7,597 research outputs found
Asymptotically fast polynomial matrix algorithms for multivariable systems
We present the asymptotically fastest known algorithms for some basic
problems on univariate polynomial matrices: rank, nullspace, determinant,
generic inverse, reduced form. We show that they essentially can be reduced to
two computer algebra techniques, minimal basis computations and matrix fraction
expansion/reconstruction, and to polynomial matrix multiplication. Such
reductions eventually imply that all these problems can be solved in about the
same amount of time as polynomial matrix multiplication
Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix
We reduce the problem of computing the rank and a nullspace basis of a
univariate polynomial matrix to polynomial matrix multiplication. For an input
n x n matrix of degree d over a field K we give a rank and nullspace algorithm
using about the same number of operations as for multiplying two matrices of
dimension n and degree d. If the latter multiplication is done in
MM(n,d)=softO(n^omega d) operations, with omega the exponent of matrix
multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K.
The softO notation indicates some missing logarithmic factors. The method is
randomized with Las Vegas certification. We achieve our results in part through
a combination of matrix Hensel high-order lifting and matrix minimal fraction
reconstruction, and through the computation of minimal or small degree vectors
in the nullspace seen as a K[x]-moduleComment: Research Report LIP RR2005-03, January 200
Asymptotic improvement of the Gilbert-Varshamov bound for linear codes
The Gilbert-Varshamov bound states that the maximum size A_2(n,d) of a binary
code of length n and minimum distance d satisfies A_2(n,d) >= 2^n/V(n,d-1)
where V(n,d) stands for the volume of a Hamming ball of radius d. Recently
Jiang and Vardy showed that for binary non-linear codes this bound can be
improved to A_2(n,d) >= cn2^n/V(n,d-1) for c a constant and d/n <= 0.499. In
this paper we show that certain asymptotic families of linear binary [n,n/2]
random double circulant codes satisfy the same improved Gilbert-Varshamov
bound.Comment: Submitted to IEEE Transactions on Information Theor
Lower bounds on the size of semidefinite programming relaxations
We introduce a method for proving lower bounds on the efficacy of
semidefinite programming (SDP) relaxations for combinatorial problems. In
particular, we show that the cut, TSP, and stable set polytopes on -vertex
graphs are not the linear image of the feasible region of any SDP (i.e., any
spectrahedron) of dimension less than , for some constant .
This result yields the first super-polynomial lower bounds on the semidefinite
extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the
positive semidefinite rank of a matrix. To this end, we establish a close
connection between arbitrary SDPs and those arising from the sum-of-squares SDP
hierarchy. For approximating maximum constraint satisfaction problems, we prove
that SDPs of polynomial-size are equivalent in power to those arising from
degree- sum-of-squares relaxations. This result implies, for instance,
that no family of polynomial-size SDP relaxations can achieve better than a
7/8-approximation for MAX-3-SAT
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