Periodic Reporting for period 4 - CGinsideNP (Complexity Inside NP - A Computational Geometry Perspective)
Reporting period: 2022-08-01 to 2023-05-31
In this project, we considered problems from computational geometry, the subfield of theoretical computer science that develops algorithms for tasks that deal with geometric data, such as points, lines, triangles, etc. Such tasks include, for example, comparing shapes for similarity, finding shortest routes in geometric networks, analyzing high-dimensional data, or visualizing large geometric networks. In the past 50 years, the field has been highly successful in developing new methods that are widely used in science and industry. However, it turns out that not for every problem that we would like to solve algorithmically, a satisfactory solution has been found. Sometimes, the computational resources required to find a solution are prohibitively large, sometimes we cannot be sure that the answer that we have found is indeed the best possible, and sometimes we do not even know if a solution exists.
The goal of the project was improve our understanding of why there are some algorithmic problems that stubbornly resist a satisfactory resolution. Is the reason just our general ignorance, or can we find deeper, more serious reasons for this phenomenon? The area of theoretical computer science that deals with these questions is called computational complexity theory, and we wanted to bring the methods from this area to bear in the realm of computational geometry. We pursued a somewhat non-traditional approach and considered problems that lie outside the realm of classic complexity theory. In particular, we focused on search problems, where we would actually like to find a solution whose existence is guaranteed, instead of on decision problems, where we must determine whether a solution exists. The field of geometric is full of problems of this flavor, and thus it provides a promising area where we can look for new insights. For some interesting problems, however, it is not known if a desired solution always exists, and we also considered such problems with the goal of extending our fundamental understanding of them.
We managed to make progress on all fronts: we developed new efficient algorithms for classic geometric problems, showing that faster, more satisfactory solutions actually exist. We also found new ways of classifying the difficulty of geometric problems that may lead to a deeper understanding of what makes a problem hard. Finally, we also found new answers for long-standing mathematical questions on certain geometric configurations, showing that sometimes the desired solution always exists (where this was not expected), and sometimes the desired solution does not exist (despite the long-held belief).
Our research is of a foundational nature. It contributes to the understanding of the phenomenon of computation and of the limits of what computers can do. However, there is also a more immediate impact: the new methods that we develop to show that efficient solutions actually exist can lead to new software and to the ability to solve larger computational tasks with our available resources. On the other hand, our impossibility results help focus the development efforts by showing what is feasible and what is not. As such, our project constitutes not only a contribution to the foundational and mathematical aspects of computer science, but also to broadening the algorithmic toolkit and to guiding the efforts for the development of practical software solutions.
The goal of the project was improve our understanding of why there are some algorithmic problems that stubbornly resist a satisfactory resolution. Is the reason just our general ignorance, or can we find deeper, more serious reasons for this phenomenon? The area of theoretical computer science that deals with these questions is called computational complexity theory, and we wanted to bring the methods from this area to bear in the realm of computational geometry. We pursued a somewhat non-traditional approach and considered problems that lie outside the realm of classic complexity theory. In particular, we focused on search problems, where we would actually like to find a solution whose existence is guaranteed, instead of on decision problems, where we must determine whether a solution exists. The field of geometric is full of problems of this flavor, and thus it provides a promising area where we can look for new insights. For some interesting problems, however, it is not known if a desired solution always exists, and we also considered such problems with the goal of extending our fundamental understanding of them.
We managed to make progress on all fronts: we developed new efficient algorithms for classic geometric problems, showing that faster, more satisfactory solutions actually exist. We also found new ways of classifying the difficulty of geometric problems that may lead to a deeper understanding of what makes a problem hard. Finally, we also found new answers for long-standing mathematical questions on certain geometric configurations, showing that sometimes the desired solution always exists (where this was not expected), and sometimes the desired solution does not exist (despite the long-held belief).
Our research is of a foundational nature. It contributes to the understanding of the phenomenon of computation and of the limits of what computers can do. However, there is also a more immediate impact: the new methods that we develop to show that efficient solutions actually exist can lead to new software and to the ability to solve larger computational tasks with our available resources. On the other hand, our impossibility results help focus the development efforts by showing what is feasible and what is not. As such, our project constitutes not only a contribution to the foundational and mathematical aspects of computer science, but also to broadening the algorithmic toolkit and to guiding the efforts for the development of practical software solutions.
