Is Constraint Satisfaction Over Two Variables Always Easy?

By the breakthrough work of Hastad, several constraint satisfaction problems are now known to have the following approximation resistance property: satisfying more clauses than what picking a random assignment would achieve is NP-hard. This is the case for example for Max E3-Sat, Max E3-Lin and Max E4-Set Splitting. A notable exception to this extreme hardness is constraint satisfaction over two variables (2-CSP); as a corollary of the celebrated Goemans-Williamson algorithm, we know that every Boolean 2-CSP has a non-trivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables. An intriguing question then is whether this is also the case for 2-CSPs over larger, non-Boolean domains. This question is still open, and is equivalent to whether the generalization of Max 2-SAT to domains of size d, can be approximated to a factor better than (1 - x1/d 2).

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