## On Valiant’s holographic algorithms

Citations: | 2 - 2 self |

### BibTeX

@MISC{Cai_onvaliant’s,

author = {Jin-yi Cai},

title = {On Valiant’s holographic algorithms},

year = {}

}

### OpenURL

### Abstract

Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups 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.

### Citations

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Computers and Intractability, A Guide to Theory of NP-Completeness
- Garey, Johnson
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(Show Context)
Citation Context ... of NP-complete problems include the above mentioned SAT, Graph 3-Coloring (given a graph, is it 3-colorable), Hamiltonicity (given a graph, does it contain a Hamiltonian circuit), and thousands more =-=[19]-=-. Typically, to solve an NP-complete problem it seems to require the examination of exponentially many possibilities. Whether this is intrinsically the case, is a major open problem in Theoretical Com... |

1500 |
Reducibility among combinatorial problems
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(Show Context)
Citation Context ...restricted versions of SAT, called 3SAT, where each clause Cj contains 3 literals, is also NP-complete. (Note however the 2SAT problem where each clause has 2 literals is in P.) Soon afterwards, Karp =-=[29]-=- proved a host of other problems to be NP-complete, making NP-completeness a ubiquitous tool to prove (relative) intractability. Levin [34] in the former Soviet Union independently discovered NP-compl... |

718 | Proof verification and hardness of approximation problems
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context ...o note that if instead of considering node deletion we consider edge deletion, this is just another way of defining the well known problem of MAX-CUT, which is NP-hard (and even NPhard to approximate =-=[22, 3, 26]-=-). The holographic algorithm by Valiant above is the first polynomial time algorithm for PL-NODE-BIPARTITION [54]. On the other hand, planar MAX-CUT is known to be in P [24]. In [6] a joint generaliza... |

657 | Some optimal inapproximability results
- Håstad
(Show Context)
Citation Context ...o note that if instead of considering node deletion we consider edge deletion, this is just another way of defining the well known problem of MAX-CUT, which is NP-hard (and even NPhard to approximate =-=[22, 3, 26]-=-). The holographic algorithm by Valiant above is the first polynomial time algorithm for PL-NODE-BIPARTITION [54]. On the other hand, planar MAX-CUT is known to be in P [24]. In [6] a joint generaliza... |

507 |
The complexity of theorem proving procedures
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(Show Context)
Citation Context ...f exponentially many possibilities. Whether this is intrinsically the case, is a major open problem in Theoretical Computer Science and in Mathematics, and is known as the P vs. NP problem [13]. Cook =-=[15]-=- was the first to introduce NP-completeness and proved that the problem SAT is NP-complete. He also proved some restricted versions of SAT, called 3SAT, where each clause Cj contains 3 literals, is al... |

500 |
Recursively enumerable sets and degrees
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(Show Context)
Citation Context ...t of computation and what is, and is not, ultimately computable. This is considered well established now and is encapsulated by the model of Turing machines, or by Gödel’s general recursive functions =-=[48]-=-. An important harbinger which motivated much of this development is Hilbert’s 10th problem, which asks for an algorithmic procedure to decide whether a general Diophantine equation (a ∗ Supported in ... |

389 |
Some simplified NP-complete graph problems
- Garey, Johnson, et al.
- 1976
(Show Context)
Citation Context ...neral approach to such “node deletion” problems. Note that numerous other planar NP-complete problems, such as Hamiltonian cycles and minimum vertex covers are NP-complete already for degree 3 (e.g., =-=[20]-=-, and [19]). For this problem, Valiant introduced another idea of applying the general machinery of the Holant. We will see that with an arbitrary real value x, the recognizer signature (x,1,1,1,1,1,1... |

329 |
Maximum matching and a polyhedron with 0,1-vertices
- Edmonds
- 1965
(Show Context)
Citation Context ... focus on ultimate computability in a logical sense, regardless of how long the computation may last, as long as it is finite. Starting in the 1960’s, a new focus was given for efficient computations =-=[14, 16, 25]-=-. To capture this notion, complexity classes were defined, the most prominent among them were the classes P and NP. Loosely speaking, P denotes the class of all problems which can be solved by an algo... |

199 |
On the computational complexity of algorithms
- Hartmanis, Stearns
- 1965
(Show Context)
Citation Context ... focus on ultimate computability in a logical sense, regardless of how long the computation may last, as long as it is finite. Starting in the 1960’s, a new focus was given for efficient computations =-=[14, 16, 25]-=-. To capture this notion, complexity classes were defined, the most prominent among them were the classes P and NP. Loosely speaking, P denotes the class of all problems which can be solved by an algo... |

