Real-world optimization problems pose major challenges to algorithmic research. For instance, (i) many important problems are believed to be intractable (i.e. NP-hard) and (ii) with the growth of data size, modern applications often require a decision making under incomplete and dynamically changing input data. After several decades of research, central problems in these domains have remained poorly understood (e.g. Is there an asymptotically most efficient binary search trees?) Existing algorithmic techniques either reach their limitation or are inherently tailored to special cases. This project attempts to untangle this gap in the state of the art and seeks new interplay across multiple areas of algorithms, such as approximation algorithms, online algorithms, fixed-parameter tractable (FPT) algorithms, exponential time algorithms, and data structures. We propose new directions from the structural perspectives that connect the aforementioned algorithmic problems to basic questions in combinatorics.
The following highlights have been made possible by this project. First, the area of parameterized approximation has kick-started strongly and become an active area in theoretical computer science. Our results are central to this progress (FOCS 2017/SICOMP 2020 & FOCS 2023). We organized a very successful Dagstuhl seminar on this topic in July 2023. Second, the project branched into an unplanned direction and led to major breakthroughs, giving fast algorithms for vertex connectivity problems (STOC 2019, SODA 2020, STOC 2021 & FOCS 2022). These are problems that were open for more than four decades and the publications from this project provided solutions to them in many ways (some of which led to a new FPT perspective in fast algorithms). Third, we made a substantial progress on a 6-decade-old open problem in rectangle graph packing (SODA 2021). Fourth, we strengthen connections between algorithms and combinatorics in various domains, including data structures, geometry, and graphs (SODA 2020, SODA 2021, SODA 2023 & SODA 2024). Our results give strong evidence that algorithmic techniques can be used to make progresses in combinatorics.
