Complexity of the constraint satisfaction problem and its variations

Many combinatorial problems, such as graph coloring and solving of linear equations, can be expressed as the constraint satisfaction problem for some constraint language. Some of these problems are solvable in polynomial time, while others are NP-complete. In 2017 the complexity of the constraint satisfaction problem was described for any constraint language on a finite set, but there are many other variants of this problem whose complexity is still not known. For instance, we could allow to use both universal and existential quantifiers, or require the solution to be surjective or balanced. Another variant is to require the input to satisfy an additional condition. As an example we could consider the problem of coloring a graph in 100 colors if we know that the graph is colorable in 3 colors. In the paper we discuss the usual constraint satisfaction as well as the complexity of these and other variants of the constraint satisfaction problem.