Periodic Reporting for period 4 - ScaleOpt (Scaling Methods for Discrete and Continuous Optimization)
Reporting period: 2022-07-01 to 2023-06-30
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.
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.
We attained most key objectives of the project and made surprising new discoveries. Publication highlights include seven papers at the flagship conferences in theoretical computer science, the Symposium on Theory of Computing and the Symposium on Foundations of Computer Science, three paper published in the Journal of the ACM (JACM), and one in Operations Research. We highlight four areas of achievements. We presented the results by giving a number of invited and plenary talks, and two invited minicourses dedicated to our new approaches to linear programming.
The most significant objective of the project was to develop a strongly polynomial algorithm for the minimum-cost generalized flow problem, or equivalently, for LPs with two nonzero entries in each row of the column matrix, a well-studied class. As the culmination of five years of concerted efforts, we resolved this in a paper with Dadush, Koh, Natura, and Olver, accepted to STOC 2024.
To answer this question, we developed a new theory of combinatorial interior point methods, exploring deep connections between discrete and continuous optimization. We further developed the theory of layered least squares interior point methods pioneered by Vavasis and Ye in the nineties. Ultimately, we developed a novel interior point method that is essentially the best possible from a strongly polynomial perspective. The running time of this method for a particular problem class boils down to analysing a simple combinatorial quantity. This is the main tool in resolving the minimum-cost generalized flow question. In this context, we also developed a new theory of circuit imbalances, geometric parameters that play important roles in a wide range of problems in combinatorics and optimization.
We also obtained a number of new results in the context of equilibrium computation and mechanism design. In particular, with Garg we resolved a longstanding open question by giving a strongly polynomial algorithm for Arrow-Debreu exchange market equilibrium. We made significant advancements on the approximability of the Nash social welfare problem.
We explored a range of connections of the above concepts, leading to new results in related areas such as network optimization, tropical geometry, machine learning, and mechanism design.
The most significant objective of the project was to develop a strongly polynomial algorithm for the minimum-cost generalized flow problem, or equivalently, for LPs with two nonzero entries in each row of the column matrix, a well-studied class. As the culmination of five years of concerted efforts, we resolved this in a paper with Dadush, Koh, Natura, and Olver, accepted to STOC 2024.
To answer this question, we developed a new theory of combinatorial interior point methods, exploring deep connections between discrete and continuous optimization. We further developed the theory of layered least squares interior point methods pioneered by Vavasis and Ye in the nineties. Ultimately, we developed a novel interior point method that is essentially the best possible from a strongly polynomial perspective. The running time of this method for a particular problem class boils down to analysing a simple combinatorial quantity. This is the main tool in resolving the minimum-cost generalized flow question. In this context, we also developed a new theory of circuit imbalances, geometric parameters that play important roles in a wide range of problems in combinatorics and optimization.
We also obtained a number of new results in the context of equilibrium computation and mechanism design. In particular, with Garg we resolved a longstanding open question by giving a strongly polynomial algorithm for Arrow-Debreu exchange market equilibrium. We made significant advancements on the approximability of the Nash social welfare problem.
We explored a range of connections of the above concepts, leading to new results in related areas such as network optimization, tropical geometry, machine learning, and mechanism design.
Our results are transformative in the context of strongly polynomial computability. We mapped the limits of strongly polynomial computability using interior point methods: on the one hand, we showed that these methods subsume all previous approaches, and on the other hand, we established a clear combinatorial characterization in terms of the straight-line complexity of when an interior point method can be strongly polynomial. This technique is used to answer an important open question in combinatorial optimization where other methods seem to have failed. Moreover, we developed powerful tools for further research and identified the next critical problems. In a different direction, in the context of mechanism design, our work on Nash Social welfare developed a surprising new technique that enabled very significant progress on the problem.