“Solving Billion-Scale Knapsack Problems”, Xingwen Zhang, Feng Qi, Zhigang Hua, Shuang Yang2020-02-02 (algorithms)⁠:

Knapsack problems (KPs) are common in industry, but solving KPs is known to be NP-hard and has been tractable only at a relatively small scale.

This paper examines KPs in a slightly generalized form and shows that they can be solved nearly optimally at scale via distributed algorithms [via sub-problems with Lagrangians optimized with dual descent]. The proposed approach can be implemented fairly easily with off-the-shelf distributed computing frameworks (eg. MPI, Hadoop, Spark).

As an example, our implementation leads to one of the most efficient KP solvers known to date—capable to solve KPs at an unprecedented scale (eg. KPs with 1 billion decision variables and 1 billion constraints can be solved within 1 hour).

The system has been deployed to production and called on a daily basis, yielding business impacts at Ant Financial.