190 |
Planar formulae and their uses
- Lichtenstein
- 1982
(Show Context)
Citation Context ...Output: The number satisfying assignments of Φ. This is a constrained satisfiability problem, where we are given a planar formula which is a Boolean conjunction of NOT-ALL-EQUAL clauses. Lichtenstein =-=[37]-=- defined the notion of planar formulae. A Boolean formula is planar if it can be represented by a planar graph where vertices represent variables and clauses, and there is an edge iff the variable or ... |

190 |
PP is as hard as the polynomial-time hierarchy
- Toda
- 1991
(Show Context)
Citation Context ...ain threshold for a given graph, etc. It is known that the trivial containment P ⊆ NP ⊆ PH ⊆ PSPACE holds. Also trivially #P can compute NP and can be computed in PSPACE. A well known theorem by Toda =-=[50]-=- says that the class #P is at least as hard as PH. Thus #P-completeness is harder than NP-completeness, assuming the standard complexity theory hypothesis that the polynomial time hierarchy does not c... |

157 |
The intrinsic computational difficulty of functions
- Cobham
- 1962
(Show Context)
Citation Context ... focus on ultimate computability in a logical sense, regardless of how long the computation may last, as long as it is finite. Starting in the 1960’s, a new focus was given for efficient computations =-=[14, 16, 25]-=-. To capture this notion, complexity classes were defined, the most prominent among them were the classes P and NP. Loosely speaking, P denotes the class of all problems which can be solved by an algo... |

149 |
Kneser’s conjecture, chromatic number, and homotopy
- Lovász
- 1978
(Show Context)
Citation Context ...e they are particularly desirable. The following theorem is a complete characterization of 2-admissibility over fields of characteristic 0. It uses rank estimates related to the Kneser Graph KG2k+1,k =-=[32, 42, 45, 17, 18, 23, 36]-=-. Theorem 5.3 Let G be a tensor of arity n. G is 2-admissible iff (1) n = 2k is even; (2) all GS = 0 except for |S| = k; and (3) for all T ⊂ [n] with |T | = k + 1, � G S = 0. (7) S⊂T,|S|=k 1 �2k� The ... |

101 |
Graph theory and crystal physics
- Kasteleyn
- 1967
(Show Context)
Citation Context ...tivated the concept of P initially.) Another surprisingly good algorithm is the Fisher-Kestelyan-Temperley method, which can count the number of perfect matchings in a planar graph in polynomial time =-=[30, 31, 49]-=-. For weighted graphs, this method can find the sum of all weights of perfect matchings M, where the weight of a perfect matching M is the product of all the weights of the matching edges in M. The co... |

76 |
Finding a Maximum Cut of a Planar Graph in Polynomial Time
- Hadlock
- 1975
(Show Context)
Citation Context ...hard to approximate [22, 3, 26]). The holographic algorithm by Valiant above is the first polynomial time algorithm for PL-NODE-BIPARTITION [54]. On the other hand, planar MAX-CUT is known to be in P =-=[24]-=-. In [6] a joint generalization was shown to be also solvable in polynomial time by holographic algorithms. PL-NODE-EDGE-BIPARTITION Input: A planar graph G = (V,E) of maximum degree 3. A non-negative... |

74 | The complexity of counting in sparse, regular, and planar graphs
- Vadhan
(Show Context)
Citation Context ...thod is again crucial. We now consider a matching problem. We note that Jerrum [28] showed that counting the number of (not necessarily perfect) matchings in a planar graph is #P-complete, and Vadhan =-=[51]-=- subsequently proved that it remains #P-complete even for planar bipartite graphs of degree six. For degree two the problem can be solved easily and one might have conjectured that all other nontrivia... |

66 |
Dimer problem in statistical mechanics — an exact result
- Temperley, Fisher
- 1961
(Show Context)
Citation Context ...tivated the concept of P initially.) Another surprisingly good algorithm is the Fisher-Kestelyan-Temperley method, which can count the number of perfect matchings in a planar graph in polynomial time =-=[30, 31, 49]-=-. For weighted graphs, this method can find the sum of all weights of perfect matchings M, where the weight of a perfect matching M is the product of all the weights of the matching edges in M. The co... |

61 |
The statistics of dimers on a lattice
- Kasteleyn
- 1961
(Show Context)
Citation Context ...tivated the concept of P initially.) Another surprisingly good algorithm is the Fisher-Kestelyan-Temperley method, which can count the number of perfect matchings in a planar graph in polynomial time =-=[30, 31, 49]-=-. For weighted graphs, this method can find the sum of all weights of perfect matchings M, where the weight of a perfect matching M is the product of all the weights of the matching edges in M. The co... |

