We study the maximum set coverage problem in the massively parallel model. In this setting, m sets that are subsets of a universe of n elements are distributed among m machines.  In each round, these machines can communicate with each other, subject to the memory constraint that no machine may use more than  \Tilde{O}(n) memory. The objective is to find the  sets whose coverage is maximized. We consider the regime where k = \Omega(m), m = O(n), and each machine has  memory. 

We study the densest subgraph problem and its variant through the lens of learning-augmented algorithms. More specifically, we show that given a partial solution-i.e., one produced by a machine learning classifier that capture at least (1-\epsilon)-fraction of nodes in the optimal subgraph-it is possible to design an extremely simple linear-time algorithm that achieves a provable (1-\epsilon)-approximation. Our approach also naturally extends to the directed densest subgraph problem and several NP-hard variants.

In this work, we provide a complete classification of small roots and low elements for a large class of rank 3 hyperbolic Coxeter groups. Our approach relies on the projective geometric representation and the normalized isotropic cone associated with these groups to characterize these elements systematically.