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]
