The project pertains to the theory of algorithms and optimization, the discipline of designing and analyzing computational tools and understanding their efficiency. Linear programming (LP) is probably the most influential optimization model with an immense impact in practice: in supply chain management, chip design, airline scheduling, and public transportation, just to name a few areas. The pursuit of more efficient LP algorithms has been a driving force in the development of optimization theory, establishing beautiful connections to discrete mathematics, geometry, and analysis.
Whereas fast and efficient LP algorithms already exist, it remains an intriguing open question to find a strongly polynomial algorithm for LP, namely, an algorithm where the number of arithmetic operations only depends on the number of variables and constraints, but not on the numerical bit-complexity of the problem description. Currently known “weakly” polynomial algorithms heavily rely on such bit-complexity information. In contrast, a strongly polynomial algorithm would only use the combinatorial structure of the problem and would be robust against numerical instability. Finding a strongly polynomial algorithm for LP was listed by Fields medalist Smale as one of the most important mathematical challenges for the twenty-first century.
The main objective of the project was to expand the domain of linear and convex programs that are solvable in strongly polynomial time. This has been achieved by mapping the key geometric and algebraic conditions that can influence the running time of LP algorithms. In particular, the project has developed new, combinatorial versions of interior point methods, a powerful class of algorithms in convex optimization. Equipped with these new tools, we were able to develop a strongly polynomial algorithm for LPs where the constraint matrix only contains two nonzero entries per column, a long-coveted goal.
Besides this, we further developed algorithmic tools at the interface of discrete and continuous optimisation, leading to new results in network optimization, equilibrium computation, tropical geometry, machine learning, and mechanism design.
