Periodic Reporting for period 4 - TIPEA (Technology Transfer between Integer Programming and Efficient Algorithms)
Berichtszeitraum: 2024-06-01 bis 2025-11-30
TIPEA was a project in theoretical computer science seeking a deeper understanding of fundamental optimization problems such as Subset Sum, Knapsack, and more generally Integer Programming. These problems are ubiquitous, arising in everyday tasks such as scheduling events without collision, industrial logistics, and even post-quantum cryptography, where the hardness of Subset Sum underpins security against quantum attacks. Since modern solvers can handle industrial instances with up to millions of variables despite the problem being NP-hard, Integer Programming has become a vital tool for industry. The goal of this project was to leverage recent advances in theoretical computer science to design faster, and in many cases best-possible algorithms for these fundamental optimization problems to lay the foundations for next-generation solvers. To this end, we developed an algorithm design approach consisting of (1) modern algorithmic techniques such as dynamic or sublinear-time algorithms to design faster subroutines, (2) mathematical structure theory such as additive combinatorics to gain problem insights, and (3) the recently developed fine-grained complexity theory to establish that our algorithms are best-possible.
We successfully applied this approach to open problems on exact, parameterized, and approximation algorithms. We demonstrated that:
- Despite decades of prior research, significant algorithmic improvements are still possible,
- Additive combinatorics is a powerful toolkit for number-theoretic algorithmic problems, and
- Combining modern algorithm design with fine-grained complexity lower bounds can determine the optimal time complexity for many problem settings.
While focusing on the classic optimization problems Subset Sum and Knapsack, we demonstrated via numerous examples that our approach is viable across a wide range of problem domains including graphs, string, geometry, and databases. We also utilized the insights of practical integer programming solvers to obtain highly-efficient implementations for curve similarity.
We successfully applied this approach to open problems on exact, parameterized, and approximation algorithms. We demonstrated that:
- Despite decades of prior research, significant algorithmic improvements are still possible,
- Additive combinatorics is a powerful toolkit for number-theoretic algorithmic problems, and
- Combining modern algorithm design with fine-grained complexity lower bounds can determine the optimal time complexity for many problem settings.
While focusing on the classic optimization problems Subset Sum and Knapsack, we demonstrated via numerous examples that our approach is viable across a wide range of problem domains including graphs, string, geometry, and databases. We also utilized the insights of practical integer programming solvers to obtain highly-efficient implementations for curve similarity.
In this project we developed a new algorithm design approach and primarily applied it to the classic optimization problems Subset Sum and Knapsack. Our main results for Subset Sum are as follows:
- Listing Solutions: We designed new algorithms for listing all Subset Sum solutions, improving upon Bellman's classic algorithm. [Bringmann, Nakos STOC'20, Bringmann, Fischer, Nakos SODA'25]
- Approximation: We determined the optimal running time for approximating Subset Sum [Bringmann, Nakos SODA'21] and designed improved approximation algorithms for the closely related problem Subset Sum Ratio [Bringmann SODA'24].
- Near-Linear Time: We classified the settings in which Subset Sum can be solved in near-linear time. [Bringmann, Wellnitz SODA'21]
- Space Complexity: We improved the space complexity of Schroeppel and Shamir's classic Subset Sum algorithm. [Nederlof, Wegrzycki STOC'21]
- Sparse Convolution: We designed improved algorithms for Sparse Convolution, a frequent subroutine of modern Subset Sum algorithms. [Bringmann, Fischer, Nakos STOC'21, Bringmann, Nakos ICALP'21, Bringmann, Fischer, Nakos SODA'22]
For Knapsack our main results are:
- New Tradeoffs: We designed various new algorithms with improved running times that achieve different tradeoffs in terms of several natural input parameters. [Polak, Rohwedder, Węgrzycki ICALP'21, Bringmann, Cassis ICALP'22, Bringmann, Cassis ESA'23, Bringmann, Dürr, Polak ESA'24, Bringmann STOC'24]
- Optimality: We achieved a landmark result with an algorithm running in near-quadratic time in terms of the largest item weight. This matches a fine-grained lower bound, making it an optimal algorithm. [Bringmann STOC'24]
Beyond optimization, we demonstrated the versatility of our algorithm design approach on exemplary problems from several other problem domains:
- Graphs: We proved new fine-grained complexity lower bounds for distance oracles [Abboud, Bringmann, Khoury, Zamir STOC'22, Abboud, Bringmann, Fischer STOC'23] and subgraph finding [Bringmann, Gorbachev STOC'25]. We designed a faster algorithm for shortest paths in graphs with negative edge weights [Bringmann, Cassis, Fischer FOCS'23].
