Maximal induced paths and minimal percolating sets in hypercubes

For a graph $G$, the \emph{$r$-bootstrap percolation} process can be described as follows: Start with an initial set $A$ of "infected'' vertices. Infect any vertex with at least $r$ infected neighbours, and continue this process until no new vertices can be infected. $A$ is said to \emph{percolate in $G$} if eventually all the vertices of $G$ are infected. $A$ is a \emph{minimal percolating set} in $G$ if $A$ percolates in $G$ and no proper subset of $A$ percolates in $G$. An induced path, $P$, in a hypercube $Q_n$ is maximal if no induced path in $Q_n$ properly contains $P$. Induced paths in hypercubes are also called snakes. We study the relationship between maximal snakes and minimal percolating sets (under 2-bootstrap percolation) in hypercubes. In particular, we show that every maximal snake contains a minimal percolating set, and that every minimal percolating set is contained in a maximal snake

Local Treewidth of Random and Noisy Graphs with Applications to Stopping Contagion in Networks

We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all k-vertex subgraphs of an n-vertex graph. When k is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive nearly tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is k, the number of initially "infected" vertices in the network. For the random graph models we consider and a certain range of parameters the running time of our algorithms on n-vertex graphs is 2^o(k) poly(n), improving upon the 2^?(k) poly(n) performance of the best-known algorithms designed for worst-case instances of these edge deletion problems

Minimum lethal sets in grids and tori under 3-neighbour bootstrap percolation

Let $\mathcal{r} ≥ 1$ be any non negative integer and let $G = (V, E)$ be any undirected graph in which a subset $D ⊆ V$ of vertices are initially infected. We consider the following process in which, at every step, each non-infected vertex with at least $\mathcal{r}$ infected neighbours becomes and an infected vertex never becomes non-infected. The problem consists in determining the minimum size $s_r (G)$ of an initially infected vertices set $D$ that eventually infects the whole graph $G$. Note that $s_1 (G)$ = 1 for any connected graph $G$. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that $s_1(G) = 1$ for any connected graph $G$. The case when $G$ is the $n × n$ grid $G_{n×n}$ and $\mathcal{r} = 2$ is well known and appears in many puzzles books, in particular due to the elegant proof that shows that $s_2(G_{n×n})$ = $n$ for all $n$ ∈ $\mathbb{N}$. We study the cases of square grids $G_{n×n}$ and tori $T_{n×n}$ when $\mathcal{r}$ ∈ {3, 4}. We show that $s_3(G_{n×n})$ = $\lceil\frac{n^2+2n+4}{3}\rceil$ for every $n$ even and that $\lceil\frac{n^2 +2n}{3}\rceil$ ≤ $s_3(G_ {n×n})$ ≤ $\lceil\frac{n^2 +2n}{3}\rceil$ + 1 for any $n$ odd. When $n$ is odd, we show that both bounds are reached, namely $s_3(G_{n×n})$ = $\lceil\frac{n^2 +2n}{3}\rceil$ if $n$ ≡ 5 (mod 6) or $n$ = 2$^p$ − 1 for any $p$ ∈ $\mathbb{N}^*$, and $s_3(G_{n×n})$ = $\lceil\frac{n^2 +2n}{3}\rceil$ + 1 if $n$ ∈ {9, 13}. Finally, for all $n$ ∈ $\mathbb{N}$, we give the exact expression of $s_4(G_{n×n})$ and of $s_r(T_{n×n})$ when $\mathcal{r}$ ∈ {3, 4}

Oblivious tight compaction in O(n) time with smaller constant

Oblivious compaction is a crucial building block for hash-based oblivious RAM. Asharov et al. recently gave a O(n) algorithm for oblivious tight compaction. Their algorithm is deterministic and asymptotically optimal, but it is not practical to implement because the implied constant is $\gg 2^{38}$. We give a new algorithm for oblivious tight compaction that runs in time $< 16014.54n$. As part of our construction, we give a new result in the bootstrap percolation of random regular graphs