Related papers: Complexity of 2D bootstrap percolation difficulty:…
The $r$-neighbour bootstrap process on a graph $G$ begins with a set of infected vertices; subsequently, healthy vertices become infected once they have at least $r$ infected neighbours. The central extremal problem in bootstrap percolation…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
We study a two-dimensional generalization of the classical Bin Packing problem, denoted as 2D Demand Bin Packing. In this context, each bin is a horizontal timeline, and rectangular tasks (representing electric appliances or computational…
In many interacting particle systems, relaxation to equilibrium is thought to occur via the growth of 'droplets', and it is a question of fundamental importance to determine the critical length at which such droplets appear. In this paper…
We numerically study bootstrap percolation on Kleinberg's spatial networks, in which the probability density function of a node to have a long-range link at distance $r$ scales as $P(r)\sim r^{\alpha}$. Setting the ratio of the size of the…
In 2-neighborhood bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least 2 already…
In modified two-neighbour bootstrap percolation in two dimensions each site of $\mathbb Z^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two…
In order to formulate mathematical conjectures likely to be true, a number of base cases must be determined. However, many combinatorial problems are NP-hard and the computational complexity makes this research approach difficult using a…
The PL geometric category of a polyhedron $P$, denoted $\hbox{plgcat}(P)$, provides a natural upper bound for the Lusternik--Schnirelmann category and it is defined as the minimum number of PL collapsible subpolyhedra of $P$ that cover $P$.…
An elegant bootstrap percolation result on the hyperrectangle graph was proved by Balogh, Bollob\'as, Morris, and Riordan using a linear algebric method in 2012. This paper studies the same problem by purely combinatorial means. The base…
In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy,…
In this paper, we study an NP-hard problem of a single machine scheduling minimizing the makespan, where the mixed-critical tasks with an uncertain processing time are scheduled. We show the derivation of F-shaped tasks from the probability…
In 2-neighborhood bootstrap percolation on a graph $G$, an infection spreads according to the following deterministic rule: infected vertices of $G$ remain infected forever and in consecutive rounds healthy vertices with at least two…
K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been…
In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a…
Bootstrap percolation on a graph with infection threshold $r\in \mathbb{N}$ is an infection process, which starts from a set of initially infected vertices and in each step every vertex with at least $r$ infected neighbours becomes…
Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a…
This paper analyzes various questions pertaining to bootstrap percolation on the $d$-dimensional Hamming torus where each node is open with probability $p$ and the percolation threshold is 2. For each $d'<d$ we find the critical exponent…
In this work we apply a highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff to treat percolation problems. The site and bond percolation are studied on a number of lattices in two and three dimensions. Quite good…
We study bootstrap percolation (BP) on hyperbolic lattices obtained by regular tilings of the hyperbolic plane. Our work is motivated by the connection between the BP transition and the dynamical transition of kinetically constrained…