Fast Distributed Approximation for Max-Cut

Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the $\mathcal{LOCAL}$ or $\mathcal{CONGEST}$ models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any $\epsilon > 0$, we develop randomized approximation algorithms achieving a ratio of $(1-\epsilon)$ to the optimum for Max-Cut on bipartite graphs in the $\mathcal{CONGEST}$ model, and on general graphs in the $\mathcal{LOCAL}$ model. We further present efficient deterministic algorithms, including a $1/3$-approximation for Max-Dicut in our models, thus improving the best known (randomized) ratio of $1/4$. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest

Mining topological structure in graphs through forest representations

We consider the problem of inferring simplified topological substructures—which we term backbones—in metric and non-metric graphs. Intuitively, these are subgraphs with ‘few’ nodes, multifurcations, and cycles, that model the topology of the original graph well. We present a multistep procedure for inferring these backbones. First, we encode local (geometric) information of each vertex in the original graph by means of the boundary coefficient (BC) to identify ‘core’ nodes in the graph. Next, we construct a forest representation of the graph, termed an f-pine, that connects every node of the graph to a local ‘core’ node. The final backbone is then inferred from the f-pine through CLOF (Constrained Leaves Optimal subForest), a novel graph optimization problem we introduce in this paper. On a theoretical level, we show that CLOF is NP-hard for general graphs. However, we prove that CLOF can be efficiently solved for forest graphs, a surprising fact given that CLOF induces a nontrivial monotone submodular set function maximization problem on tree graphs. This result is the basis of our method for mining backbones in graphs through forest representation. We qualitatively and quantitatively confirm the applicability, effectiveness, and scalability of our method for discovering backbones in a variety of graph-structured data, such as social networks, earthquake locations scattered across the Earth, and high-dimensional cell trajectory dat