Computational Complexity of Holant Problems

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...

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Vydáno v:SIAM journal on computing Ročník 40; číslo 4; s. 1101 - 1132
Hlavní autoři: Cai, Jin-Yi, Lu, Pinyan, Xia, Mingji
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Témata:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Boolean
Computer science; control theory; systems
Exact sciences and technology
Information retrieval. Graph
Mathematical analysis
Mathematics
Miscellaneous
Ordinary differential equations
Sciences and techniques of general use
Theorems
Theoretical computing
Variables
68Q25
holographic reduction
Constraint
68Q17
Boolean function
Symmetric function
Special function
Homomorphism
Computational complexity
Holant problems
Dichotomy
P-hardness
Input
Polynomial interpolation
ISSN:0097-5397, 1095-7111
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Shrnutí: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 ($\mathrm{Holant}^*$ 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 ($\mathrm{Holant}^c$ 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.
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ISSN:0097-5397
1095-7111
DOI:10.1137/100814585