Related papers: On additive complexity of a sequence of matrices
We call a matrix blocky if, up to row and column permutations, it can be obtained from an identity matrix by repeatedly applying one of the following operations: duplicating a row, duplicating a column, or adding a zero row or column.…
We propose an algorithm for solving bound-constrained mathematical programs with complementarity constraints on the variables. Each iteration of the algorithm involves solving a linear program with complementarity constraints in order to…
We obtain a nontrivial bound on the number of solutions to the equation $A^{x_1} + A^{x_2} = A^{x_3} + A^{x_4}$, $1 \le x_1,x_2,x_3,x_4 \le \tau$, with a fixed $n\times n$ matrix $A$ over a finite field ${\mathbb F}_q$ of $q$ elements of…
The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is…
We propose a new encoding of the first-order connection method as a Boolean satisfiability problem. The encoding eschews tree-like presentations of the connection method in favour of matrices, as we show that tree-like calculi have a number…
Boolean network models of strongly connected modules are capable of capturing the high regulatory complexity of many biological gene regulatory circuits. We study numerically the previously introduced basin entropy, a parameter for the…
We develop an automated framework for proving lower bounds on the bilinear complexity of matrix multiplication over finite fields. Our approach systematically combines orbit classification of the restricted first matrix and dynamic…
We present quantum complexity lower and upper bounds for independent set problems in graphs. In particular, we give quantum algorithms for computing a maximal and a maximum independent set in a graph. We present applications of these…
Roughgarden, Vassilvitskii, and Wang (JACM 18) recently introduced a novel framework for proving lower bounds for Massively Parallel Computation using techniques from boolean function complexity. We extend their framework in two different…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix $M$ that has a $1$ in the entry $(i,j)$ iff there is a path from the $j$th source to the $i$th sink. The smallest…
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of…
We refine the complexity landscape for enumeration problems by introducing very low classes defined by using Boolean circuits as enumerators. We locate well-known enumeration problems, e.g., from graph theory, Gray code enumeration, and…
We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication. We characterize such semisimple group algebras of the minimal bilinear complexity and show…
We give an upper bound on the number of cycles in a simple graph in terms of its degree sequence, and apply this bound to resolve several conjectures of Kir\'aly and Arman and Tsaturian and to improve upper bounds on the maximum number of…
We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a non-elementary…
The subsumption problem with respect to terminologies in the description logic ALC is EXPTIME-complete. We investigate the computational complexity of fragments of this problem by means of allowed Boolean operators. Hereto we make use of…
We present new results on Boolean matrix factorization and a new algorithm based on these results. The results emphasize the significance of factorizations that provide from-below approximations of the input matrix. While the previously…
In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of boundary is given either as a list of…
In these notes we focus a bit on the complex case for some families of matrices and equivalences between them.