Section: New Results

Network and Graph Algorithms

Rumor spreading and conductance

Participant : George Giakkoupis.

In [16], we study the completion time of the PUSH-PULL variant of rumor spreading, also known as randomized broadcast. We show that if a network has $n$ nodes and conductance $\phi$ then, with high probability, PUSH-PULL will deliver the message to all nodes in the graph within $O(logn/\phi )$ many communication rounds. This bound is best possible. We also give an alternative proof that the completion time of PUSH-PULL is bounded by a polynomial in $logn/\phi$, based on graph sparsification. Although the resulting asymptotic bound is not optimal, this proof shows an interesting and, at the outset, unexpected connection between rumor spreading and graph sparsification. Finally, we show that if the degrees of the two endpoints of each edge in the network differ by at most a constant factor, then both PUSH and PULL alone attain the optimal completion time of $O(logn/\phi )$, with high probability.

This work was done in collaboration with Flavio Chierichetti (Sapienza University of Rome), Silvio Lattanzi (Google Research), and Alessandro Panconesi (Sapienza University of Rome).

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 [21], we study 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 $O(logn/\phi$) 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 show that for any connected regular graph the cover time is $O(nlogn)$ with high probability, independently of the expansion. Both bounds are asymptotically tight. Since our bounds coincide with the worst-case time bounds for Push rumor spreading on regular graphs until all nodes are informed, this raises the question whether CoBra walks and Push rumor spreading perform similarly in general. We answer this negatively by separating the cover time of CoBra walks and the rumor spreading time of Push by a super-polylogarithmic factor on a family of tree-like regular graphs.

This work was done in collaboration with Petra Berenbrink and Peter Kling from the University of Hamburg.

The quadratic shortest path problem: Complexity, approximability, and solution methods

Participant : Davide Frey.

In work [20] we considered the problem of finding a shortest path in a directed graph with a quadratic objective function (the QSPP). We show that the QSPP cannot be approximated unless $𝖯=\mathrm{𝖭𝖯}$. For the case of a convex objective function, we presented an $n$-approximation algorithm, where $n$ is the number of nodes in the graph, and we proved $\mathrm{𝖠𝖯𝖷}$-hardness. Furthermore, we proved that even if only adjacent arcs play a part in the quadratic objective function, the problem still cannot be approximated unless $𝖯=\mathrm{𝖭𝖯}$. In order to solve the general problem we first proposed a mixed integer programming formulation, and then devised an efficient exact Branch-and-Bound algorithm for the general QSPP. This algorithm computes lower bounds by considering a reformulation scheme that is solvable through a number of minimum-cost-flow problems. We carried out computational experiments solving to optimality different classes of instances with up to 1000 nodes.

This work was carried out in collaboration with Borzou Rostami from Polytechnique Montréal, Adreé Chasssein and Michael Hopf from TU Kaiserslautern, Christoph Buchheim from TU Dortmund, Federico Malucelli from Politecnico di Milano, and Marc Goerigk from Lancaster University.

Weighting past on the geo-aware state deployment problem

Participant : François Taïani.

The geographical barrier between mobile devices and mobile application servers (typically hosted in the Cloud) imposes an unavoidable latency and jitter that negatively impacts the performance of modern mobile systems. Fog Computing architectures can mitigate this impact if there is a middleware service able to correctly partition and deploy the state of an application at optimal locations. Geo-aware state deployment is challenging as it must consider the mobility of the devices and the dependencies arising when multiple devices concurrently manipulate the same application state. In [28], we propose a range of new object-graph-based strategies for geo-aware state deployment. In particular, our investigation focuses on understanding the role of preserving previously observed associations between state items on application performance.

This work was performed in collaboration with Diogo Lima and Hugo Miranda from the University of Lisbon (Portugal).

Mind the gap: Autonomous detection of partitioned MANET systems using opportunistic aggregation

Participants : Simon Bouget, David Bromberg, François Taïani.

Mobile Ad-hoc Networks (MANETs) use limited-range wireless communications and are thus exposed to partitions when nodes fail or move out of reach of each other. Detecting partitions in MANETs is unfortunately a nontrivial task due to their inherently decentralized design and limited resources such as power or bandwidth. In [32], we propose a novel and fully decentralized approach to detect partitions (and other large membership changes) in MANETs that is both accurate and resource efficient. We monitor the current composition of a MANET using the lightweight aggregation of compact membership-encoding filters. Changes in these filters allow us to infer the likelihood of a partition with a quantifiable level of confidence. We first present an analysis of our approach, and show that it can detect close to 100% of partitions under realistic settings, while at the same time being robust to false positives due to churn or dropped packets. We perform a series of simulations that compare against alternative approaches and confirm our theoretical results, including above 90% accurate detection even under a 40% message loss rate.

This work was performed in collaboration with Etienne Rivière from UC Louvain (Belgium) and Hugues Mercier from University of Neuchâtel (Switzerland).