On the hardness of approximating the permanent of structured matrices

Codenotti, Bruno; Shparlinski, Igor E.; Winterhof, Arne

doi:10.1007/s00037-002-0174-3

Cited by 5 publications

(4 citation statements)

References 26 publications

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“…Even the computation of the permanent of 0-1 matrices restricted to have only three ones per row is ♯P-complete [9]. However, for circulant matrices, one can exactly calculate the permanent in polynomial time [10]- [13]. Later these results were strengthened in various ways in [14]- [17].…”

Section: A Permanents and Bethe Permanentsmentioning

confidence: 99%

Bounding the Bethe and the Degree-$M$ Bethe Permanents

Smarandache¹,

Haenggi²

2015

Preprint

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In [1], it was conjectured that the permanent of a P-lifting θ ↑P of a matrix θ of degree M is less than or equal to the M th power of the permanent perm(θ), i.e., perm(θ ↑P ) perm(θ) M and, consequently, that the degree-M Bethe permanent perm M,B (θ) of a matrix θ is less than or equal to the permanent perm(θ) of θ, i.e., perm M,B (θ) perm(θ). In this paper, we prove these related conjectures and show in addition a few properties of the permanent of block matrices that are lifts of a matrix. As a corollary, we obtain an alternative proof of the inequality perm B (θ) perm(θ) on the Bethe permanent of the base matrix θ that uses only the combinatorial definition of the Bethe permanent (the first proof was given by Gurvits in [2]).

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Section: A Permanents and Bethe Permanentsmentioning

confidence: 99%

Bounding the Bethe and the Degree-$M$ Bethe Permanents

Smarandache¹,

Haenggi²

2015

Preprint

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“…On the other hand, for certain special structured classes of matrices one can exactly calculate the permanent in "polynomial time". The most studied example of such a class is probably the circulant matrices, which, as discussed in [7], can be thought of as the borderline between the easy and hard cases.…”

Section: N] the Permanent Of A Ismentioning

confidence: 99%

“…The most studied example of such a class is probably the circulant matrices, which, as discussed in [7], can be thought of as the borderline between the easy and hard cases.…”

mentioning

confidence: 99%

“…Later Codenotti, Resta and various coauthors improved these results in various ways; e.g. in [2] showing how to evaluate sparse circulant matrices of size ≤ 200; in [4,5] showing that the permanents of circulants with only three 1s per row can be evaluated in polynomial time; in [6] showing how the permanents of some special sparse circulants can be expressed in terms of determinants and are therefore solvable in polynomial time; in [2] showing that the permanents of dense circulants are hard to calculate and in [7] that even approximating the permanent of an arbitrary circulant modulo a prime p is "hard" unless P #P = BPP.…”

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confidence: 99%

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Permanents of Circulants: a Transfer Matrix Approach

Golin¹,

Leung²,

Wang³

2006

2006 Proceedings of the Third Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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Calculating the permanent of a (0, 1) matrix is a #P -complete problem but there are some classes of structured matrices for which the permanent is calculable in polynomial time. The most well-known example is the fixed-jump (0, 1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constantcoefficient fixed-order recurrence relation.In this note we show how, by interpreting the problem as calculating the number of cycle-covers in a directed circulant graph, it is straightforward to reprove Minc's result using combinatorial methods. This is a two step process: the first step is to show that the cyclecovers of directed circulant graphs can be evaluated using a transfer matrix argument. The second is to show that the associated transfer matrices, while very large, actually have much smaller characteristic polynomials than would a-priori be expected.An important consequence of this new viewpoint is that, in combination with a new recursive decomposition of circulant-graphs, it permits extending Minc's result to calculating the permanent of the much larger class of circulant matrices with non-fixed (but linear) jumps.

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Noisy interpolation of sparse polynomials in finite fields

Shparlinski

Winterhof

2005

AAECC

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On the hardness of approximating the permanent of structured matrices

Cited by 5 publications

References 26 publications

Bounding the Bethe and the Degree-$M$ Bethe Permanents

Bounding the Bethe and the Degree-$M$ Bethe Permanents

Permanents of Circulants: a Transfer Matrix Approach

Noisy interpolation of sparse polynomials in finite fields

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