Technology Transfer between Integer Programming and Efficient Algorithms

ended

This project aims to resolve challenging integer programming problems in exact and approximate settings, with a focus on Knapsack-type problems (such as Subset Sum, Partition, and Knapsack). To this end, we will develop a unified approach of algorithm design as a combination of algorithmic tools, structural theory, and conditional lower bounds. Specific tasks include: - utilizing recent advances in efficient algorithms, since although Knapsack-type algorithms are NP-hard their main challenges ask for polynomial improvements in running time, - leveraging structural results from additive combinatorics for the design of algorithms for problems of additive nature, such as Knapsack-type problems, and - using and expanding fine-grained complexity theory to explain the limits of algorithms by proving conditional lower bounds based on plausible conjectures. In particular, our combination of modern algorithmic techniques and structural results on the one hand, and conditional lower bounds on the other hand, allows us to aim at best-possible algorithms (conditional on plausible conjectures). We also plan to transfer techniques in the other direction (from integer programming to efficient algorithms), by using the insights of practical integer programming solvers to obtain highly-efficient implementations for selected polynomial-time problems. Designing best-possible algorithms for one of the Knapsack-type problems will already be groundbreaking, and complete resolution of our goals would be dramatic algorithmic progress with consequences in computer science, optimization, and operations research.