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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
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
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, 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, Explicit proofs, and the Complexity Barrier, under preparation, to be available at the above website soon
"... 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. 1 ..."
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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. 1
Orbits and Arithmetical Circuit Lower Bounds
"... Abstract. The orbit of a regular function f over a field F under action by a matrix group G is the collection of functions f(Ex) for E ∈ G. We show that some lower bounds of Bürgisser and Lotz [BL03] and Shpilka and Wigderson [SW99] in restricted arithmetical circuit/formula models extend to orbits, ..."
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Abstract. The orbit of a regular function f over a field F under action by a matrix group G is the collection of functions f(Ex) for E ∈ G. We show that some lower bounds of Bürgisser and Lotz [BL03] and Shpilka and Wigderson [SW99] in restricted arithmetical circuit/formula models extend to orbits, where E does not count against the complexity bounds and is not subject to the (same) restriction(s). Our “orbit model ” and a second “linearcombination model ” aim to bridge the gap between the boundedcoefficient linear/bilinear circuit model of [Mor73,NW95,Raz03,BL03] and the arbitrarycoefficient case. We extend sizedepth tradeoff methods of Lokam [Lok01] to the latter model. Variants of the BaurStrassen “Derivative Lemma ” are developed, including one that can be iterated for sums of higher partial derivatives. 1
A Lower Bound on Computing Blocking Flows in Graphs
"... We show that the Maximum Blocking Flow problem on a graph of n vertices cannot be computed in time o(n 1/8 ) with polynomially many processors on an unbounded fanin PRAM without bit operations, even when the bitlengths of the inputs are restricted to be of size O(n 2 ). 1 ..."
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We show that the Maximum Blocking Flow problem on a graph of n vertices cannot be computed in time o(n 1/8 ) with polynomially many processors on an unbounded fanin PRAM without bit operations, even when the bitlengths of the inputs are restricted to be of size O(n 2 ). 1
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