Some observations on holographic algorithms

Valiant, Leslie G.

doi:10.1007/s00037-017-0160-4

Cited by 7 publications

(5 citation statements)

References 27 publications

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“…This result gives a way forward for the development of the structural theory of holographic algorithms on higher domain sizes in the same vein as Cai et al's work on domain size 2. In [24], Valiant gave examples of holographic algorithms on domain size 3, but holographic algorithms on higher domain sizes have yet to be explored. Our result shows that for domain size k, we can focus on understanding changes of basis in M 2 log 2 k ×k (C) rather than over an infinite set of dimensions, just as the collapse theorem of Cai and Lu [8] showed that on the Boolean domain, they could focus on understanding changes of basis in GL 2 (C).…”

Section: Our Results and Techniquesmentioning

confidence: 99%

“…In his new framework, multiple strands of computation get combined in a "holographic" mixture with exponential, custom-built cancellations specified by a choice of basis vectors to produce the final answer. With this change-of-basis approach, Valiant [21,22,23,24] found polynomial-time solutions to a number of counting problems, minor variants of which are known to be intractable, and noted that the only criterion for their existence was the solvability of certain finite systems of polynomial equations. As a notable example, whereas counting the number of satisfying assignments to planar, read-twice, monotone 3-CNFs is #P-complete and even the same problem modulo 2 is NP-hard under randomized reductions, the same problem modulo 7, known as # 7 Pl-Rtw-Mon-3CNF, has a polynomial-time holographic solution [21].…”

Section: Introduction 1matchgates and Holographic Algorithmsmentioning

confidence: 99%

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Basis collapse for holographic algorithms over all domain sizes

Chen

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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The theory of holographic algorithms introduced by Valiant represents a novel approach to achieving polynomial-time algorithms for seemingly intractable counting problems via a reduction to counting planar perfect matchings and a linear change of basis. Two fundamental parameters in holographic algorithms are the domain size and the basis size. Roughly, the domain size is the range of colors involved in the counting problem at hand (e.g. counting graph k-colorings is a problem over domain size k), while the basis size captures the dimensionality of the representation of those colors. A major open problem has been: for a given k, what is the smallest for which any holographic algorithm for a problem over domain size k "collapses to" (can be simulated by) a holographic algorithm with basis size ? Cai and Lu showed in 2008 that over domain size 2, basis size 1 suffices, opening the door to an extensive line of work on the structural theory of holographic algorithms over the Boolean domain. Cai and Fu later showed for signatures of full rank that over domain sizes 3 and 4, basis sizes 1 and 2, respectively, suffice, and they conjectured that over domain size k there is a collapse to basis size log 2 k . In this work, we resolve this conjecture in the affirmative for signatures of full rank for all k.

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Section: Our Results and Techniquesmentioning

confidence: 99%

Section: Introduction 1matchgates and Holographic Algorithmsmentioning

confidence: 99%

Basis collapse for holographic algorithms over all domain sizes

Chen

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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“…But for #CSP over higher domains, the situation is different. Over the general domain of size q ≥ 3, there are only a few holographic algorithms with matchgates [27,1]. But it can be argued that they are problems that actually get transformed to a Boolean domain #CSP problems.…”

Section: For Any Finite Set Of Constraint Functions F Over Boolean mentioning

confidence: 99%

On blockwise symmetric matchgate signatures and higher domain #CSP

Yang

Yin

2019

Information and Computation

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For any n ≥ 3 and q ≥ 3, we prove that the Equality function (= n ) on n variables over a domain of size q cannot be realized by matchgates under holographic transformations. This is a consequence of our theorem on the structure of blockwise symmetric matchgate signatures. This has the implication that the standard holographic algorithms based on matchgates, a methodology known to be universal for #CSP over the Boolean domain, cannot produce P-time algorithms for planar #CSP over any higher domain q ≥ 3.

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“…The possibility that accidental or freak objects in the enumeration exist cannot be discounted if the objects in the enumeration have not been studied systematically." Indeed, if any "freak" object exists in this framework, it would collapse #P to P. Therefore, over the past 10 to 15 years, this technique has been intensely studied in order to gain a systematic understanding of the limit of the trio of holographic reductions, matchgates, and the FKT algorithm [6,7,15,16,32,37,38,42,46]. Without settling the P versus #P question, the best hope is to achieve a complexity classification.…”

Section: Introductionmentioning

confidence: 99%

FKT is Not Universal — A Planar Holant Dichotomy for Symmetric Constraints

Cai

Guo

et al. 2021

Theory Comput Syst

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We prove a complexity classification for Holant problems defined by an arbitrary set of complex-valued symmetric constraint functions on Boolean variables. This is to specifically answer the question: Is the Fisher-Kasteleyn-Temperley (FKT) algorithm under a holographic transformation (Valiant, SIAM J. Comput. 37(5), 1565–1594 2008) a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable on general graphs? There are problems that are #P-hard on general graphs but polynomial-time solvable on planar graphs. For spin systems (Kowalczyk 2010) and counting constraint satisfaction problems (#CSP) (Guo and Williams, J. Comput. Syst. Sci. 107, 1–27 2020), a recurring theme has emerged that a holographic reduction to FKT precisely captures these problems. Surprisingly, for Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In particular, a straightforward formulation of a dichotomy for planar Holant problems along the above recurring theme is false. A dichotomy theorem for #CSPd, which denotes #CSP where every variable appears a multiple of d times, has been an important tool in previous work. However the proof for the #CSPd dichotomy violates planarity, and it does not generalize to the planar case easily. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSPd dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP2. In this case, the putative form of the dichotomy is true. (This is presented in Part II of the paper.) We manage to prove the planar Holant dichotomy relying only on this planar #CSP2 dichotomy, without resorting to a more general planar #CSPd dichotomy for d ≥ 3. A special case of the new polynomial-time computable problems is counting perfect matchings (#PM) over k-uniform hypergraphs when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, which is also a consequence of our dichotomy. When k = 2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5. It is worth noting that it is the gcd, and not a bound on hyperedge sizes, that is the criterion for tractability.

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Some observations on holographic algorithms

Cited by 7 publications

References 27 publications

Basis collapse for holographic algorithms over all domain sizes

Basis collapse for holographic algorithms over all domain sizes

On blockwise symmetric matchgate signatures and higher domain #CSP

FKT is Not Universal — A Planar Holant Dichotomy for Symmetric Constraints

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