#16603: permanental_minor_vector, matching polynomial
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Reporter: pernici | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.3
Component: linear algebra | Resolution:
Keywords: | Merged in:
Authors: Mario Pernici | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/pernici/ticket/16603 | b0d714572f15dd915b1caabbb11f5f96f4020c85
Dependencies: | Stopgaps:
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Description changed by pernici:
Old description:
> ADD DESCRIPTION
> added function `permanental_minor_vector`, which computes
> the polynomial of the sums of the permanental minors in an efficient
> way, see http://arxiv.org/abs/1406.5337
>
> As a benchmark, consider
> http://www.jaapspies.nl/mathfiles/dancing.sage
> on my computer `dance(10)` took 3 hours using sage-6.2, 0.08 seconds
> with this patch.
>
> In the case of banded matrices, the speedup is greater, since
> for fixed bandwidth the new algorithm is polynomial; here is a
> permanent
> which cannot be computed with sage-6.2
>
> sage: from sage.matrix.matrix_misc import permanental_minor_vector
> sage: n, w = 100, 5
> sage: m = matrix([[i*j + 1 if abs(i-j) <= w else 0
> ....: for i in range(n)] for j in range(n)])
> sage: time p = permanental_minor_vector(m, 1)
> CPU times: user 1.11 s, sys: 20 ms, total: 1.13 s
> Wall time: 1.07 s
>
> This can be used to compute the matching polynomial of bipartite graphs
> much faster than matching_polynomial for certain graphs.
>
> For example, using the attached file to compute the reduced adjacency
> matrix of
> a 2d grid (n1, n2)
>
> sage: from grid import sq_mat
> sage: from sage.matrix.matrix_misc import permanental_minor_vector
> sage: m = matrix(sq_mat(6, 6))
> sage: time a = permanental_minor_vector(m)
> Wall time: 16.3 ms
> sage: g = BipartiteGraph(m)
> sage: time p = g.matching_polynomial()
> Wall time: 52 s
>
> For n1,n2=10,10 permanental_minor_vector(m) takes 0.7s
> For n1,n2=50,6 it takes 0.8s
>
> priority: major
> type: enhancement
> component: linear algebra
New description:
ADD DESCRIPTION
added function `permanental_minor_vector`, which computes
the polynomial of the sums of the permanental minors in an efficient
way, see http://arxiv.org/abs/1406.5337
As a benchmark, consider
http://www.jaapspies.nl/mathfiles/dancing.sage
on my computer `dance(10)` took 3 hours using sage-6.2, 0.08 seconds
with this patch.
In the case of banded matrices, the speedup is greater, since
for fixed bandwidth the new algorithm is polynomial; here is a permanent
which cannot be computed with sage-6.2
sage: from sage.matrix.matrix_misc import permanental_minor_vector
sage: n, w = 100, 5
sage: m = matrix([[i*j + 1 if abs(i-j) <= w else 0
....: for i in range(n)] for j in range(n)])
sage: time p = permanental_minor_vector(m, 1)
CPU times: user 1.11 s, sys: 20 ms, total: 1.13 s
Wall time: 1.07 s
This can be used to compute the matching polynomial of bipartite graphs
much faster than matching_polynomial for certain graphs.
For example, using the attached file to compute the reduced adjacency
matrix of
a 2d grid (n1, n2)
sage: from grid import sq_mat
sage: from sage.matrix.matrix_misc import permanental_minor_vector
sage: m = matrix(sq_mat(6, 6))
sage: time a = permanental_minor_vector(m)
Wall time: 16.3 ms
sage: g = BipartiteGraph(m)
sage: time p = g.matching_polynomial()
Wall time: 52 s
For n1,n2=10,10 permanental_minor_vector(m) takes 0.7s
For n1,n2=50,6 it takes 0.8s
priority: major
type: enhancement
component: linear algebra
--
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Ticket URL: <http://trac.sagemath.org/ticket/16603#comment:6>
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