Related papers: On the Complexity of Computing Two Nonlinearity Me…
Monotone Boolean functions are a structurally important class of Boolean functions, but their restricted form imposes strong limitations on achievable nonlinearity. In this paper, we investigate whether evolutionary computation can evolve…
We analyze the properties of arguably the simplest bilinear stochastic multiplicative process, proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise. By construction, it…
This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterised by a finite set F of non-negative functions that…
Here we consider an approach for fast computing the algebraic degree of Boolean functions. It combines fast computing the ANF (known as ANF transform) and thereafter the algebraic degree by using the weight-lexicographic order (WLO) of the…
It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…
Many problems in computer science and applied mathematics require rounding a vector $\mathbf{w}$ of fractional values lying in the interval $[0,1]$ to a binary vector $\mathbf{x}$ so that, for a given matrix $\mathbf{A}$,…
Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with…
We present three algorithms to compute the complexity $\Vert n\Vert$ of all natural numbers $ n\le N$. The first of them is a brute force algorithm, computing all these complexities in time $O(N^2)$ and space $O(N\log^2 N)$. The main…
Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where…
Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…
In 1991, Edelsbrunner and Tan gave an O(n^2) algorithm for finding the MinMax Length triangulation of a set of points in the plane. In this paper we resolve one of the open problems stated in that paper, by showing that finding a MaxMin…
We study a Faulty Congested Clique model, in which an adversary may fail nodes in the network throughout the computation. We show that any task of $O(n\log{n})$-bit input per node can be solved in roughly $n$ rounds, where $n$ is the size…
We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are…
The {\it matrix-chain multiplication} problem is a classic problem that is widely taught to illustrate dynamic programming. The textbook solution runs in $\theta(n^3)$ time. However, there is a complex $O(n \log n)$-time method \cite{HU82},…
We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give algorithms with…
First of all we give some reasons that "natural proofs" built not a barrier to prove P $\not=$ NP using Boolean complexity. Then we investigate the approximation method for its extension to prove super-polynomial lower bounds for the…
We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced…
We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem.…
The classical approach to measure the expressive power of deep neural networks with piecewise linear activations is based on counting their maximum number of linear regions. This complexity measure is quite relevant to understand general…
We construct a reduction which proves that the fooling set number and the determinantal rank of a Boolean matrix are NP-hard to compute.