A new line of attack on the dichotomy conjecture

• Authors: Gábor Kun;Mario Szegedy
• Venue: Proceedings of the forty-first annual ACM symposium on Theory of computing
• Year: 2009

The well known dichotomy conjecture of Feder and Vardi states that for every finite family Γ of constraints CSP(Γ) is either polynomially solvable or NP-hard. Bulatov and Jeavons reformulated this conjecture in terms of the properties of the algebra Pol(Γ), where the latter is the collection of those n-ary operations (n= 1,2,...) that keep all constraints in Γ invariant. We show that the algebraic condition boils down to whether there are arbitrarily resilient functions in Pol(Γ). Using this characterization and a result of Dinur, Friedgut and Regev, we give an entirely new and transparent proof to the Hell-Nesetril theorem, which states that for a simple, connected and undirected graph H, the problem CSP(H) is NP-hard if and only if H is non-bipartite. We also introduce another notion of resilience (we call it strong resilience), and we use it to characterize CSP problems that 'do not have the ability to count.' Very recently this class has been shown to be equivalent with the the class of bounded width problems, i.e. the class of CSPs that be described by existential k-pebble games. What emerges from our research, is that certain important algebraic conditions that are usually expressed via identities have equivalent definitions that rely on asymptotic properties of term operations. Our new notions have a potential to show hardness of CSPs (as demonstrated on the Hell-Nesetril theorem), or to prove their tractability.