Favorite Theorems: Dichotomy

April Edition

A constraint satisfaction problem has a group of constraints applied to a set of variables and we want to know if there is a setting of the variables that make all the constraints true. In CNF-Satisfiability the variables are Boolean and the constraints are ORs of variables and their negations. In graph coloring, the variables are the colors of the nodes and the constraints, corresponding to edges, are two variables must be different. These problems lie in NP, just guess the values of the variables and check the constraints. They are often NP-complete. They are sometimes in P, like 2-coloring graphs. But they are never in between--all such problems are either in P or NP-complete.

A Dichotomy Theorem for Nonuniform CSPs by Andrei Bulatov

A Proof of the CSP Dichotomy Conjecture by Dmitriy Zhuk

Ladner's Theorem states that if P \(\neq\) NP then there exists a set in NP that is not in P and not NP-complete. Ladner's proof works by blowing holes in Satisfiability, an unsatisfying construction as it gives us a set that is NP-complete on some input lengths and easy on others. One could hope that some version of a constraint satisfaction problem could lead to a more natural intermediate set but dichotomy theorems tell us we need to look elsewhere.

In 1978, Thomas Schaefer gave a dichotomy theorem for satisfiability problems, basically CSP problems over Boolean variables. In 1990, Pavol Hell and Jaroslav Nešetřil showed a dichotomy result for homomorphisms of undirected graphs as described in my 2017 blog post. In 1998 Tomás Feder and Moshe Vardi formalized the constraint satisfaction dichotomy conjecture and expressed it as homomorphisms of directed graphs. The blog post described a claimed but later retracted solution to the dichotomy conjecture. Bulatov and Zhuk announced independent and different correct proofs later that year. In 2020 Zhuk received the Presburger Award for his paper (Bulatov was too senior for the award).