## Section: New Results

### Network and Graph Algorithms

#### Tight bounds on vertex connectivity under sampling

Participant : George Giakkoupis.

A fundamental result by Karger (SODA 1994) states that for any $\lambda $-edge-connected graph with $n$ nodes, independently sampling each edge with probability $p=\Omega (log(n)/\lambda )$ results in a graph that has edge connectivity $\Omega \left(\lambda p\right)$, with high probability. In [15], we proved the analogous result for vertex connectivity, when either vertices or edges are sampled. We showed that for any $k$-vertex-connected graph $G$ with $n$ nodes, if each node is independently sampled with probability $p=\Omega \left(\sqrt{log\left(n\right)/k}\right)$, then the subgraph induced by the sampled nodes has vertex connectivity $\Omega \left(k{p}^{2}\right)$, with high probability. If edges are sampled with probability $p=\Omega (log(n)/k)$then the sampled subgraph has vertex connectivity $\Omega \left(kp\right)$, with high probability. Both bounds are existentially optimal.

This work was done in collaboration with Keren Censor-Hillel (Technion), Mohsen Ghaffari (MIT), Bernhard Haeupler (Carnegie Mellon University), and Fabian Kuhn (University of Freiburg).

#### Tight bounds for coalescing-branching random walks on regular graphs

Participant : George Giakkoupis.

A *coalescing-branching random walk (Cobra)* is a natural extension to the standard random walk on a graph.
The process starts with one pebble at an arbitrary node.
In each round of the process every pebble splits into $k$ pebbles, which are sent to $k$ random neighbors.
At the end of the round all pebbles at the same node coalesce into a single pebble.
The process is also similar to randomized rumor spreading, with each informed node pushing the rumor to $k$ random neighbors each time it receives a copy of the rumor.
Besides its mathematical interest, this process is relevant as an information dissemination primitive and a basic model for the spread of epidemics.

In [25] we studied the *cover time* of Cobra walks, which is the time until each node has seen at least one pebble.
Our main result is a bound of $O({\phi}^{-1}logn)$ rounds with high probability on the cover time of a Cobra walk with $k=2$ on any regular graph with $n$ nodes and conductance $\phi $.
This bound improves upon all previous bounds in terms of graph expansion parameters.
Moreover, we showed that for any connected regular graph the cover time is $O(nlogn)$ with high probability, independently of the expansion.
Both bounds are asymptotically tight.

This work was done in collaboration with Petra Berenbrink (University of Hamburg), Peter Kling (University of Hamburg).