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12
A Complete Dichotomy Rises from the Capture of Vanishing Signatures
"... We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complexvalued symmetric constraint functions F on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions taking values in a field of characteristi ..."
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We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complexvalued symmetric constraint functions F on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions taking values in a field of characteristic zero. We define and characterize all symmetric vanishing signatures. They turned out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion. The Holant problem defined by a set of constraint functions F is solvable in polynomial time if it satisfies this tractability criterion, and is #Phard otherwise. The tractability criterion can be intuitively stated as follows: the set F is tractable if (1) every function in F has arity at most 2, or (2) F is transformable to an affine type, or (3) F is transformable to a product type, or (4) F is vanishing, combined with the right type of binary functions, or (5) F belongs to a special category of vanishing type Fibonacci gates. The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and
Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
, 1007
"... We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a c ..."
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Cited by 7 (3 self)
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We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of Bürgisser on the VNPcompleteness of the partial permanent. In particular, we show that the partial permanent cannot be VNPcomplete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.
The Complexity of Planar Boolean #CSP with Complex Weights
"... We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generali ..."
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Cited by 6 (3 self)
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We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3, 3) to counting the number of Eulerian orientations over planar 4regular graphs to show the latter is #Phard. This strengthens a theorem by Huang and Lu to the planar setting.
A Holant Dichotomy: Is the FKT Algorithm Universal?
"... We prove a complexity dichotomy for complexweighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specically to answer the question: Is the FKT algorithm under a holographic transformation [38] a universal strategy to obtain polynom ..."
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We prove a complexity dichotomy for complexweighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specically to answer the question: Is the FKT algorithm under a holographic transformation [38] a universal strategy to obtain polynomialtime algorithms for problems over planar graphs that are intractable in general? This dichotomy is a culmination of previous ones, including those for Spin Systems [25], Holant [21, 6], and #CSP [20]. In the study of counting complexity, such as #CSP, there are problems which are #Phard over general graphs but polynomialtime solvable over planar graphs. A recurring theme has been that a holographic reduction to FKT precisely captures these problems. Surprisingly, for planar Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In particular, a straightforward formulation of a dichotomy for planar Holant problems along the above recurring theme is false. In previous work, an important tool was a dichotomy for #CSPd, which denotes #CSP where every variable appears a multiple of d times. However the very rst step in the #CSPd
Simulating Special but Natural Quantum Circuits
"... We identify a subclass of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures a ..."
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We identify a subclass of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures at most O(log n) qubits can be simulated by classical randomized polynomial time algorithms. This does not dequantize Shor’s algorithm (as the latter measures n qubits) but our work also highlights a new potentially hard function for cryptographic applications. Our main technical contribution is (to the best of our knowledge) a new exact characterization of certain sums of Fouriertype coefficients (with exponentially many summands). One of the key problems in complexity theory is to determine the power of the complexity class BQP. Recall that this is the set of languages accepted by uniform polynomial size quantum circuits with bounded twosided error. It is essentially the quantum version of the complexity class BPP. Just as BPP corresponds to what is feasible on a classical computer with randomness, BQP corresponds
Dichotomy for Holant* Problems with a Function on Domain Size 3
, 2012
"... Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the Boolean domain. In this paper, we give the first dichotomy ..."
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Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the Boolean domain. In this paper, we give the first dichotomy theorem for Holant problems for domain size> 2. We discover unexpected tractable families of counting problems, by giving new polynomial time algorithms. This paper also initiates holographic reductions in domains of size> 2. This is our main algorithmic technique, and is used for both tractable families and hardness reductions. The dichotomy theorem is the following: For any complexvalued symmetric function F with arity 3 on domain size 3, we give an explicit criterion on F, such that if F satisfies the criterion then the problem Holant∗(F) is computable in polynomial time, otherwise Holant∗(F) is #Phard.
This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. Contemporary Mathematics Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
"... Abstract. We deploy algebraic complexity theoretic techniques to construct symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials havi ..."
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Abstract. We deploy algebraic complexity theoretic techniques to construct symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of Bürgisser on the VNPcompleteness of the partial permanent. In particular, we show that the partial permanent cannot be VNPcomplete in a finite field of characteristic 2 unless the polynomial hierarchy collapses. 1.
Simulating Special but Natural Quantum Circuits
"... We identify a subclass of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures a ..."
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We identify a subclass of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures at most O(log n) qubits can be simulated by classical randomized polynomial time algorithms. This does not dequantize Shor’s algorithm (as the latter measures n qubits) but our work also highlights a new potentially hard function for cryptographic applications. Our main technical contribution is (to the best of our knowledge) a new exact characterization of certain sums of Fouriertype coefficients (with exponentially many summands).
The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2∗
, 2014
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