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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 41 (16 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
Graph Homomorphisms with Complex Values: A Dichotomy Theorem
"... Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined as ZA(G) = Aξ(u),ξ(v), ξ:V →[m] (u,v)∈E where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting ..."
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Cited by 31 (14 self)
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Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined as ZA(G) = Aξ(u),ξ(v), ξ:V →[m] (u,v)∈E where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting vertex covers and kcolorings. We study the computational complexity of ZA(G) for arbitrary complex valued symmetric matrices A. Building on work by Dyer and Greenhill [6], Bulatov and Grohe [2], and especially the recent beautiful work by Goldberg,
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 17 (7 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
TimeSpace Tradeoffs for Counting NP Solutions Modulo Integers
 In Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km sat ..."
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Cited by 15 (5 self)
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We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODpSat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6Sat, as well as MODmSat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.
Dichotomy for Holant Problems of Boolean Domain
"... Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ (F), where F is a set of constraint functions on Boolean variables and taking comp ..."
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Cited by 13 (4 self)
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Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ (F), where F is a set of constraint functions on Boolean variables and taking complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #Phard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems over the complex domain, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (nonzero) vector whose inner product with itself is zero.
The Complexity of Symmetric Boolean Parity Holant Problems (Extended Abstract)
"... Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizatio ..."
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Cited by 12 (3 self)
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Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizations such as Constraint Satisfaction Problems, and directed and undirected Graph Homomorphism Problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant Problems. These additionally allow for the expression of matching problems, which have had pivotal roles in complexity theory. As our main result we prove the dichotomy theorem that, for the class ⊕P, every set of boolean symmetric Holant signatures of any arities that is not polynomial time computable is ⊕Pcomplete. The result exploits some special properties of the class ⊕P and characterizes four distinct tractable subclasses within ⊕P. It leaves open the corresponding questions for NP, #P and #kP for k ̸ = 2. 1
Constrained Codes as Networks of Relations
"... Abstract — We address the wellknown problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a twodimensional ..."
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Cited by 11 (3 self)
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Abstract — We address the wellknown problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a twodimensional constrained coding system is still an elusive research challenge. The only known exception in the twodimensional case is an exact (however, not rigorous) solution to the (1, ∞)RLL system on the hexagonal lattice. Furthermore, only exponentialtime algorithms are known for the related problem of counting the exact number of constrained twodimensional information arrays. We present the first known rigorous technique that yields an exact capacity of a twodimensional constrained coding system. In addition, we devise an efficient (polynomial time) algorithm for counting the exact number of constrained arrays of any given size. Our approach is a composition of a number of ideas and techniques: describing the capacity problem as a solution to a counting problem in networks of relations, graphtheoretic tools originally developed in the field of statistical mechanics, techniques for efficiently simulating quantum circuits, as well as ideas from the theory related to the spectral distribution of Toeplitz matrices. Using our technique we derive a closed form solution to the capacity related to the PathCover constraint in a twodimensional triangular array (the resulting calculated capacity is 0.72399217...). PathCover is a generalization of the well known onedimensional (0, 1)RLL constraint for which the capacity is known to be 0.69424... Index Terms — capacity of constrained systems, capacity of twodimensional constrained systems, holographic reductions, networks of relations, FKT method, spectral distribution of Toeplitz matrices I.
Holographic algorithms: the power of dimensionality resolved
 In: Automata, Languages and Programming. In: Lecture Notes in Comput. Sci
, 2007
"... Valiant’s theory of holographic algorithms is a novel methodology to achieve exponential speedups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that ..."
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Cited by 11 (4 self)
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Valiant’s theory of holographic algorithms is a novel methodology to achieve exponential speedups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that 2dimensional bases are universal for holographic algorithms. 1
A Complete Dichotomy Rises from the Capture of Vanishing Signatures
"... We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complexvalued symmetric constraint functions F on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions taking values in a field of characteristi ..."
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Cited by 10 (6 self)
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We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complexvalued symmetric constraint functions F on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions taking values in a field of characteristic zero. We define and characterize all symmetric vanishing signatures. They turned out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion. The Holant problem defined by a set of constraint functions F is solvable in polynomial time if it satisfies this tractability criterion, and is #Phard otherwise. The tractability criterion can be intuitively stated as follows: the set F is tractable if (1) every function in F has arity at most 2, or (2) F is transformable to an affine type, or (3) F is transformable to a product type, or (4) F is vanishing, combined with the right type of binary functions, or (5) F belongs to a special category of vanishing type Fibonacci gates. The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and
Signature Theory in Holographic Algorithms
"... In the theory of holographic algorithms proposed by Valiant, computation is expressed and processed in terms of signatures. We substantially develop the signature theory in holographic algorithms. This theory is developed in terms of drealizability and dadmissibility. For the class of 2realizable ..."
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Cited by 10 (3 self)
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In the theory of holographic algorithms proposed by Valiant, computation is expressed and processed in terms of signatures. We substantially develop the signature theory in holographic algorithms. This theory is developed in terms of drealizability and dadmissibility. For the class of 2realizable signatures we prove a Birkhofftype theorem which determines this class. It gives a complete structural understanding of the relationship between 2realizability and 2admissibility. This is followed by characterization theorems for 1realizability and 1admissibility. Finally, using this theory of general (i.e., unsymmetric) It is generally conjectured that many combinatorial problems in the class NP or #P are not computable in polynomial time. The prevailing opinion is that these problems seem to require the accounting or processing of exponentially many potential solution fragments to the problem. However it is rather natural, and it should not cause any surprise, that the answer to such a problem can in general be