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Understanding the MulmuleySohoni approach to P vs
 NP, Bulletin of the EATCS
, 2002
"... We explain the essence of K. Mulmuley and M. Sohoni, “Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems ” [MS02] for a general complexitytheory audience. We evaluate the power and prospects of the new approach. The emphasis is not on probing the deep mathematics that u ..."
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We explain the essence of K. Mulmuley and M. Sohoni, “Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems ” [MS02] for a general complexitytheory audience. We evaluate the power and prospects of the new approach. The emphasis is not on probing the deep mathematics that underlies this work, but rather on helping computational complexity theorists not versed in its background to understand the combinatorics involved. 1
A Lower Bound for the Shortest Path Problem
"... We show that the Shortest Path Problem cannot be solved in o(log n) time on an unbounded fanin PRAM without bit operations using poly(n) processors even when the bitlengths of the weights on the edges are restricted to be of size O(log 3 n). This shows that the matrixbased repeated squaring al ..."
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We show that the Shortest Path Problem cannot be solved in o(log n) time on an unbounded fanin PRAM without bit operations using poly(n) processors even when the bitlengths of the weights on the edges are restricted to be of size O(log 3 n). This shows that the matrixbased repeated squaring algorithm for the Shortest Path Problem is optimal in the unbounded fanin PRAM model without bit operations. 1
On P vs. NP, Geometric Complexity Theory, and the Riemann Hypothesis
, 2009
"... Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems suggested in a series of articles we call GCTlocal [27], GCT18 [30][35], and GCTflip [28]. A high level overview of this research plan and the results obtained so far was presented in a series of three lectures i ..."
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Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems suggested in a series of articles we call GCTlocal [27], GCT18 [30][35], and GCTflip [28]. 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 911, 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. For those who are interested in a short mathematical overview, the first lecture (chapter) of this article gives this. The video lectures for this series are available at:
On P vs NP, geometric complexity theory, and the flip I: a high–level view
, 2007
"... Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems through algebraic geometry and representation theory. This article gives a highlevel exposition of the basic plan of GCT based on the principle, called the flip, without assuming any background in algebraic geomet ..."
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Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems through algebraic geometry and representation theory. This article gives a highlevel exposition of the basic plan of GCT based on the principle, called the flip, without assuming any background in algebraic geometry or representation theory.
On P vs. NP, Geometric Complexity Theory, Explicit proofs, and the Complexity Barrier
, 2009
"... 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. ..."
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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.
On P vs. NP and Geometric Complexity Theory Dedicated to Sri Ramakrishna
, 2011
"... This article gives an overview of the geometric complexity theory (GCT) approach towards the P vs. NP and related problems focussing on its main complexity theoretic results. These are: (1) two concrete lower bounds, which are currently the best known lower bounds in the context of the P vs. NC and ..."
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This article gives an overview of the geometric complexity theory (GCT) approach towards the P vs. NP and related problems focussing on its main complexity theoretic results. These are: (1) two concrete lower bounds, which are currently the best known lower bounds in the context of the P vs. NC and permanent vs. determinant problems, (2) the Flip Theorem, which formalizes the self referential paradox in the P vs. NP problem, and (3) the Decomposition Theorem, which decomposes the arithmetic P vs. NP and permanent vs. determinant problems into subproblems without self referential difficulty, consisting of positivity hypotheses in algebraic geometry and representation theory and easier hardness hypotheses. 1
This document in subdirectoryRS/98/14/ The hardness of speedingup Knapsack
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• Manindra Agrawal and his students Neeraj Kayal and Nitin Saxena at ITT Kanpur have given
"... This has been an exciting summer for computational complexity. ..."
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The hardness of speedingup Knapsack
, 1998
"... ... rounds even by using 2 p n processors. We extend the result to the PRAM model without bitoperations. These results are consistent with Mulmuley's [6] recent result on the separation of the stronglypolynomial class and the corresponding N C class in the arithmetic PRAM model. ..."
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... rounds even by using 2 p n processors. We extend the result to the PRAM model without bitoperations. These results are consistent with Mulmuley's [6] recent result on the separation of the stronglypolynomial class and the corresponding N C class in the arithmetic PRAM model.