Improved bounds for matroid partition and intersection algorithms
SIAM Journal on Computing
Efficient theoretic and practical algorithms for linear matroid intersection problems
Journal of Computer and System Sciences
The linear delta-matroid parity problem
Journal of Combinatorial Theory Series B
An algebraic approach to network coding
IEEE/ACM Transactions on Networking (TON)
Matroid Matching Via Mixed Skew-Symmetric Matrices
Combinatorica
Deterministic network coding by matrix completion
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
The complexity of matrix completion
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Information Theory and Network Coding
Information Theory and Network Coding
Matrices and Matroids for Systems Analysis
Matrices and Matroids for Systems Analysis
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
SIAM Journal on Computing
Graph Connectivities, Network Coding, and Expander Graphs
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
IEEE Transactions on Information Theory
A Random Linear Network Coding Approach to Multicast
IEEE Transactions on Information Theory
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Ivanoys, Karpinski and Saxena (2010) have developed a deterministic polynomial time algorithm for finding scalars x1, …, xn that maximize the rank of the matrix B0+x1B1+…+xnBn for given matrices B0, B1, …, Bn, where B1, …, Bn are of rank one. Their algorithm runs in O(m4.37n) 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 one unless n is much larger than m. Our algorithm makes use of an efficient completion method for mixed matrices by Harvey, Karger and Murota (2005). 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 B0+x1B1+…+xnBn for given skew-symmetric matrices B0, B1, …, Bn, where B1, …, Bn 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 the linear delta-covering algorithm of Geelen, Iwata and Murota (2003).