201 research outputs found
Almost Settling the Hardness of Noncommutative Determinant
In this paper, we study the complexity of computing the determinant of a
matrix over a non-commutative algebra. In particular, we ask the question,
"over which algebras, is the determinant easier to compute than the permanent?"
Towards resolving this question, we show the following hardness and easiness of
noncommutative determinant computation.
* [Hardness] Computing the determinant of an n \times n matrix whose entries
are themselves 2 \times 2 matrices over a field is as hard as computing the
permanent over the field. This extends the recent result of Arvind and
Srinivasan, who proved a similar result which however required the entries to
be of linear dimension.
* [Easiness] Determinant of an n \times n matrix whose entries are themselves
d \times d upper triangular matrices can be computed in poly(n^d) time.
Combining the above with the decomposition theorem of finite dimensional
algebras (in particular exploiting the simple structure of 2 \times 2 matrix
algebras), we can extend the above hardness and easiness statements to more
general algebras as follows. Let A be a finite dimensional algebra over a
finite field with radical R(A).
* [Hardness] If the quotient A/R(A) is non-commutative, then computing the
determinant over the algebra A is as hard as computing the permanent.
* [Easiness] If the quotient A/R(A) is commutative and furthermore, R(A) has
nilpotency index d (i.e., the smallest d such that R(A)d = 0), then there
exists a poly(n^d)-time algorithm that computes determinants over the algebra
A.
In particular, for any constant dimensional algebra A over a finite field,
since the nilpotency index of R(A) is at most a constant, we have the following
dichotomy theorem: if A/R(A) is commutative, then efficient determinant
computation is feasible and otherwise determinant is as hard as permanent.Comment: 20 pages, 3 figure
The topological period-index problem over 6-complexes
By comparing the Postnikov towers of the classifying spaces of projective
unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral
sequence, we deduce a lower bound on the topological index in terms of the
period, and solve the topological version of the period-index problem in full
for finite CW complexes of dimension at most 6. Conditions are established
that, if they were met in the cohomology of a smooth complex 3-fold variety,
would disprove the ordinary period-index conjecture. Examples of
higher-dimensional varieties meeting these conditions are provided. We use our
results to furnish an obstruction to realizing a period-2 Brauer class as the
class associated to a sheaf of Clifford algebras, and varieties are constructed
for which the total Clifford invariant map is not surjective. No such examples
were previously known.Comment: To appear in J. To
On the Complexity of Random Quantum Computations and the Jones Polynomial
There is a natural relationship between Jones polynomials and quantum
computation. We use this relationship to show that the complexity of evaluating
relative-error approximations of Jones polynomials can be used to bound the
classical complexity of approximately simulating random quantum computations.
We prove that random quantum computations cannot be classically simulated up to
a constant total variation distance, under the assumption that (1) the
Polynomial Hierarchy does not collapse and (2) the average-case complexity of
relative-error approximations of the Jones polynomial matches the worst-case
complexity over a constant fraction of random links. Our results provide a
straightforward relationship between the approximation of Jones polynomials and
the complexity of random quantum computations.Comment: 8 pages, 4 figure
Approximating the Permanent of a Matrix with Deep Rejection Sampling
Computing the permanent of a matrix is a famous #P-hard problem with a wide range of applications. The fastest known exact algorithms for the problem require an exponential number of operations, and all known fully polynomial randomized approximation schemes are rather complicated to implement and have impractical time complexities. The most promising recent advancements on approximating the permanent are based on rejection sampling and upper bounds for the permanent.
In this thesis, we improve the current state of the art by developing the deep rejection sampling method, which combines an exact algorithm with the rejection sampling method. The algorithm precomputes a dynamic programming table that tightens the initial upper bound used by the rejection sampling method. In a sense, the table is used to jump-start the sampling process.
We give a high probability upper bound for the time complexity of the deep rejection sampling method for random (0, 1)-matrices in which each entry is 1 with probability p. For matrices with p < 1/5, our high probability bound is stronger than in previous work. In addition to that, we empirically observe that our algorithm outperforms earlier rejection sampling methods by testing it with different parameters against other algorithms on multiple classes of matrices. The improvements in sampling times are especially notable in cases in which the ratios of the permanental upper bounds and the exact value of the permanent are huge
Reductions in computational complexity using Clifford algebras
International audienceA number of combinatorial problems are treated using properties of abelian nilpotent- and idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of , where is an appropriate nilpotent adjacency matrix, the -cycles in any finite graph are recovered. Within the algebra context (i.e., considering the number of multiplications performed within the algebra), these problems are reduced to matrix multiplication, which is in complexity class . The Hamiltonian cycle problem is one of many problems moved from classes NP-complete and -complete to class in this context. Other problems considered include the set covering problem, counting the edge-disjoint cycle decompositions of a finite graph, computing the permanent of an arbitrary matrix, computing the girth and circumference of a graph, and finding the longest path in a graph
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