretical computer science blogs

Media and aggregators

8th August

9th August

10th August

11th August

Real-time searches

Other

Additions to the above list of links are of course very welcome.

Timeline

Bibliography

Other links

Further reading

Given the current interest in the subject matters discussed on this page, it would be good to start collecting a list of references where people can learn more about such topics. Please add liberally to the list below.

Retrieved from "https://michaelnielsen.org/polymath/index.php?title=Deolalikar_P_vs_NP_paper&oldid=3395"