57 |
Recognizing primes in random polynomial time
- Adleman, Huang
- 1988
(Show Context)
Citation Context ... difficulty of recognizing primes and the related (but separate) problem of integer factorization already fascinated Gauss, and are closely tied to the structural theory of the distribution of primes =-=[21, 1, 2]-=-. The precise formulation of the concept of computation can be traced to the work of Gödel, Turing, and other logicians in the 1930’s, who were particularly concerned with foundational questions in ma... |

54 |
The node-deletion problem for hereditary properties is NP-complete
- Yannakakis, Lewis
- 1980
(Show Context)
Citation Context ... The cardinality of a smallest subset V ′ ⊂ V such that the deletion of V ′ and its incident edges results in a bipartite graph. This problem is known to be NP-complete for maximum degree 6 [33]. See =-=[35]-=- for a general approach to such “node deletion” problems. Note that numerous other planar NP-complete problems, such as Hamiltonian cycles and minimum vertex covers are NP-complete already for degree ... |

53 |
Matrices and Matroids for Systems Analysis
- Murota
- 2000
(Show Context)
Citation Context ...The coefficient ǫπ of this monomial is the parity of the number of overlapping pairs of edges, in the sense defined earlier. The following theorem states the Grassmann-Plücker identities. Theorem 2.1 =-=[46]-=- For any n × n skew-symmetric matrix M, and for any I = {i1,... ,iK} ⊆ [n] and J = {j1,... ,jL} ⊆ [n], the following is called the Grassmann-Plücker identities, L� (−1) l K� Pf(jl,i1,... ,iK)Pf(j1,...... |

45 |
Two-dimensional monomer-dimer systems are computationally intractable
- Jerrum
- 1987
(Show Context)
Citation Context ...plicated, but follows generally the same methodology as Valiant’s algorithm for PL-NODE-BIPARTITION. The interpolation method is again crucial. We now consider a matching problem. We note that Jerrum =-=[28]-=- showed that counting the number of (not necessarily perfect) matchings in a planar graph is #P-complete, and Vadhan [51] subsequently proved that it remains #P-complete even for planar bipartite grap... |

36 |
Quantum circuits that can be simulated classically in polynomial time
- Valiant
(Show Context)
Citation Context ...lass of general unsymmetric signatures. 3s2 Preliminaries In this section we give some basic definitions. Terminolgies from the theory of holographic algorithms have been mostly introduced by Valiant =-=[52, 54, 53]-=-, but we also include some modifications from [7, 8]. Let G = (V,E,W) be a weighted undirected graph, where V is the set of vertices represented by integers k1 < k2 < ... < kn, E is the set of edges, ... |

33 |
878-Approximation algorithms for MAX
- Goemans, Williamson
- 1994
(Show Context)
Citation Context ...o note that if instead of considering node deletion we consider edge deletion, this is just another way of defining the well known problem of MAX-CUT, which is NP-hard (and even NPhard to approximate =-=[22, 3, 26]-=-). The holographic algorithm by Valiant above is the first polynomial time algorithm for PL-NODE-BIPARTITION [54]. On the other hand, planar MAX-CUT is known to be in P [24]. In [6] a joint generaliza... |

32 |
Residual entropy of square ice
- Lieb
- 1967
(Show Context)
Citation Context ...orientation had to have incoming and outgoing degree two at every node. The question of determining how the number of such orientations grows for various planar repeating structures has been analyzed =-=[38, 39, 40, 41, 5, 58]-=-. However, more generally the range of natural counting problems for graphtheoretic problems for which there are polynomial time algorithms is very limited, including for “ice” problems. #PL-3-NAE-ICE... |

28 | Holographic algorithms
- Valiant
(Show Context)
Citation Context ...in size, and difficult to handle. But whenever we find a suitable solution, we get an exotic polynomial time algorithm. Searching for these signatures is what Valiant called the “enumerative” form in =-=[57]-=-. Quoting Valiant [57]: “The objects enumerated are sets of polynomial systems such that the solvability of any one member would give a polynomial time algorithm for a specific problem. ... the situat... |

26 | The complexity of planar counting problems
- Marathe, Radhakrishnan, et al.
- 1998
(Show Context)
Citation Context ...over, for many connectives other than NOT-ALL-EQUAL (e.g., EXACTLY-ONE) the unrestricted or the planar decision problems are still NP-complete, and the corresponding counting problems are #P-complete =-=[27]-=-. To solve this problem by a holographic algorithm, we represent each NOT-ALL-EQUAL gate with the recognizer signature (0,1,1,1,1,1,1,0), and represent a Boolean variable having fan-out k with the gen... |

