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Holant Problems for Regular Graphs with Complex Edge Functions
[article]

2011
*
arXiv
*
pre-print

We prove a

arXiv:1001.0464v3
fatcat:55y66y4d2zbehhrcu5plindw6q
*complexity*dichotomy theorem*for**Holant**Problems*on 3-*regular**graphs**with*an arbitrary*complex*-valued*edge**function*. ... of the proof*for*computational*complexity*. ... In this paper we give a dichotomy theorem*for*the*complexity*of*Holant**Problems*on 3-*regular**graphs**with*arbitrary signature g = [x, y, z], where x, y, z ∈ C. ...##
###
Holant Problems for 3-Regular Graphs with Complex Edge Functions

2016
*
Theory of Computing Systems
*

Then the

doi:10.1007/s00224-016-9671-7
fatcat:7owxvnbfpbdurbuhmwxxlv4yrm
*Holant**Problem*on 3-*regular**graphs**with*g = [a, 1, b] is #P-hard except in the following cases,*for*which the*problem*is in P. ... In this paper we give a dichotomy theorem*for*the*complexity*of*Holant**problems*on 3-*regular**graphs**with*arbitrary signature g = [x, y, z], where x, y, z ∈ C. ... Introduction In this paper we consider the following subclass of*Holant**Problems*[4, 5] . An input*regular**graph*G = (V, E) is given, where every e ∈ E is labeled*with*a (symmetric)*edge**function*g. ...##
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The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems

2014
*
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
*

In fact, we prove that counting

doi:10.1109/focs.2014.70
dblp:conf/focs/CaiGW14
fatcat:4cqezlzvx5ahfoanofrt6xgsey
*edge*κcolorings is #P-hard over planar r-*regular*multigraphs*for*all κ ≥ r ≥ 3. The*problem*is polynomial-time computable in all other parameter settings. ... A special case of this result is that counting*edge*κcolorings is #P-hard over planar 3-*regular*multigraphs*for*all κ ≥ 3. ... We are very grateful to Bjorn Poonen and especially Aaron Levin*for*sharing their expertise on Runge's method, and in particular*for*the auxiliary*function*g 2 (x, y) in the proof of Lemma V.2. ...##
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The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems

2016
*
Research in the Mathematical Sciences
*

In fact, we prove that counting

doi:10.1186/s40687-016-0067-8
fatcat:bgxkyqjo3zbqdmesxty7z5xmea
*edge*κcolorings is #P-hard over planar r-*regular*multigraphs*for*all κ ≥ r ≥ 3. The*problem*is polynomial-time computable in all other parameter settings. ... A special case of this result is that counting*edge*κcolorings is #P-hard over planar 3-*regular*multigraphs*for*all κ ≥ 3. ... We are very grateful to Bjorn Poonen and especially Aaron Levin*for*sharing their expertise on Runge's method, and in particular*for*the auxiliary*function*g 2 (x, y) in the proof of Lemma V.2. ...##
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The Complexity of Counting Edge Colorings for Simple Graphs
[article]

2020
*
arXiv
*
pre-print

Furthermore, we show that

arXiv:2010.04910v1
fatcat:oelmy754mvhwrelzezxkvitl2y
*for*planar r-*regular*simple*graphs*where r ∈{3, 4, 5} counting*edge*colorings*with*ąp̨p̨ą colors*for*any κ≥ r is also #P-complete. ... We prove that*for*any κ≥ r ≥ 3 counting κ-*edge*colorings on r-*regular*simple*graphs*is #P-complete. ...*For*the reason of Turing computability we assume all signatures take*complex*algebraic values. A*Holant**problem**Holant*(F) is parameterized by a set of signatures F. ...##
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The Complexity of Planar Boolean #CSP with Complex Weights
[article]

2013
*
arXiv
*
pre-print

We also obtain a dichotomy theorem

arXiv:1212.2284v2
fatcat:n42fthkjn5e63flp66hm2pzpfq
*for*a symmetric arity 4 signature*with**complex*weights in the planar*Holant*framework, which we use in the proof of our #CSP dichotomy. ... We prove a*complexity*dichotomy theorem*for*symmetric*complex*-weighted Boolean #CSP when the constraint*graph*of the input must be planar. ... We also thank him*for*his careful reading and insightful comments on a draft of this work as well as*for*the proof of Lemma 4.2. ...##
###
The Complexity of Planar Boolean #CSP with Complex Weights
[chapter]

2013
*
Lecture Notes in Computer Science
*

We also obtain a dichotomy theorem

doi:10.1007/978-3-642-39206-1_44
fatcat:ormuxbgvfjbr5kuve4ybzzgjmi
*for*a symmetric arity 4 signature*with**complex*weights in the planar*Holant*framework, which we use in the proof of our #CSP dichotomy. ... We prove a*complexity*dichotomy theorem*for*symmetric*complex*-weighted Boolean #CSP when the constraint*graph*of the input must be planar. ... We also thank him*for*his careful reading and insightful comments on a draft of this work as well as*for*the proof of Lemma 4.2. ...##
###
The Complexity of Symmetric Boolean Parity Holant Problems

