Permuting matrices to avoid forbidden submatrices

Forbidden submatrices in 0-1 matrices

On linear forbidden submatrices

In this paper we study the extremal problem of finding how many 1 entries an n by n 0-1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0-1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0-1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0-1 matrix) li...

Avoiding Forbidden Submatrices by Row Deletions

We initiate a systematic study of the Row Deletion(B) problem on matrices: For a fixed “forbidden submatrix” B, the question is, given an input matrix A (both A and B have entries chosen from a finite-size alphabet), to remove a minimum number of rows such that A has no submatrix which is equivalent to a row or column permutation of B. An application of this question can be found, e.g., in the ...

Forbidden Submatrices: Some New Bounds and Constructions

We explore an extremal hypergraph problem for which both the vertices and edges are ordered. Given a hypergraph F (not necessarily simple), we consider how many edges a simple hypergraph (no repeated edges) on m vertices can have while forbidding F as a subhypergraph where both hypergraphs have fixed vertex and edge orderings. A hypergraph of n edges on m vertices can be encoded as an m × n (0,...

Reconstruction of matrices from submatrices

For an arbitrary matrix A of n×n symbols, consider its submatrices of size k×k, obtained by deleting n−k rows and n−k columns. Optionally, the deleted rows and columns can be selected symmetrically or independently. We consider the problem of whether these multisets determine matrix A. Following the ideas of Krasikov and Roditty in the reconstruction of sequences from subsequences, we replace t...