25 |
Algorithmic Number Theory, Volume I: Efficient Algorithms
- Bach, Shallit
- 1996
(Show Context)
Citation Context ...mber theory. At the same time, the correctness and efficiency of this and similar algorithms demand proofs in a purely structural sense, and use quite a bit more structural results from number theory =-=[4]-=-. As another example, the computational difficulty of recognizing primes and the related (but separate) problem of integer factorization already fascinated Gauss, and are closely tied to the structura... |

25 |
Holographic Algorithms (Extended Abstract
- Valiant
- 2004
(Show Context)
Citation Context ... vs. NP. Of course it is quite possible that the theory of holographic algorithms does not in the end lead to any collapse of complexity classes. But even in this eventuality, as Valiant suggested in =-=[54]-=-, “any proof of P �= NP may need to explain, and not only to imply, the unsolvability” of NP-hard problems using this approach. In Section 2 we will start with some basic definitions on holographic al... |

24 |
Exact solution of the problem of the entropy of two-dimensional ice
- Lieb
- 1967
(Show Context)
Citation Context ...orientation had to have incoming and outgoing degree two at every node. The question of determining how the number of such orientations grows for various planar repeating structures has been analyzed =-=[38, 39, 40, 41, 5, 58]-=-. However, more generally the range of natural counting problems for graphtheoretic problems for which there are polynomial time algorithms is very limited, including for “ice” problems. #PL-3-NAE-ICE... |

20 | Holographic algorithms: From art to science
- Cai, Lu
(Show Context)
Citation Context ...not a priori stated for a planar graph. It is in the process of forming the matchgrid that we obtain a planar graph. There are several more problems that have been solved using holographic algorithms =-=[54, 57, 7, 6, 10]-=-. Interested readers are referred to these papers for more details. 4 Symmetric Signatures As seen from the examples, the general outline of the design of a holographic algorithm consists of two parts... |

20 |
The structure and entropy of ice and of other crystals with some randomness of atomic arrangement
- Pauling
- 1935
(Show Context)
Citation Context ...n of an undirected graph G is an assignment of a direction to each of its edges. An ice problem involves counting the number of orientations such that certain local constraints are satisfied. Pauling =-=[47]-=- initially proposed such a model for planar square lattices, where the constraint was that an orientation had to have incoming and outgoing degree two at every node. The question of determining how th... |

18 | Some Results on Matchgates and Holographic Algorithms
- Cai, Choudhary
- 2006
(Show Context)
Citation Context ...pproximate [22, 3, 26]). The holographic algorithm by Valiant above is the first polynomial time algorithm for PL-NODE-BIPARTITION [54]. On the other hand, planar MAX-CUT is known to be in P [24]. In =-=[6]-=- a joint generalization was shown to be also solvable in polynomial time by holographic algorithms. PL-NODE-EDGE-BIPARTITION Input: A planar graph G = (V,E) of maximum degree 3. A non-negative integer... |

17 |
A combinatorial proof of Kneser’s conjecture
- Matouˇsek
(Show Context)
Citation Context ...e they are particularly desirable. The following theorem is a complete characterization of 2-admissibility over fields of characteristic 0. It uses rank estimates related to the Kneser Graph KG2k+1,k =-=[32, 42, 45, 17, 18, 23, 36]-=-. Theorem 5.3 Let G be a tensor of arity n. G is 2-admissible iff (1) n = 2k is even; (2) all GS = 0 except for |S| = k; and (3) for all T ⊂ [n] with |T | = k + 1, � G S = 0. (7) S⊂T,|S|=k 1 �2k� The ... |

16 |
Exact solution of the F model of an antiferroelectric
- Lieb
- 1967
(Show Context)
Citation Context ...orientation had to have incoming and outgoing degree two at every node. The question of determining how the number of such orientations grows for various planar repeating structures has been analyzed =-=[38, 39, 40, 41, 5, 58]-=-. However, more generally the range of natural counting problems for graphtheoretic problems for which there are polynomial time algorithms is very limited, including for “ice” problems. #PL-3-NAE-ICE... |

16 |
Exact solution of the two-dimensional Slater KDP model of a ferroelectric
- Lieb
- 1967
(Show Context)
Citation Context |

