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

Michael Kowalczyk, Jin-Yi Cai
2011 arXiv   pre-print
We prove a 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.  ... 
arXiv:1001.0464v3 fatcat:55y66y4d2zbehhrcu5plindw6q

Holant Problems for 3-Regular Graphs with Complex Edge Functions

Michael Kowalczyk, Jin-Yi Cai
2016 Theory of Computing Systems  
Then the 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.  ... 
doi:10.1007/s00224-016-9671-7 fatcat:7owxvnbfpbdurbuhmwxxlv4yrm

The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems

Jin-Yi Cai, Heng Guo, Tyson Williams
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
In fact, we prove that counting 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.  ... 
doi:10.1109/focs.2014.70 dblp:conf/focs/CaiGW14 fatcat:4cqezlzvx5ahfoanofrt6xgsey

The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems

Jin-Yi Cai, Heng Guo, Tyson Williams
2016 Research in the Mathematical Sciences  
In fact, we prove that counting 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.  ... 
doi:10.1186/s40687-016-0067-8 fatcat:bgxkyqjo3zbqdmesxty7z5xmea

The Complexity of Counting Edge Colorings for Simple Graphs [article]

Jin-Yi Cai, Artem Govorov
2020 arXiv   pre-print
Furthermore, we show that 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.  ... 
arXiv:2010.04910v1 fatcat:oelmy754mvhwrelzezxkvitl2y

The Complexity of Planar Boolean #CSP with Complex Weights [article]

Heng Guo, Tyson Williams
2013 arXiv   pre-print
We also obtain a dichotomy theorem 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.  ... 
arXiv:1212.2284v2 fatcat:n42fthkjn5e63flp66hm2pzpfq

The Complexity of Planar Boolean #CSP with Complex Weights [chapter]

Heng Guo, Tyson Williams
2013 Lecture Notes in Computer Science  
We also obtain a dichotomy theorem 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.  ... 
doi:10.1007/978-3-642-39206-1_44 fatcat:ormuxbgvfjbr5kuve4ybzzgjmi

The Complexity of Symmetric Boolean Parity Holant Problems

Heng Guo, Pinyan Lu, Leslie G. Valiant
2013 SIAM journal on computing (Print)  
Such dichotomy results have been proved 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.  ... 
doi:10.1137/100815530 fatcat:afi6vxkdgvg5lmfnzp5bwcb6ti

Approximate Counting via Correlation Decay on Planar Graphs [chapter]

Yitong Yin, Chihao Zhang
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
Very recently, a dichotomy theorem [7] is proved 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.  ... 
doi:10.1137/1.9781611973105.4 dblp:conf/soda/YinZ13 fatcat:mul6kf3gujbqjpboqazfhp4hm4

Counting perfect matchings and the eight-vertex model [article]

Jin-Yi Cai, Tianyu Liu
2019 arXiv   pre-print
We study the approximation 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( | ).  ... 
arXiv:1904.10493v1 fatcat:4vmzzxzesjdi5iq6i6cz3ih3ly

Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP [article]

Jin-Yi Cai and Pinyan Lu and Mingji Xia
2010 arXiv   pre-print
Moreover, 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  ... 
arXiv:1008.0683v1 fatcat:fajxmqnyovhynkj6tchbxog55a

A Dichotomy for Real Weighted Holant Problems

Sangxia Huang, Pinyan Lu
2012 2012 IEEE 27th Conference on Computational Complexity  
Recently, 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] .  ... 
doi:10.1109/ccc.2012.16 dblp:conf/coco/HuangL12 fatcat:rmv26udkwvfhxn3vdt47xc5fkq

Complexity Dichotomies of Counting Problems [article]

Pinyan Lu
2011 Electronic colloquium on computational complexity  
Recently, we proposed and explored a novel alternative framework, called 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.  ... 
dblp:journals/eccc/Lu11 fatcat:sbtez2cixbauhgi42mviinbzii

A Dichotomy for Real Weighted Holant Problems

Sangxia Huang, Pinyan Lu
2015 Computational Complexity  
Recently, 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] .  ... 
doi:10.1007/s00037-015-0118-3 fatcat:dhmgsxvvk5gh3k5ed5zus4jgqu

The Complexity of Symmetric Boolean Parity Holant Problems [chapter]

Heng Guo, Pinyan Lu, Leslie G. Valiant
2011 Lecture Notes in Computer Science  
Such dichotomy results have been proved 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.  ... 
doi:10.1007/978-3-642-22006-7_60 fatcat:jvp6wwbo3jcurg4qwiax7r5f7q
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