computational complexity of holant problems

ISCAS OpenIR

	computational complexity of holant problems
	Cai Jin-Yi; Lu Pinyan; Xia Mingji
	2011
发表期刊	SIAM Journal on Computing
ISSN	0097-5397
卷号	40 期号:4 页码:1101-1132
摘要	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 (Holant^c 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.; 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 (Holant^c 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.
收录类别	EI ; SCI
关键词	Boolean Functions Interpolation Real Variables
部门归属	(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
学科领域	Computer Science ; Mathematics
资助者	NSFCCF-0830488, CCF-0914969, CCF-0511679; Chinese Academy of Sciences; NSFC61003030, 60970003
语种	英语
WOS记录号	WOS:000294296100006
引用统计	被引频次：49[WOS] [WOS记录] [WOS相关记录]
内容类型	期刊论文
URI标识	http://ir.iscas.ac.cn/handle/311060/16071
专题	中国科学院软件研究所
推荐引用方式 GB/T 7714	Cai Jin-Yi,Lu Pinyan,Xia Mingji. computational complexity of holant problems[J]. SIAM Journal on Computing,2011,40(4):1101-1132.
APA	Cai Jin-Yi,Lu Pinyan,&Xia Mingji.(2011).computational complexity of holant problems.SIAM Journal on Computing,40(4),1101-1132.
MLA	Cai Jin-Yi,et al."computational complexity of holant problems".SIAM Journal on Computing 40.4(2011):1101-1132.