Fast Deterministic Algorithms for Matrix Completion Problems

Abstract

Ivanyos, Karpinski, and Saxena [SIAM J. Comput., 39 (2010), pp. 3736--3751] have developed a deterministic polynomial time algorithm for finding scalars $x_1, \dots, x_n$ that maximize the rank of the matrix $B_0 + x_1B_1 + \dots + x_nB_n$ for given matrices $B_0, B_1, \dots, B_n$, where $B_1, \dots, B_n$ are of rank one. Their algorithm runs in $O(m^{4.37}n)$ time, where $m$ is the larger of the row size and the column size of the input matrices. In this paper, we present a new deterministic algorithm that runs in $O((m+n)^{2.77})$ time, which is faster than the previous algorithm unless $n$ is much larger than $m$. Our algorithm makes use of an efficient completion method for mixed matrices. As an application of our completion algorithm, we devise a deterministic algorithm for the multicast problem with linearly correlated sources. We also consider a skew-symmetric version: maximize the rank of the matrix $B_0+ x_1B_1 + \dots + x_nB_n$ for given skew-symmetric matrices $B_0, B_1, \dots, B_n$, where $B_1, \dots, B_n$ are of rank two. We design the first deterministic polynomial time algorithm for this problem based on the concept of mixed skew-symmetric matrices and a linear delta-covering problem.