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Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 19 (10 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
Bases Collapse in Holographic Algorithms
 Electronic Colloquium on Computational Complexity Report
, 2007
"... Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) c ..."
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Cited by 7 (2 self)
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Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #Pcomplete, and counting mod 2 is NPhard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2. 1
From Holant To #CSP And Back: Dichotomy For Holant c Problems
"... We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted Hcolorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results fro ..."
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Cited by 7 (4 self)
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We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted Hcolorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results from one framework can be used to derive results in another, and this happens in both directions. Holographic reductions discover an underlying unity, which is only revealed when these counting problems are investigated in the complex domain C. We prove three complexity dichotomy theorems, leading to a general theorem for Holant c problems. This is the natural class of Holant problems where one can assign constants 0 or 1. More specifically, given any signature grid on G = (V, E) over a set F of symmetric functions, we completely classify the complexity to be in P or #Phard, according to F, of X Y fv(σ E(v)), σ:E→{0,1} v∈V where fv ∈ F ∪ {0, 1} (0, 1 are the unary constant 0, 1 functions). Not only is holographic reduction the main tool, but also the final dichotomy can be only naturally stated in the language of holographic transformations. The proof goes through another dichotomy theorem on boolean complex weighted
Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP
"... Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a un ..."
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Cited by 4 (1 self)
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Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #Phard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary realvalued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #Phard on general graphs but tractable on planar graphs, or (3) those which are #Phard even on planar graphs. The classification criteria
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Cited by 2 (2 self)
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.
Holographic Reduction, Interpolation and Hardness
"... We prove a dichotomy theorem for a class of counting problems expressible by Boolean signatures. The proof methods are holographic reductions and interpolations. We show that interpolatability provides a universal strategy to prove #Phardness for this class of problems. For these problems whenever ..."
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Cited by 1 (0 self)
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We prove a dichotomy theorem for a class of counting problems expressible by Boolean signatures. The proof methods are holographic reductions and interpolations. We show that interpolatability provides a universal strategy to prove #Phardness for this class of problems. For these problems whenever holographic reductions followed by interpolations fail to prove #Phardness, we can show that the problems are actually solvable in polynomial time. 1
List of Publications and Articles Submitted for
"... 2011 To my parents and my wife ii iii Curriculum Vitae ..."
The Complexity of Planar Boolean #CSP with Complex Weights ⋆
"... Abstract. We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This ..."
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Abstract. We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3, 3) to counting the number of Eulerian orientations over planar 4regular graphs to show the latter is #Phard. This strengthens a theorem by Huang and Lu to the planar setting. 1