- Strings: We designed improved approximation algorithms for the Edit Distance problem. [Bringmann, Cassis, Fischer, Nakos STOC'22, Bringmann, Cassis, Fischer, Nakos ICALP'22, Bringmann, Cassis, Fischer, Kociumaka SODA'24]
- Geometry: We showed fine-grained complexity lower bounds for computing the translation-invariant Hausdorff distance [Bringmann, Nusser SoCG'21], the translation-invariant earth mover's distance [Bringmann, Staals, Węgrzycki, van Wordragen SoCG'24], and the diameter of geometric intersection graphs [Bringmann, Kisfaludi-Bak, Künnemann, Nusser, Parsaeian SoCG'22].
- Databases: We determined the optimal time complexity for direct access on join queries. [Bringmann, Carmeli, Mengel PODS'22]
On the implementation side of the project we have designed and implemented practical algorithms for translation-invariant curve similarity measures. [Bringmann, Künnemann, Nusser ESA'20]
- Listing Solutions: We designed new algorithms for listing all Subset Sum solutions, improving upon Bellman's classic algorithm. [Bringmann, Nakos STOC'20, Bringmann, Fischer, Nakos SODA'25]
- Approximation: We determined the optimal running time for approximating Subset Sum [Bringmann, Nakos SODA'21] and designed improved approximation algorithms for the closely related problem Subset Sum Ratio [Bringmann SODA'24].
- Near-Linear Time: We classified the settings in which Subset Sum can be solved in near-linear time. [Bringmann, Wellnitz SODA'21]
- Space Complexity: We improved the space complexity of Schroeppel and Shamir's classic Subset Sum algorithm. [Nederlof, Wegrzycki STOC'21]
- Sparse Convolution: We designed improved algorithms for Sparse Convolution, a frequent subroutine of modern Subset Sum algorithms. [Bringmann, Fischer, Nakos STOC'21, Bringmann, Nakos ICALP'21, Bringmann, Fischer, Nakos SODA'22]
For Knapsack our main results are:
- New Tradeoffs: We designed various new algorithms with improved running times that achieve different tradeoffs in terms of several natural input parameters. [Polak, Rohwedder, Węgrzycki ICALP'21, Bringmann, Cassis ICALP'22, Bringmann, Cassis ESA'23, Bringmann, Dürr, Polak ESA'24, Bringmann STOC'24]
- Optimality: We achieved a landmark result with an algorithm running in near-quadratic time in terms of the largest item weight. This matches a fine-grained lower bound, making it an optimal algorithm. [Bringmann STOC'24]
Beyond optimization, we demonstrated the versatility of our algorithm design approach on exemplary problems from several other problem domains:
- Graphs: We proved new fine-grained complexity lower bounds for distance oracles [Abboud, Bringmann, Khoury, Zamir STOC'22, Abboud, Bringmann, Fischer STOC'23] and subgraph finding [Bringmann, Gorbachev STOC'25]. We designed a faster algorithm for shortest paths in graphs with negative edge weights [Bringmann, Cassis, Fischer FOCS'23].
- Strings: We designed improved approximation algorithms for the Edit Distance problem. [Bringmann, Cassis, Fischer, Nakos STOC'22, Bringmann, Cassis, Fischer, Nakos ICALP'22, Bringmann, Cassis, Fischer, Kociumaka SODA'24]
- Geometry: We showed fine-grained complexity lower bounds for computing the translation-invariant Hausdorff distance [Bringmann, Nusser SoCG'21], the translation-invariant earth mover's distance [Bringmann, Staals, Węgrzycki, van Wordragen SoCG'24], and the diameter of geometric intersection graphs [Bringmann, Kisfaludi-Bak, Künnemann, Nusser, Parsaeian SoCG'22].
- Databases: We determined the optimal time complexity for direct access on join queries. [Bringmann, Carmeli, Mengel PODS'22]
On the implementation side of the project we have designed and implemented practical algorithms for translation-invariant curve similarity measures. [Bringmann, Künnemann, Nusser ESA'20]