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SIAM Journal on Computing

The Complexity of Boolean Holant Problems with Nonnegative Weights

Jiabao Lin, Hanpin Wang

Published Jan 1, 2018• 25 citations
Publisher

Abstract

Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Boolean Holant problems with nonnegative algebraic weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric and no auxiliary function is assumed to be available. Let $\mathcal{F}$ be a set of nonnegative functions defined on the Boolean domain. We show that the problem Holant$(\mathcal{F})$ is computable in polynomial time if the function set $\mathcal{F}$ satisfies one of three conditions: (1) every function in $\mathcal{F}$ is a tensor product of functions of arity at most 2; (2) every function in $\mathcal{F}$ is affine; (3) $\mathcal{F}$ is holographically transformable to a product type by some real orthogonal matrix. Otherwise the problem is #P-hard.

DOI
10.1137/17m113304x

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Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Boolean Holant problems with nonnegative algebraic weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric and no auxiliary function is assumed to be available. Let F\mathcal{F}F be a set of nonnegative functions defined on the Boolean domain. We show that the problem Holant(F)(\mathcal{F})(F) is computable in polynomial time if the function set F\mathcal{F}F satisfies one of three conditions: (1) every function in F\mathcal{F}F is a tensor product of functions of arity at most 2; (2) every function in F\mathcal{F}F is affine; (3) F\mathcal{F}F is holographically transformable to a product type by some real orthogonal matrix. Otherwise the problem is #P-hard.
Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric. Holant problems are indeed read-twice #\##CSPs. Intuitively, some #\##CSPs that are #\##P-hard become tractable when restricting to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set F\mathcal{F}F satisfies the condition, then F\mathcal{F}F is of affine type or product type. Otherwise (a) Holant(F)\mathrm{Holant}(\mathcal{F})Holant(F) is #\##P-hard; or (b) every function in F\mathcal{F}F is a tensor product of functions of arity at most 2; or (c) F\mathcal{F}F is transformable to a product type by some real orthogonal matrix. Holographic transformations play an important role in both the hardness proof and the characterization of tractability.
Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ (F), where F is a set of constraint functions on Boolean variables and output complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #P-hard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems over the complex domain, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (non-zero) vector whose inner product with itself is zero. 1
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∗\mathrm{Holant}^*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\mathrm{Holant}^cHolantc 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.
Holant problems capture a class of Sum-of-Product computations such as counting matchings. It is inspired by holographic algorithms and is equivalent to tensor networks, with counting CSP being a special case. A complexity classification for Holant problems is more difficult to prove, not only because it logically implies a classification for counting CSP, but also due to the deeper reason that there exist more intricate polynomial time tractable problems in the broader framework. We discover a new family of constraint functions ℒ which define polynomial time computable counting problems. These do not appear in counting CSP, and no newly discovered tractable constraints can be symmetric. It has a delicate support structure related to error-correcting codes. Local holographic transformations is fundamental in its tractability. We prove a complexity dichotomy theorem for all Holant problems defined by any real valued constraint function set on Boolean variables and contains two 0–1 pinning functions. Previously, dichotomy for the same framework was only known for symmetric constraint functions. The set ℒ supplies the last piece of tractability. We also prove a dichotomy for a variant of counting CSP as a technical component toward this Holant dichotomy.
We prove a complexity dichotomy for Holant problems on the boolean domain with arbitrary sets of real-valued constraint functions. These constraint functions need not be symmetric nor do we assume any auxiliary functions as in previous results. It is proved that for every set F\mathcal{F}F of real-valued constraint functions, Holant(F)(\mathcal{F})(F) is either P-time computable or #P-hard. The classification has an explicit criterion. This is the culmination of much research on this problem, and it uses previous results and techniques from many researchers. Some particularly intriguing concrete functions f6f_6f6​, f8f_8f8​ and their associated families with extraordinary closure properties related to Bell states in quantum information theory play an important role in this proof.
We prove a complexity dichotomy for Holant problems on the boolean domain with arbitrary sets of real-valued constraint functions. These constraint functions need not be symmetric nor do we assume any auxiliary functions. It is proved that for every set F of real-valued constraint functions, Holant(F) is either P-time computable or #P-hard. The classification has an explicit criterion. This is a culmination of much research on this problem, and it uses many previous results and techniques. Dealing with some concrete functions plays an important role in this proof. In particular, two functions, called f6 and f8, and their associated families exhibit intriguing and extraordinary closure properties related to Bell states in quantum information theory.