Periodic Reporting for period 1 - POCOCOP (Polynomial-time Computation: Opening the Blackboxes in Constraint Problems)
Période du rapport: 2023-03-01 au 2024-08-31
The overall goal of POCOCOP is the systematic understanding of polynomial-time computation: ways to solve computational problems so that the number of steps required in the worst case before obtaining a solution grows only polynomially with the size of instances of the problem (avoiding, in particular, exponential explosion of the time needed). The main motivation for this goal is that the class of polynomial-time computable problems is widely considered the most important and robust class for the study of efficient computation, and computation is ubiquitous in the world.
Our main focus lies on Constraint Satisfaction Problems, that is, computational problems where values have to be assigned to given variables subject to constraints: for example, solving systems of equations or sudokus. For such problems, we strive for the ideal trinity: a uniform notion of reduction allowing us to compare their complexity; a uniform description of those problems that can be solved in polynomial time (and hence, also of those problems for which this is not the case); and a uniform polynomial-time algorithm solving them.
A significant unification has previously been achieved for “basic” Constraint Satisfaction Problems that capture exact solvability of problems that allow only finite sets of values for the variables. POCOCOP concentrates on extensions of these problems in three dimensions: from finite sets of values towards infinite ones; from finding exact solutions to approximating solutions; and from finding just any solution to finding an optimal one.
Our approach towards achieving this goal is the opening of three “blackboxes” containing Zhuk’s algorithm, Linear Programming, and the PCP theorem. Each of these boxes contains a powerful technology (an algorithm in the former two cases, and a hardness tool in the latter), which we take from the box, improve, and apply outside its traditional scope.
Our main focus lies on Constraint Satisfaction Problems, that is, computational problems where values have to be assigned to given variables subject to constraints: for example, solving systems of equations or sudokus. For such problems, we strive for the ideal trinity: a uniform notion of reduction allowing us to compare their complexity; a uniform description of those problems that can be solved in polynomial time (and hence, also of those problems for which this is not the case); and a uniform polynomial-time algorithm solving them.
A significant unification has previously been achieved for “basic” Constraint Satisfaction Problems that capture exact solvability of problems that allow only finite sets of values for the variables. POCOCOP concentrates on extensions of these problems in three dimensions: from finite sets of values towards infinite ones; from finding exact solutions to approximating solutions; and from finding just any solution to finding an optimal one.
Our approach towards achieving this goal is the opening of three “blackboxes” containing Zhuk’s algorithm, Linear Programming, and the PCP theorem. Each of these boxes contains a powerful technology (an algorithm in the former two cases, and a hardness tool in the latter), which we take from the box, improve, and apply outside its traditional scope.
We created a team of high-potential young researchers from a wide array of backgrounds including theoretical computer science, universal algebra, and model theory. Additionally, we recruited Dmitriy Zhuk, who is one of the very leading scientists worldwide in the field and the bearer of the wisdom of one of the three blackboxes. We distributed the team members among the three sites so that all directions are present at each of them. This way, the knowledge required for the three dimensions of extension of Constraint Satisfaction Problems is available at all three sites, and the exchange of knowledge between the sites is facilitated. For the latter, we established several online learning seminars as well as one POCOCOP meeting of the entire team per semester. This was complemented by numerous scientific visits of smaller task forces between the sites. A special highlight was the additional organization of the CSP World Congress in 2023 and 2024 which included the entire team plus more than thirty leading experts in the field.
A significant part of our research was devoted to the theory of exact finite-domain Constraint Satisfaction Problems since it is the starting point of POCOCOP. Highlights of our achievements in this direction include a simplified and improved proof of the most important result in the area, a classification of polynomial-time solvable problems in this class. The content of Zhuk’s blackbox thus got simpler and potentially better applicable beyond finite domains.
