Holographic algorithms: From art to science

Cai, Jin-Yi; Lu, Pinyan

doi:10.1016/j.jcss.2010.06.005

Cited by 85 publications

(115 citation statements)

References 27 publications

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“…(We chose these basis transformations not "out of blue", but rather they are informed by an underlying signature theory of holographic algorithms [3,4]. But for brevity of exposition we state these transformations as is without discussing the background.…”

Section: Interpolation Methodsmentioning

confidence: 99%

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Holographic reduction, interpolation and hardness

Cai

Xia

2012

comput. complex.

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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 #P-hardness for this class of problems. For these problems whenever holographic reductions followed by interpolations fail to prove #P-hardness, we can show that the problems are actually solvable in polynomial time.

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Section: Interpolation Methodsmentioning

confidence: 99%

“…. This uses some signature theory of holographic algorithms [3,4]. Under this holographic reduction, the signatures [0, 1,1] or [1,0,1] respectively are transformed to some new signature […”

Section: Introductionmentioning

confidence: 99%

Holographic reduction, interpolation and hardness

Cai

Xia

2012

comput. complex.

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“…The seemingly mysterious number 7 was subsequently explained by Cai and Lu [4], who showed that the k-SAT version of Valiant's problem is tractable modulo any prime factor of 2 k − 1.…”

Section: Counting Modulomentioning

confidence: 99%

Counting Homomorphisms to Square-Free Graphs, Modulo 2

Göbel

Goldberg

Richerby

2016

ACM Trans. Comput. Theory

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We study the problem ⊕HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph H. A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (non-modular) counting, so subtle dichotomy theorems can arise. We show the following dichotomy: for any H that contains no 4-cycles, ⊕HomsToH is either in polynomial time or is ⊕P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.

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“…For one thing the use of the three element basis b3 from [V08] puts them outside the collapse theorem of Cai and Lu [CL09], and hence outside any known classification such as [CL07]. (Of course, the possibility has not yet been excluded that #P-complete problems can be solved even within the scope of this collapse theorem or classification.)…”

Section: Introductionmentioning

confidence: 99%

Some observations on holographic algorithms

Valiant

2017

comput. complex.

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Abstract. We define the notion of diversity for families of finite functions, and express the limitations of a simple class of holographic algorithms in terms of limitations on diversity. We go on to describe polynomial time holographic algorithms for computing the parity of the following quantities for degree three planar undirected graphs: the number of 3-colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case the parity can be computed for any slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes. These holographic algorithms use bases of three components, rather than two.

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Holographic algorithms: From art to science

Cited by 85 publications

References 27 publications

Holographic reduction, interpolation and hardness

Holographic reduction, interpolation and hardness

Counting Homomorphisms to Square-Free Graphs, Modulo 2

Some observations on holographic algorithms

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