  The problem of selecting a small-size representative summary of a large
dataset is a cornerstone of machine learning, optimization and data science.
Motivated by applications to recommendation systems and other scenarios with
query-limited access to vast amounts of data, we propose a new rigorous
algorithmic framework for a standard formulation of this problem as a
submodular maximization subject to a linear (knapsack) constraint. Our
framework is based on augmenting all partial Greedy solutions with the best
additional item. It can be instantiated with negligible overhead in any model
of computation, which allows the classic \greedy algorithm and its variants to
be implemented. We give such instantiations in the offline (Greedy+Max),
multi-pass streaming (Sieve+Max) and distributed (Distributed+Max) settings.
Our algorithms give ($1/2-\epsilon$)-approximation with most other key
parameters of interest being near-optimal. Our analysis is based on a new set
of first-order linear differential inequalities and their robust approximate
versions. Experiments on typical datasets (movie recommendations, influence
maximization) confirm scalability and high quality of solutions obtained via
our framework. Instance-specific approximations are typically in the 0.6-0.7
range and frequently beat even the $(1-1/e) \approx 0.63$ worst-case barrier
for polynomial-time algorithms.