The second main direction of research was the theory of infinite-domain Constraint Satisfaction Problems. Here POCOCOP contributed to a technique, called smooth approximations, which unifies previous complexity classifications and allows one to tackle classes of problems that were out of reach before. Our result for one such class (so-called Hypergraph-SAT problems) also opens, for the first time in this context, the Zhuk blackbox in order to obtain a polynomial-time algorithm.
The third main direction of research was towards a unified theory for the three extensions of the exact finite domain framework. The main achievement in this direction is the foundation of a unified framework and a unified theory of reductions that simultaneously incorporates extensions towards approximation (so-called Promise CSPs) and optimization (so-called Valued CSPs). In particular, the hardness result forming the PCP blackbox has been tackled within our framework.
Our results and further information are available on our website https://pococop.eu(s’ouvre dans une nouvelle fenêtre).
A significant part of our research was devoted to the theory of exact finite-domain Constraint Satisfaction Problems since it is the starting point of POCOCOP. Highlights of our achievements in this direction include a simplified and improved proof of the most important result in the area, a classification of polynomial-time solvable problems in this class. The content of Zhuk’s blackbox thus got simpler and potentially better applicable beyond finite domains.
The second main direction of research was the theory of infinite-domain Constraint Satisfaction Problems. Here POCOCOP contributed to a technique, called smooth approximations, which unifies previous complexity classifications and allows one to tackle classes of problems that were out of reach before. Our result for one such class (so-called Hypergraph-SAT problems) also opens, for the first time in this context, the Zhuk blackbox in order to obtain a polynomial-time algorithm.
The third main direction of research was towards a unified theory for the three extensions of the exact finite domain framework. The main achievement in this direction is the foundation of a unified framework and a unified theory of reductions that simultaneously incorporates extensions towards approximation (so-called Promise CSPs) and optimization (so-called Valued CSPs). In particular, the hardness result forming the PCP blackbox has been tackled within our framework.
Our results and further information are available on our website https://pococop.eu(s’ouvre dans une nouvelle fenêtre).
Our results go beyond the state of the art by introducing and applying novel concepts (such as plurimorphisms for Valued Promise CSPs), methods (smooth approximations), approaches (resilience problems in Database Theory via VCSPs), and research directions (approximation via algebra).
The results bring us closer to a unified theory of computation for a large class of problems. The potential broader impact of such a theory is a better understanding of which problems can be efficiently solved and how, and for which computational problems we cannot hope for their solution, no matter how fast computers will get in the future. Another impact is to overcome the ad hoc concepts that are currently dominating theoretical computer science, and changing the way the field is perceived and taught.
The results obtained so far suggest several exciting research directions. One is to improve the newly emerged framework unifying optimization and approximation, ideally fully opening the PCP blackbox. A unified theory of reductions that would also incorporate infinite domains is so far elusive; however, we discovered a new potential approach via category theory and will pursue this idea further. Finally, some progress has been made on a unified algorithm, but it is far from satisfactory. A recent negative result suggests that it cannot be solely a combination of classical numerical algorithms (such as Linear Programming) even for finite-domain Constraint Satisfaction Problems, and we will explore new candidates.
The results bring us closer to a unified theory of computation for a large class of problems. The potential broader impact of such a theory is a better understanding of which problems can be efficiently solved and how, and for which computational problems we cannot hope for their solution, no matter how fast computers will get in the future. Another impact is to overcome the ad hoc concepts that are currently dominating theoretical computer science, and changing the way the field is perceived and taught.
The results obtained so far suggest several exciting research directions. One is to improve the newly emerged framework unifying optimization and approximation, ideally fully opening the PCP blackbox. A unified theory of reductions that would also incorporate infinite domains is so far elusive; however, we discovered a new potential approach via category theory and will pursue this idea further. Finally, some progress has been made on a unified algorithm, but it is far from satisfactory. A recent negative result suggests that it cannot be solely a combination of classical numerical algorithms (such as Linear Programming) even for finite-domain Constraint Satisfaction Problems, and we will explore new candidates.