16 |
Expressiveness of matchgates
- Valiant
- 2002
(Show Context)
Citation Context ...lass of general unsymmetric signatures. 3s2 Preliminaries In this section we give some basic definitions. Terminolgies from the theory of holographic algorithms have been mostly introduced by Valiant =-=[52, 54, 53]-=-, but we also include some modifications from [7, 8]. Let G = (V,E,W) be a weighted undirected graph, where V is the set of vertices represented by integers k1 < k2 < ... < kn, E is the set of edges, ... |

16 | Holographic circuits - Valiant |

15 | On the theory of matchgate computations
- Cai, Choudhary, et al.
(Show Context)
Citation Context ...In this section we give some basic definitions. Terminolgies from the theory of holographic algorithms have been mostly introduced by Valiant [52, 54, 53], but we also include some modifications from =-=[7, 8]-=-. Let G = (V,E,W) be a weighted undirected graph, where V is the set of vertices represented by integers k1 < k2 < ... < kn, E is the set of edges, and W denotes the weights of the edges. We represent... |

15 | On the structure of t-designs
- Graham, Li, et al.
- 1980
(Show Context)
Citation Context ...e they are particularly desirable. The following theorem is a complete characterization of 2-admissibility over fields of characteristic 0. It uses rank estimates related to the Kneser Graph KG2k+1,k =-=[32, 42, 45, 17, 18, 23, 36]-=-. Theorem 5.3 Let G be a tensor of arity n. G is 2-admissible iff (1) n = 2k is even; (2) all GS = 0 except for |S| = k; and (3) for all T ⊂ [n] with |T | = k + 1, � G S = 0. (7) S⊂T,|S|=k 1 �2k� The ... |

15 | Completeness for parity problems - Valiant - 2005 |

14 |
Aufgabe 360, Jahresbericht der Deutschen
- Kneser
- 1955
(Show Context)
Citation Context |

13 | Valiant’s Holant Theorem and Matchgate Tensors - Cai, Choudhary |

12 |
Exactly Solved Models in Statistical
- Baxter
- 1982
(Show Context)
Citation Context |

12 |
On symmetric signatures in holographic algorithms
- Cai, Lu
- 2007
(Show Context)
Citation Context ...denoted as [0,1,1,1]. Symmetric signatures are often convenient for the design of holographic algorithms, as they have easily interpretable combinatorial meanings. For symmetric signatures Cai and Lu =-=[9]-=- have achieved a complete characterization of their realizability. These tell us exactly what signatures can be realized over some bases. Theorem 4.1 �� A symmetric � � �� signature [x0,x1,...,xm] for... |

12 | The Complexity of Planar Counting Problems
- Hunt, Marathe, et al.
- 1998
(Show Context)
Citation Context ...over, for many connectives other than NOT-ALL-EQUAL (e.g., EXACTLY-ONE) the unrestricted or the planar decision problems are still NP-complete, and the corresponding counting problems are #P-complete =-=[27]-=-. To solve this problem by a holographic algorithm, we represent each NOT-ALL-EQUAL gate with the recognizer signature (0,1,1,1,1,1,1,0), and represent a Boolean variable having fan-out k with the gen... |

11 |
Node-deletion NP-complete problems
- Krishnamoorthy, Deo
- 1979
(Show Context)
Citation Context ...3. Output: The cardinality of a smallest subset V ′ ⊂ V such that the deletion of V ′ and its incident edges results in a bipartite graph. This problem is known to be NP-complete for maximum degree 6 =-=[33]-=-. See [35] for a general approach to such “node deletion” problems. Note that numerous other planar NP-complete problems, such as Hamiltonian cycles and minimum vertex covers are NP-complete already f... |

9 |
On theory and applications of BIB designs with repeated blocks
- Foody, Hedayat
- 1977
(Show Context)
Citation Context |

9 |
Hilbert’s Tenth Problem
- Matijasevich
- 1993
(Show Context)
Citation Context ...ariables with integer coefficients) has an integer solution. This problem was finally shown by Matiyasevich in 1970 to admit no algorithm which can always answer correctly in a finite number of steps =-=[43, 44]-=-. Thus, starting with the work by Gödel, Turing, and others, in answering Hilbert’s Entscheidungsproblem, computability theory was born. However, in computability theory, we focus on ultimate computab... |

8 |
Holographic algorithms: the power of dimensionality resolved
- Cai, Lu
- 2007
(Show Context)
Citation Context ...an infinite set of possibilities in which an exotic or “freak” object may materialize leading to the collapse of complexity classes. However, it has been shown that a universal bases collapse theorem =-=[12]-=- holds for holographic algorithms: Any holographic algorithm using two basis vectors of arbitrary size k can be simulated by a holographic algorithm using another basis of size 1. This cuts off a pote... |