Algorithms in the Public-Private Model to Sub-Additive functions

In this work we consider the Public-Private model of graphs introduced by Chierichetti et al [1]. In the first part, we briefly introduce the model and give the description of the shortest-path algorithm as described in their paper. In the second part we describe algorithms for Community Detection(Densest Subgraph problem), Max-Cut and Vertex Cover . We also briefly explain how the above algorithms can be extended to a small class of sub-additive functions. Finally, we give details of the experimental analysis of this algorithm run on real-world social network data. I. The Public-Private Model The public-private model was introduced in [1] to capture graph problems on social networks. In this model we have a known public graph G = (V, E) and for each node u ∈ V we have a hidden private graph Gu = (V, Eu). For each (v, w) ∈ Eu both v and w must be at most a distance of two from u in the union graph G ∪ Gu. Without loss of generality, we can also assume G ∩ Gu = ∅. In social networks, nodes represent people/users and edges represent connections between them (e.g. friendship, shared group membership, etc.). Here we differentiate between edges which are known to the public, E, and those which are hidden by the user, Eu. For example, a Facebook user can hide their friend list while other users allow their friend lists to be publicly available. A 2012 study [2] showed that out of 1.4 million New York City Facebook users, 52.6% hid their friend list. To see how this simple public-private model captures phenomena in social networks, we consider a few examples. If a user u hides their friend list, then the private graph Gu is a star centered at u. Similarly, if u is part of a private group, then Gu will include a clique containing all members of the group. Note that in some online social networks, such as Facebook, two users can be in the same group even if they are not “friends” and we put an edge between them since they can interact with each other in the group. When solving problems in this graph model, we assume the public graph G is known in advance and a private graph Gu for some user u will be revealed. The solution we find should apply to the union graph G ∪ Gu. Observe that it could be very time consuming to compute some function on the entire union graph each time a user reveals their public graph. So we wish to do some preprocessing once on the public graph to answer queries efficiently when private graphs are revealed. Specifically, our goal is to preprocess the public graph G using poly(|E|) time and Õ(|V|) space. Then, when Gu is revealed, we should answer queries in time/space that is Õ(|Eu|) and poly(lg |V|). II. All Pairs Shortest Path In this section, we will describe the algorithm for the All Pairs Shortest Path(APSP) problem in the Public-Private model. APSP is an important problem in social networks since many learning