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 $n$-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than $2^{n^c}$, for some constant $c > 0$. 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-$O(1)$ 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