Periodic Reporting for period 4 - ALGOCom (Novel Algorithmic Techniques through the Lens of Combinatorics)
Reporting period: 2022-08-01 to 2024-01-31
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.
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.
The overarching goal of this project is to explore the transfer of techniques between combinatorics and algorithms. As benchmarks, we aim at making concrete progress on long-standing problems that arise specific domains (such as the dynamic optimality conjecture for binary search trees), as well as developing theoretical foundations for new research paradigms (such as parameterized approximation). These directions lead to the following results (that are more relevant to the project).
1) Parameterized Approximation.
We show that the Gap Exponential Time Hypothesis (Gap-ETH), postulating the inapproximability of 3SAT in subexponential time, would imply strongest possible parameterized inapproximability results. This leads to a research program that aims at replacing Gap-ETH with weaker, more standard complexity assumptions. More applications of Gap-ETH have been found over the years since our paper appeared in FOCS 2017 (the paper now received 124 citations according to Google Scholar). Given the significance of exponential time complexity, we explore exponential-time approximation results for Maximum Independent Set and show connections to parameters of Probabilistically Checkable Proofs (PCP). This was published in Algorithmica 2018.
Our FOCS 2023 paper presents a parameterized approximation algorithm that resolves more than 10 clustering problems (and unifying, simplifying many existing results).
A parameterization perspective has been used in extremal results about vertex sparsifiers (SODA 2021), which led to a breakthrough by another group of researchers on dynamic graph problems.
We also explored the connection between treewidth parameter and approximation algorithm systematically, leading to publications in TALG 2024, ESA 2023 and APPROX 2018.
2) Unplanned Breakthroughs on Vertex Connectivity
The PI decided to fund Sorrachai to explore synergies between this project and Danupon Nanongkai's ERC project. This led to several unexpected breakthroughs in resolving decades old problems around vertex connectivity -- measuring the network's resilience against node failures. These results have been published in STOC 2019, SODA 2020, STOC 2021, and FOCS 2022.
Both the STOC 2021 and FOCS 2022 papers rely on techniques that were developed from parameterization perspectives, thus providing strong evidence to the directions outlined in this project (unexpectedly). In particular, the former relies on kernelization idea and the latter on our SODA 2021 paper on parameterized graph sparsification.
3) Rectangle graph coloring
We made the first progress on a 6-decade-old problem about rectangle graph coloring (a long-standing extremal question about rectangle graphs). This relies on the perspectives from approximation algorithms (the LP rounding techniques), so it presents a showcase of how algorithmic techniques can be used in extremal combinatorics.
4) Combinatorics and algorithms.
Finally, we address the key component of the project -- the interplay between algorithms and combinatorics. We discover new connections between combinatorics and algorithms in many domains: (i) We use local search framework (a popolar approach in algorithms) to make progress on an extremal combinatorics question (SODA 2020, STACS 2019) and (ii) we formulated and studied a new extremal question arising in the problem of sorting (SODA 2024 & SODA 2023).
1) Parameterized Approximation.
We show that the Gap Exponential Time Hypothesis (Gap-ETH), postulating the inapproximability of 3SAT in subexponential time, would imply strongest possible parameterized inapproximability results. This leads to a research program that aims at replacing Gap-ETH with weaker, more standard complexity assumptions. More applications of Gap-ETH have been found over the years since our paper appeared in FOCS 2017 (the paper now received 124 citations according to Google Scholar). Given the significance of exponential time complexity, we explore exponential-time approximation results for Maximum Independent Set and show connections to parameters of Probabilistically Checkable Proofs (PCP). This was published in Algorithmica 2018.
Our FOCS 2023 paper presents a parameterized approximation algorithm that resolves more than 10 clustering problems (and unifying, simplifying many existing results).
A parameterization perspective has been used in extremal results about vertex sparsifiers (SODA 2021), which led to a breakthrough by another group of researchers on dynamic graph problems.
We also explored the connection between treewidth parameter and approximation algorithm systematically, leading to publications in TALG 2024, ESA 2023 and APPROX 2018.
2) Unplanned Breakthroughs on Vertex Connectivity
The PI decided to fund Sorrachai to explore synergies between this project and Danupon Nanongkai's ERC project. This led to several unexpected breakthroughs in resolving decades old problems around vertex connectivity -- measuring the network's resilience against node failures. These results have been published in STOC 2019, SODA 2020, STOC 2021, and FOCS 2022.
Both the STOC 2021 and FOCS 2022 papers rely on techniques that were developed from parameterization perspectives, thus providing strong evidence to the directions outlined in this project (unexpectedly). In particular, the former relies on kernelization idea and the latter on our SODA 2021 paper on parameterized graph sparsification.
3) Rectangle graph coloring
We made the first progress on a 6-decade-old problem about rectangle graph coloring (a long-standing extremal question about rectangle graphs). This relies on the perspectives from approximation algorithms (the LP rounding techniques), so it presents a showcase of how algorithmic techniques can be used in extremal combinatorics.
4) Combinatorics and algorithms.
Finally, we address the key component of the project -- the interplay between algorithms and combinatorics. We discover new connections between combinatorics and algorithms in many domains: (i) We use local search framework (a popolar approach in algorithms) to make progress on an extremal combinatorics question (SODA 2020, STACS 2019) and (ii) we formulated and studied a new extremal question arising in the problem of sorting (SODA 2024 & SODA 2023).
As outlined above, we obtained many results that have gone beyond the state of the art in many directions. The project has now reached its end, but we are still actively pursuing directions that arise in the project. We believe that we will make further substantial progresses, e.g. on the dynamic optimality conjecture and parameterized approximation. We will acknowledge the ERC if these ideas lead to new results.