MinrankA

MinrankA is a parameter used in linear algebra and combinatorial optimization to denote the minimum possible rank of a matrix that completes or conforms to a given sparsity pattern A. In many contexts A is a binary matrix indicating where nonzero entries may occur. The notation minrankA often refers to the minimum rank, over a specified field F, among all matrices X that satisfy: X_ij ≠ 0 whenever A_ij ≠ 0 and X_ij = 0 whenever A_ij = 0. Some definitions also require diagonal entries to be nonzero or impose additional constraints.

Special case: graphs. If A encodes the adjacency pattern of a graph, then minrank over F is the minimum rank of any matrix consistent with that pattern: off-diagonal nonzero entries correspond to edges, zero entries to non-edges, with diagonal entries typically unrestricted. This graph-theoretic version is studied in relation to problems in index coding, network coding, and linear representations of graphs.

Properties and complexity. The value of minrankA depends on the underlying field F; different fields can yield different results. Computing minrankA is generally NP-hard, though exact results are known for particular graph families or pattern classes. Researchers also study bounds and approximate methods, using relaxations such as nuclear norm minimization or other rank-constrained optimization techniques to obtain estimates.

Applications. MinrankA provides insights into how the structure of A constrains linear representations, with implications for information theory, coding theory, and communication networks. It informs lower and upper bounds on information-flow problems and helps compare linear versus nonlinear representations of combinatorial objects.

See also. Rank minimization; matrix completion; minrank of a graph; index coding.