7 research outputs found
On giant components and treewidth in the layers model
Given an undirected -vertex graph and an integer , let
denote the random vertex induced subgraph of generated by ordering
according to a random permutation and including in those
vertices with at most of their neighbors preceding them in this order.
The distribution of subgraphs sampled in this manner is called the \emph{layers
model with parameter} . The layers model has found applications in studying
-degenerate subgraphs, the design of algorithms for the maximum
independent set problem, and in bootstrap percolation.
In the current work we expand the study of structural properties of the
layers model.
We prove that there are -regular graphs for which with high
probability has a connected component of size . Moreover,
this connected component has treewidth . This lower bound on the
treewidth extends to many other random graph models. In contrast, is
known to be a forest (hence of treewidth~1), and we establish that if is of
bounded degree then with high probability the largest connected component in
is of size . We also consider the infinite two-dimensional
grid, for which we prove that the first four layers contain a unique infinite
connected component with probability
On Percolation and NP-Hardness
The edge-percolation and vertex-percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational hardness of problems whose inputs are obtained by applying percolation to worst-case instances. Specifically, we show that a number of classical N P-hard graph problems remain essentially as hard on percolated instances as they are in the worst-case (assuming NP !subseteq BPP). We also prove hardness results for other NP-hard problems such as Constraint Satisfaction Problems, where random deletions are applied to clauses or variables.
We focus on proving the hardness of the Maximum Independent Set problem and the Graph Coloring problem on percolated instances. To show this we establish the robustness of the corresponding parameters alpha(.) and Chi(.) to percolation, which may be of independent interest. Given a graph G, let G\u27 be the graph obtained by randomly deleting edges of G. We show that if alpha(G) is small, then alpha(G\u27) remains small with probability at least 0.99. Similarly, we show that if Chi(G) is large, then Chi(G\u27) remains large with probability at least 0.99
On Giant Components and Treewidth in the Layers Model
Given an undirected -vertex graph and an integer , let
denote the random vertex induced subgraph of generated by ordering
according to a random permutation and including in those
vertices with at most of their neighbors preceding them in this order.
The distribution of subgraphs sampled in this manner is called the \emph{layers
model with parameter} . The layers model has found applications in studying
-degenerate subgraphs, the design of algorithms for the maximum
independent set problem, and in bootstrap percolation.
In the current work we expand the study of structural properties of the
layers model.
We prove that there are -regular graphs for which with high
probability has a connected component of size . Moreover,
this connected component has treewidth . This lower bound on the
treewidth extends to many other random graph models. In contrast, is
known to be a forest (hence of treewidth~1), and we establish that if is of
bounded degree then with high probability the largest connected component in
is of size . We also consider the infinite two-dimensional
grid, for which we prove that the first four layers contain a unique infinite
connected component with probability
Infinite and Giant Components in the Layers Percolation Model
In this work we continue the investigation launched in \cite{feige2013layers} of the structural properties of the structural properties of the \emph{Layers model}, a dependent percolation model. Given an undirected graph and an integer , let
denote the random vertex-induced subgraph of , generated by ordering
according to Uniform clocks and including in those vertices
with at most of their neighbors having a faster clock. The distribution
of subgraphs sampled in this manner is called the \emph{layers model with
parameter} . The layers model has found applications in the study of -degenerate
subgraphs, the design of algorithms for the maximum independent set problem
and in the study of bootstrap percolation.
We prove that every infinite locally finite tree with no leaves, satisfying that the degree of the vertices grow sub-exponentially in their distance from the root, has an infinite connected component. In contrast, we show that for any locally finite graph , every connected component of is finite.
We also consider random graphs with a given degree sequence and show that if the minimal degree is at least 3 and the maximal degree is bounded, then has a giant component. Finally, we also consider and show that if is sufficiently large, then contains an infinite cluster