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Register a new userReport on Mathematical Aspects of P vs. NP and its Variants.
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Joshua Grochow
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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This is a report on a workshop held August 1 to August 5, 2011 at the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University, Providence, Rhode Island, organized by Saugata Basu, Joseph M. Landsberg, and J. Maurice Rojas. We provide overviews of the more recent results presented at the workshop, including some works-in-progress as well as tentative and intriguing ideas for new directions. The main themes we discuss are representation theory and geometry in the Mulmuley-Sohoni Geometric Complexity Theory Program, and number theory and other ideas in the Blum-Shub-Smale model.
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Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and related problems. A high level overview of this research plan and the results obtained so far was presented in a series of three lectures in the Institute of Advanced study, Princeton, Feb 9-11, 2009. This article contains the material covered in those lectures after some revision, and gives a mathematical overview of GCT. No background in algebraic geometry, representation theory or quantum groups is assumed.
Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems. This article gives its complexity theoretic overview without assuming any background in algebraic geometry or representation theory.
This article provide new approach to solve P vs NP problem by using cardinality of bases function. About NP-Complete problems, we can divide to infinite disjunction of P-Complete problems. These P-Complete problems are independent of each other in disjunction. That is, NP-Complete problem is in infinite dimension function space that bases are P-Complete. The other hand, any P-Complete problem have at most a finite number of P-Complete basis. The reason is that each P problems have at most finite number of Least fixed point operator. Therefore, we cannot describe NP-Complete problems in P. We can also prove this result from incompleteness of P.
Mahaneys Theorem states that, assuming $mathsf{P} eq mathsf{NP}$, no NP-hard set can have a polynomially bounded number of yes-instances at each input length. We give an exposition of a very simple unpublished proof of Manindra Agrawal whose ideas appear in Agrawal-Arvind (Geometric sets of low information content, Theoret. Comp. Sci., 1996). This proof is so simple that it can easily be taught to undergraduates or a general graduate CS audience - not just theorists! - in about 10 minutes, which the author has done successfully several times. We also include applications of Mahaneys Theorem to fundamental questions that bright undergraduates would ask which could be used to fill the remaining hour of a lecture, as well as an application (due to Ikenmeyer, Mulmuley, and Walter, arXiv:1507.02955) to the representation theory of the symmetric group and the Geometric Complexity Theory Program. To this author, the fact that sparsity results on NP-complete sets have an application to classical questions in representation theory says that they are not only a gem of classical theoretical computer science, but indeed a gem of mathematics.
The proof of Todas celebrated theorem that the polynomial hierarchy is contained in $P^{# P}$ relies on the fact that, under mild technical conditions on the complexity class $C$, we have $exists C subset BP cdot oplus C$. More concretely, there is a randomized reduction which transforms nonempty sets and the empty set, respectively, into sets of odd or even size. The customary method is to invoke Valiants and Vaziranis randomized reduction from NP to UP, followed by amplification of the resulting success probability from $1/poly(n)$ to a constant by combining the parities of $poly(n)$ trials. Here we give a direct algebraic reduction which achieves constant success probability without the need for amplification. Our reduction is very simple, and its analysis relies on well-known properties of the Legendre symbol in finite fields.
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