Since the project concerns foundational research in a theoretical field, the main conduits of dissemination of our results lie in the usual venues of scientific communication: we attended scientific conferences where we presented our results by giving talks and contributing to the published proceedings. We published our results in peer-reviewed scientific journals that are relevant for the field, and we performed scientific visits, where we interacted with our colleagues about the topics of the project. In fact, our project work resulted in more than 50 scientific publications, and we are also currently working on a book that presents some of our ideas. Throughout, we made sure to abide by the principle of open-access publishing, making sure that the latest version of our results was always available in a public repository without any access restrictions.
Concerning the fundamental complexity theoretic aspects of geometric problems, we considered an interesting new complexity class that was recently discovered, UEoPL, and we managed to identify new problems that lie in it and that may shed more insight into the nature of this class. We also managed to show that several simple questions about high-dimensional point sets are actually hard to solve efficiently, showing that we need to make compromises if we want fast solutions for them.
In the other direction, we also developed new efficient algorithms for fundamental geometric problems, showing that better methods actually exist. Two concrete results are 1) a faster algorithm for maintaining the connectivity structure in a network of wireless sensors as new sensors come online and old sensors fail, a widely studied problem that turns out to be quite challenging; and 2) a new algorithm for finding an approximate "Tverberg partition" in high dimensions, a task that is important for analyzing high-dimensional data sets.
More fundamentally, we studied basic mathematical questions for the underlying geometric objects. Most interestingly, we managed to show that 1) large bichromatic separated matchings in convex point sets always exist, answering a question that arose in many diverse areas, from combinatorics to computational biology to the theory of formal languages; and 2) that is not always possible to partition the edges of a large drawing with many intersections into "few" well-behaved pieces that are all without intersections, a task that often needs to be accomplished when visualizing large geometric networks. Here, the negative solution comes quite as a surprise, and it shows that additonal methods
We also obtained results on many related questions, leading to a deeper understanding of the many interrelated problems in geometry and its adjacent fields. We also found many promising new directions for research that we would like to pursue in the future.
Concerning the fundamental complexity theoretic aspects of geometric problems, we considered an interesting new complexity class that was recently discovered, UEoPL, and we managed to identify new problems that lie in it and that may shed more insight into the nature of this class. We also managed to show that several simple questions about high-dimensional point sets are actually hard to solve efficiently, showing that we need to make compromises if we want fast solutions for them.
In the other direction, we also developed new efficient algorithms for fundamental geometric problems, showing that better methods actually exist. Two concrete results are 1) a faster algorithm for maintaining the connectivity structure in a network of wireless sensors as new sensors come online and old sensors fail, a widely studied problem that turns out to be quite challenging; and 2) a new algorithm for finding an approximate "Tverberg partition" in high dimensions, a task that is important for analyzing high-dimensional data sets.
More fundamentally, we studied basic mathematical questions for the underlying geometric objects. Most interestingly, we managed to show that 1) large bichromatic separated matchings in convex point sets always exist, answering a question that arose in many diverse areas, from combinatorics to computational biology to the theory of formal languages; and 2) that is not always possible to partition the edges of a large drawing with many intersections into "few" well-behaved pieces that are all without intersections, a task that often needs to be accomplished when visualizing large geometric networks. Here, the negative solution comes quite as a surprise, and it shows that additonal methods
We also obtained results on many related questions, leading to a deeper understanding of the many interrelated problems in geometry and its adjacent fields. We also found many promising new directions for research that we would like to pursue in the future.
As explained above, we believe that throughout the project, we managed to make significant progress beyond the state of the art. Nonetheless, there are still many new questions to be explored. For example: What can we say about the new complexity class UEoPL? Can we identify more interesting difficult problems for it? What about the foundational mathematical results that we have obtained? Can they be improved? What about dynamic disk graphs? Can we get even faster performance? We believe that all these questions deserve detailed further study, and we are looking forward to investigating them in a new project.