A new line of attack on the dichotomy conjecture
European journal of combinatorics (Print)
The well known dichotomy conjecture of Feder and Vardi states that for every 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 P ol(Γ), where the latter is the collection of those m-ary operations (m = 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 P ol(Γ). Equivalently,
... we can express this in the terms of the PCP theory: CSP(Γ) is NP-hard iff all long code tests created from Γ that passes with zero error admits only juntas 1 . Then, using this characterization and a result of Dinur, Friedgut and Regev, we give an entirely new and transparent proof to the Hell-Nešetřil theorem, which states that for a simple connected 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 in the investigation of CSP problems that 'do not have the ability to count.' The complexity of this class is unknown. Several authors conjectured that CSP problems without the ability to count have bounded width, or equivalently, that they can be characterized by existential k-pebble games. The resolution of this conjecture would be a major step towards the resolution of the dichotomy conjecture. We show that CSP problems without the ability to count are exactly the ones with strongly resilient term operations, which might give a handier tool to attack the conjecture than the known algebraic characterizations.