Enumeration of 0/1-matrices avoiding some 2x2 matrices

We enumerate the number of 0/1-matrices avoiding 2x2 submatrices satisfying certain conditions. We also provide corresponding exponential generating functions.Comment: 14 pages, 3 figures; Some references and related works are added in v

Supermodularity on chains and complexity of maximum constraint satisfaction

In the maximum constraint satisfaction problem ($\mathrm{Max \; CSP}$), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximise the number (or the total weight) of satisfied constraints. This problem is $\mathrm{NP}$-hard in general so it is natural to study how restricting the allowed types of constraints affects the complexity of the problem. In this paper, we show that any $\mathrm{Max \; CSP}$ problem with a finite set of allowed constraint types, which includes all constants (i.e. constraints of the form $x=a$), is either solvable in polynomial time or is $\mathrm{NP}$-complete. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known combinatorial property of supermodularity

Large homogeneous submatrices

A matrix is homogeneous if all of its entries are equal. Let $P$ be a $2\times 2$ zero-one matrix that is not homogeneous. We prove that if an $n\times n$ zero-one matrix $A$ does not contain $P$ as a submatrix, then $A$ has an $cn\times cn$ homogeneous submatrix for a suitable constant $c>0$. We further provide an almost complete characterization of the matrices $P$ (missing only finitely many cases) such that forbidding $P$ in $A$ guarantees an $n^{1-o(1)}\times n^{1-o(1)}$ homogeneous submatrix. We apply our results to chordal bipartite graphs, totally balanced matrices, halfplane-arrangements and string graphs.Comment: 20 pages, 1 figure, moved Theorem 1.6 to introduction, added its proof + a new application of Theorem 1.

Large homogeneous submatrices

A matrix is homogeneous if all of its entries are equal. Let P be a 2 × 2 zero-one matrix that is not homogeneous. We prove that if an n × n zero-one matrix A does not contain P as a submatrix, then A has a cn × cn homogeneous submatrix for a suitable constant c > 0. We further provide an almost complete characterization of the matrices P (missing only finitely many cases) such that forbidding P in A guarantees an n1 - o(1) × n1 - o(1) homogeneous submatrix. We apply our results to chordal bipartite graphs, totally balanced matrices, halfplane arrangements, and string graphs. © 2020 author

Permuting matrices to avoid forbidden submatrices

This paper attaches a frame to a natural class of combinatorial problems and points out that this class includes many important special cases. A matrix M is said to avoid a set of matrices if M does not contain any element of as (ordered) submatrix. For a fixed set of matrices, we consider the problem of deciding whether the rows and columns of a matrix can be permuted in such a way that the resulting matrix M avoids all matrices in . We survey several known and new results on the algorithmic complexity of this problem, mostly dealing with (0,1)-matrices. Among others, we will prove that the problem is polynomial time solvable for many sets containing a single, small matrix and we will exhibit some example sets for which the problem is NP-complete