2013
*
SIAM journal on computing (Print)
*

Such dichotomy results have been proved

doi:10.1137/100815530
fatcat:afi6vxkdgvg5lmfnzp5bwcb6ti
*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*. ... An F-gate is a tuple (H, F, π), where H = (V, E, D) is a*graph*where the*edge*set consists of*regular**edges*E and dangling*edges*D. The labelling π assigns a*function*from F to each internal node. ...##
###
Approximate Counting via Correlation Decay on Planar Graphs
[chapter]

2013
*
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
*

Very recently, a dichotomy theorem [7] is proved

doi:10.1137/1.9781611973105.4
dblp:conf/soda/YinZ13
fatcat:mul6kf3gujbqjpboqazfhp4hm4
*for**Holant**problems**with**complex*-valued*functions*on general*graphs*, concluding a long series of dichotomies on*Holant**problems*. ... The core of our algorithm is a fixed-parameter tractable algorithm which computes the exact values of the*Holant**problems**with**regular*constraint*functions*on*graphs*of bounded treewidth. ... We would like to thank Jin-Yi Cai, Heng Guo, and Pinyan Lu*for*the in-depth discussions. Thank Alistair Sinclair and Leslie Valiant*for*their comments and interests. ...##
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Counting perfect matchings and the eight-vertex model
[article]

2019
*
arXiv
*
pre-print

We study the approximation

arXiv:1904.10493v1
fatcat:4vmzzxzesjdi5iq6i6cz3ih3ly
*complexity*of the partition*function*of the eight-vertex model on general 4-*regular**graphs*. ... We also identify a region of the parameter space where approximating the partition*function*on planar 4-*regular**graphs*is feasible but on general 4-*regular**graphs*is equivalent to approximately counting ... bipartite*graph*= ( , , )*for*the*Holant**problem**Holant*( | ). ...##
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Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP
[article]

2010
*
arXiv
*
pre-print

Moreover,

arXiv:1008.0683v1
fatcat:fajxmqnyovhynkj6tchbxog55a
*problems*in category (2) are tractable on planar*graphs*precisely by holographic algorithms*with*matchgates. ... community*for*decades. ... Acknowledgments We thank the following colleagues*for*their interests and helpful comments: Xi Chen, Martin Dyer, Alan Frieze, Sean Hallgren, Leslie Goldberg, Sorin Istrail, Richard Lipton, Jason Morton ...##
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A Dichotomy for Real Weighted Holant Problems

2012
*
2012 IEEE 27th Conference on Computational Complexity
*

Recently,

doi:10.1109/ccc.2012.16
dblp:conf/coco/HuangL12
fatcat:rmv26udkwvfhxn3vdt47xc5fkq
*complexity*dichotomy*for*a variety of sub-families of*Holant*such as #CSP,*Graph*Homomorphism,*Holant** and*Holant*c were proved. ... This is the first time a dichotomy is obtained*for*general*Holant**Problems*without any auxiliary*functions*. ... A couple of recent works studied the*complexity*of*Holant*on*regular**graphs*where all the vertices take a same*function*[30] , [32] - [34] . ...##
###
Complexity Dichotomies of Counting Problems
[article]

2011
*
Electronic colloquium on computational complexity
*

Recently, we proposed and explored a novel alternative framework, called

dblp:journals/eccc/Lu11
fatcat:sbtez2cixbauhgi42mviinbzii
*Holant**Problems*. It is a refinement*with*a more explicit role*for*constraint*functions*. ... Both*graph*homomorphism and #CSP can be viewed as special sub-frameworks of*Holant**Problems*. ... Compared to #CSP, it is a refinement*with*a more explicit role*for*the constraint*functions*. Both*graph*homomorphism and #CSP can be viewed as special cases of*Holant**Problems*. ...##
###
A Dichotomy for Real Weighted Holant Problems

2015
*
Computational Complexity
*

Recently,

doi:10.1007/s00037-015-0118-3
fatcat:dhmgsxvvk5gh3k5ed5zus4jgqu
*complexity*dichotomy*for*a variety of sub-families of*Holant*such as #CSP,*Graph*Homomorphism,*Holant** and*Holant*c were proved. ... This is the first time a dichotomy is obtained*for*general*Holant**Problems*without any auxiliary*functions*. ... A couple of recent works studied the*complexity*of*Holant*on*regular**graphs*where all the vertices take a same*function*[30] , [32] - [34] . ...##
###
The Complexity of Symmetric Boolean Parity Holant Problems
[chapter]

2011
*
Lecture Notes in Computer Science
*

Such dichotomy results have been proved

doi:10.1007/978-3-642-22006-7_60
fatcat:jvp6wwbo3jcurg4qwiax7r5f7q
*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*. ... An F-gate is a tuple (H, F, π), where H = (V, E, D) is a*graph*where the*edge*set consists of*regular**edges*E and dangling*edges*D. The labelling π assigns a*function*from F to each internal node. ...
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