The complexity of matrix completion
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
SIAM Journal on Computing
Algebraic algorithms for linear matroid parity problems
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Fast deterministic algorithms for matrix completion problems
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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Tutte associates a V by V skew-symmetric matrix T, having indeterminate entries, with a graph G=(V,E). This matrix, called the Tutte matrix, has rank exactly twice the size of a maximum cardinality matching of G. Thus, to find the size of a maximum matching it suffices to compute the rank of T. We consider the more general problem of computing the rank of T + K where K is a real V by V skew-symmetric matrix. This modest generalization of the matching problem contains the linear matroid matching problem and, more generally, the linear delta-matroid parity problem. We present a tight upper bound on the rank of T + K by decomposing T + K into a sum of matrices whose ranks are easy to compute.