Subject: Computer Science
; Mathematics
Title: computational complexity of holant problems
Author: Cai Jin-Yi
; Lu Pinyan
; Xia Mingji
Keyword: Boolean functions
; Interpolation
; Real variables
Source: SIAM Journal on Computing
Issued Date: 2011
Volume: 40, Issue: 4, Pages: 1101-1132 Indexed Type: EI
; SCI
Department: (1) Computer Sciences Department University of Wisconsin Madison WI 53706 United States; (2) Beijing University Beijing China; (3) Microsoft Research Asia Beijing China; (4) State Key Laboratory of Computer Science Institute of Software Chinese Academy of Sciences Beijing China
Sponsorship: NSFCCF-0830488, CCF-0914969, CCF-0511679; Chinese Academy of Sciences; NSFC61003030, 60970003
Abstract: We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant problems. Compared to counting constraint satisfaction problems (#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. We prove complexity dichotomy theorems in this framework. Our dichotomy theorems apply to local constraint functions, which are symmetric functions on Boolean input variables and evaluate to arbitrary real or complex values. We discover surprising tractable subclasses of counting problems, which could not easily be specified in the #CSP framework. When all unary functions are assumed to be free (Holant * problems), the tractable ones consist of functions that are degenerate, or of arity at most two, or holographic transformations of Fibonacci gates. When only two special unary functions, the constant zero and constant one functions, are assumed to be free (Holantc problems), we further identify three special families of tractable cases. Then we prove that all other cases are #P-hard. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations. © 2011 Society for Industrial and Applied Mathematics. English Abstract: We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant problems. Compared to counting constraint satisfaction problems (#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. We prove complexity dichotomy theorems in this framework. Our dichotomy theorems apply to local constraint functions, which are symmetric functions on Boolean input variables and evaluate to arbitrary real or complex values. We discover surprising tractable subclasses of counting problems, which could not easily be specified in the #CSP framework. When all unary functions are assumed to be free (Holant * problems), the tractable ones consist of functions that are degenerate, or of arity at most two, or holographic transformations of Fibonacci gates. When only two special unary functions, the constant zero and constant one functions, are assumed to be free (Holantc problems), we further identify three special families of tractable cases. Then we prove that all other cases are #P-hard. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations. © 2011 Society for Industrial and Applied Mathematics. Language: 英语
WOS ID: WOS:000294296100006
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Content Type: 期刊论文
URI: http://ir.iscas.ac.cn/handle/311060/16071
Appears in Collections: 软件所图书馆_期刊论文
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Recommended Citation:
Cai Jin-Yi,Lu Pinyan,Xia Mingji. computational complexity of holant problems[J]. SIAM Journal on Computing,2011-01-01,40(4):1101-1132.
