## Periodic Reporting for period 4 - CSP-Infinity (Homogeneous Structures, Constraint Satisfaction Problems, and Topological Clones)

Reporting period: 2021-04-01 to 2021-09-30

The project investigates the theoretical boundaries of efficient computation. Which computational problems can be solved with a reasonable amount of computational resources, and for which computational problems is this hopeless? We primarily studied so-called constraint satisfaction problems, which is a very large class of problems that appear for example in scheduling, artificial intelligence, temporal and spatial reasoning, computational linguistics, phylogenetic reconstruction, and optimisation. The project explores the power of existing algorithmic paradigms (such as constraint propagation or least fixed point computation) but also developed powerful theoretical results about computational hardness. A major aspect of the project are complete classifications for large classes of problems; these classifications indicate which problems are easy and which are hard to solve. These complexity classifications lead to the discovery of completely new and unexpected polynomial-time algorithms. Such algorithms and more generally insights into the nature of polynomial-time computation has an impact in all of the application fields mentioned above.

The project work concentrated on the following two directions:

Constraint satisfaction problems over `numeric domains’ such as the integers or the rationals, and `qualitative’ constraint satisfaction problems that resemble the well-studied constraint satisfaction problems over finite domains. In both of these directions, great progress has been made concerning the fundamental theory of such problems in general. Moreover, the general theory has been linked to concrete classes of computational problems

that are relevant in application areas.

One example of such a class is the logic MMSNP, a fragment of existential second-order logic that is studied in knowledge representation and in database theory. We verified the so-called `tractability conjecture’ for such problems using results from model theory, universal algebra, and Ramsey theory. On the algorithmic side, found a new polynomial-time reduction to CSPs over finite-domains so that the algorithms from the famous Bulatov-Zhuk dichotomy result from 2017 can be applied to prove polynomial-time tractability results.

Constraint satisfaction problems over `numeric domains’ such as the integers or the rationals, and `qualitative’ constraint satisfaction problems that resemble the well-studied constraint satisfaction problems over finite domains. In both of these directions, great progress has been made concerning the fundamental theory of such problems in general. Moreover, the general theory has been linked to concrete classes of computational problems

that are relevant in application areas.

One example of such a class is the logic MMSNP, a fragment of existential second-order logic that is studied in knowledge representation and in database theory. We verified the so-called `tractability conjecture’ for such problems using results from model theory, universal algebra, and Ramsey theory. On the algorithmic side, found a new polynomial-time reduction to CSPs over finite-domains so that the algorithms from the famous Bulatov-Zhuk dichotomy result from 2017 can be applied to prove polynomial-time tractability results.

Concerning numeric domains, we have applied the general theory to obtain a complete classification of the computational complexity for all constraint satisfaction problems that are definable over the order of the integers, covering CSPs that have been studied in scheduling. Moreover, we showed that Tropical CSPs lie in the intersection of NP and coNP.

A completely new field of research has been opened by the study of Valued CSPs on infinite domains, which allows to model many optimisation problems in a common framework. We developed a strategy to efficiently reduce certain infinite-domain Valued CSPs to finite-domain Valued CSPs, that can then be solved in polynomial time by linear programming relaxation techniques.

In the direction of qualitative CSPs, we successfully applied our universal-algebraic and Ramsey-theoretic techniques to network satisfaction problems for relation algebras, a field which has seen little interaction with the theoretical CSP community lately. We have also obtained a complete classification result for a large class of spatial reasoning CSPs.

A completely new field of research has been opened by the study of Valued CSPs on infinite domains, which allows to model many optimisation problems in a common framework. We developed a strategy to efficiently reduce certain infinite-domain Valued CSPs to finite-domain Valued CSPs, that can then be solved in polynomial time by linear programming relaxation techniques.

In the direction of qualitative CSPs, we successfully applied our universal-algebraic and Ramsey-theoretic techniques to network satisfaction problems for relation algebras, a field which has seen little interaction with the theoretical CSP community lately. We have also obtained a complete classification result for a large class of spatial reasoning CSPs.