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Dichotomy for Holant Problems of Boolean Domain

Jin‐Yi Cai, Pinyan Lu, Mingji Xia

Published Aug 14, 2010• 34 citations
OpenAlex

Abstract

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

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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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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.

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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 $\mathcal{F}$ 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.

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Holant problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions. Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement.

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Holographic Algorithm with Matchgates Is Universal for Planar $\#$CSP Over Boolean Domain

Jin‐Yi Cai, Zhiguo Fu

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).

arXiv:1603.07046•Mar 23, 2016•1 citation
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 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 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.
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 problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions. Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement.
Holographic Algorithm with Matchgates Is Universal for Planar #\##CSP Over Boolean Domain
We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (#\##CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) #\##P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) #\##P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).