Related papers: Forbidden Configurations and Boundary Cases
$R_+^{n\times n}$ denotes the set of $n\times n$ non-negative matrices. For $A\in R_+^{n\times n}$ let $\Omega(A)$ be the set of all matrices that can be formed by permuting the elements within each row of $A$. Formally: $$\Omega(A)=\{B\in…
We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$…
This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We…
A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial…
For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B.…
The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. By Minc's conjecture, there exists a reachable upper bound on the permanent of 2-dimensional (0,1)-matrices. In this paper we obtain some…
Obtaining a non-trivial (super-linear) lower bound for computation of the Fourier transform in the linear circuit model has been a long standing open problem for over 40 years. An early result by Morgenstern from 1973, provides an $\Omega(n…
We study the impact of forbidding short cycles to the edge density of $k$-planar graphs; a $k$-planar graph is one that can be drawn in the plane with at most $k$ crossings per edge. Specifically, we consider three settings, according to…
First, we prove tight bounds of $n 2^{\frac{1}{(t-2)!}\alpha(n)^{t-2} \pm O(\alpha(n)^{t-3})}$ on the extremal function of the forbidden pair of ordered sequences $(1 2 3 \ldots k)^t$ and $(k \ldots 3 2 1)^t$ using bounds on a class of…
This paper defines a linear representation for nonlinear maps $F:\mathbb{F}^n\rightarrow\mathbb{F}^n$ where $\mathbb{F}$ is a finite field, in terms of matrices over $\mathbb{F}$. This linear representation of the map $F$ associates a…
Complex orthogonal designs (CODs) are used to construct space-time block codes. COD $\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p \times n$ matrix, where nonzero entries are filled by $\pm z_i$ or $\pm z^*_i$, $i = 1, 2,..., k$, such…
Finite dimensional linear spaces (both complex and real) with indefinite scalar product [.,.] are considered. Upper and lower bounds are given for the size of an indecomposable matrix that is normal with respect to this scalar product in…
We study some properties of the growth rate of $\mathcal{L}(\mathcal{A},\mathcal{F})$, that is, the language of words over the alphabet $\mathcal{A}$ avoiding the set of forbidden factors $\mathcal{F}$. We first provide a sufficient…
The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the…
In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of…
A $0$-$1$ matrix $M$ is saturating for a $0$-$1$ matrix $P$ if $M$ does not contain a submatrix that can be turned into $P$ by changing some $1$ entries to $0$ entries, and changing an arbitrary $0$ to $1$ in $M$ introduces such a submatrix…
In this paper, we consider the problem of generating a set of counterfactual explanations for a group of instances, with the one-for-many allocation rule, where one explanation is allocated to a subgroup of the instances. For the first…
In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…
The classical branch-and-bound algorithm for the integer feasibility problem has exponential worst case complexity. We prove that it is surprisingly efficient on reformulated problems, in which the columns of the constraint matrix are…
In this paper we introduce a problem that bridges forbidden subposet and forbidden subconfiguration problems. The sets $F_1,F_2, \dots,F_{|P|}$ form a copy of a poset $P$, if there exists a bijection $i:P\rightarrow \{F_1,F_2,…