The Complexity of Holant Problems over Boolean Domain with Non-negative Weights

Lin, Jiabao; Wang, Hanpin

doi:10.48550/arxiv.1611.00975

Cited by 2 publications

(2 citation statements)

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“…In addition, there is, of course, plain HOLANT: the full Holant problem with no freely-available signatures [7,23].…”

Section: Existing Results About the Holant Problemmentioning

confidence: 99%

“…Additionally, all functions are required to be symmetric, meaning their value depends only on the Hamming weight of the input [8]. Additional dichotomies exist for the plain HOLANT problem with no freely available functions, but these, too, restrict the functions sets considered: either to symmetric functions [7] or to real-valued functions only [23]. A full dichotomy for all Holant problems, as well as a dichotomy for HOLANT c without the symmetry restriction, have so far remained elusive.…”

Section: Introductionmentioning

confidence: 99%

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A new Holant dichotomy inspired by quantum computation

Backens

2017

Preprint

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Holant problems are a framework for the analysis of counting complexity problems on graphs. This framework is simultaneously general enough to encompass many other counting problems on graphs and specific enough to allow the derivation of dichotomy results, partitioning all problem instances into those which can be solved in polynomial time and those which are #P-hard. The Holant framework is based on the theory of holographic algorithms, which was originally inspired by concepts from quantum computation, but this connection appears not to have been explored before.Here, we employ quantum information theory to explain existing results in a concise way and to derive a dichotomy for a new family of problems, which we call HOLANT + . This family sits in between the known families of HOLANT * , for which a full dichotomy is known, and HOLANT c , for which only a restricted dichotomy is known. Using knowledge from entanglement theoryboth previously existing work and new results of our own -we prove a full dichotomy theorem for HOLANT + , which is very similar to the restricted HOLANT c dichotomy. Other than the dichotomy for #R 3 -CSP, ours is the first Holant dichotomy in which the allowed functions are not restricted and in which only a finite number of functions are assumed to be freely available.

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“…In addition, there is, of course, plain HOLANT: the full Holant problem with no freely-available signatures [7,23].…”

Section: Existing Results About the Holant Problemmentioning

confidence: 99%