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Friday, December 23, 2005
All I Want for Christmas is a Proof that P≠NP
Posted by Lance
For the first time in my lifetime, Christmas and Chanukah land on the same day. So to all my readers, Happy Holidays!And what do we find under our theory Christmas tree? Our first present is a new randomized polynomial-time algorithm for linear programming from Kelner and Spielman. The card reads
In this paper, we present the first randomized polynomial-time simplex method. Like the other known polynomial-time algorithms for linear programming, the running time of our algorithm depends polynomially on the bit-length of the input. We do not prove an upper bound on the diameter of polytopes. Rather we reduce the linear programming problem to the problem of determining whether a set of linear constraints defines an unbounded polyhedron. We then randomly perturb the right-hand sides of these constraints, observing that this does not change the answer, and use a shadow-vertex simplex method to try solve the perturbed problem. When the shadow-vertex method fails, it suggests a way to alter the distributions of the perturbations, after which we apply the method again. We prove that the number of iterations of this loop is polynomial with high probability.Our next present is a list-decodable code from Guruswami and Rudra. Back in October I posted about the Parvaresh-Vardy code that list decode a 1-ε fraction of errors using a code of rate O(ε/log(1/ε)). Guruswami and Rudra, for any constant δ>0, create a code with rate ε-δ.Many more presents under the tree, presumably many STOC and Complexity submissions. No proofs (at least correct ones) that P≠NP this year but maybe Santa will come through for us in time for next Christmas.
8:47 AM #
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The Kelner-Spielman paper is great
but I think people should
know the following. There are
two main reasons for people's
interest in poly-time simplex type algorithms.
The first is to explain why simplex
performs so well on many real-world
problems. Second, simplex seems to
be the only candidate that
could lead to a strongly polynomial
time algorithm for LP, this being
the main theoretical motivation. Unfortunately
the Kelner-Spielman paper does
several things in addition to
running the shadow vertex simplex
that causes a dependence on numbers.
They have to avoid degeneracy which
is explicitly introduced by their
approach, by doing perturbation.
They need to sample implicity
from an affine transformation that
makes the polytope rounded. Both
steps cause the algorithm to
depend on the bits of the matrix
and it is unclear how this
can be avoided by their approach.
It is nice cute paper following
some geometric ideas in
Spielman-Teng smoothed analysis
for LP. However, there is some
danger here that
people might think that the
main problem is solved. It doesn't
seem that way (IMHO). It would
be nice if other more knowleadgeable
folk can add to the discussion.
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I think Christmas and Chanukah land on the same day every 19 years.
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"...but maybe Santa will come through for us in time for next Christmas"
Then it won't be Christmas, it will be Armageddon.
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Happy Holidays Lance,
Thanks for a great year of blogging!
Amar
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Isn't the Kelner-Spielman algorithm closer to ellipsoid than it is to simplex? I mean, in the ellipsoid algorithm, you guess a bound on the optimal solution (which is where the dependence on input numbers comes in) and then check for feasibility. Here, (I think) they guess a bound on the optimal solution and then check for feasibility via boundedness. A real "simplex" algorithm should have the property that it takes some series of steps along a single polytope. I see how their algorithm is related to a simplex like algorithm, but I really don't see how it can be called a simplex algorithm.
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I think there are some misconceptions in the above posts about the Kelner-Spielman algorithm that ought to be remedied.
1. The perturbation to avoid degeneracy that they use does not introduce any additional dependence on the precision of the inputs. As noted in Megiddo's original paper, it can be performed purely formally in terms of a symbolic parameter. The only reason that their algorithm isn't strongly polynomial is that it has to repeatedly revise the probability distribution that it uses for its simplex pivots, and the number of times that it needs to do so depends on the bit-length of the input. It is not obvious that this dependency can be eliminated, but it is also not obvious that it cannot be (by a different pivot rule). This seems to be real progress towards a strongly polynomial algorithm. In particular, it provides a combinatorial algorithmic approach that can work without proving the Hirsch conjecture. This removes a big roadblock...
2. Their algorithm really is a simplex algorithm. They don't walk along the *feasible* vertices of a single polytope, but they do walk along the potential (feasible or infeasible) vertices of a single polytope. Many other simplex algorithms attempt to do this too (e.g., the self-dual simplex method). It's just that nobody had previously been able to design such an algorithm to run in poly time. The geometric stuff that the poster said looked like ellipsoid appeared in the analysis, not in the algorithm--the algorithm really is just a sequence of combinatorial pivots.
Independently of the question of strong polynomiality, it has been a very long open question whether any simplex method can run in poly time. Their paper resolves that, which is really exciting.
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"All I Want for Christmas is a Proof that P?NP"
For 2006 ... be nice. I have several elves randomly typing on PCs. Other elves have designed a Turing enumerator for generating proofs. What else can I do? Help me help you!
Dr. S. Claus.
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id like to begin to my question with many apologies
i am an electrical engineering student dealing with optimization in my project. I need to find the best method for solving quadratic programming problem with constraints. Up to now I found active set method, simplex and interior point. From the complexity view of the methods seems interior point is the best one. Is it true? is there a nother type of algorithm?
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