Abstract:
By the breakthrough work of Håstad, 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 - 1/d^2). In an attempt to make progress towards this question, in this paper we prove, firstly, that a slight restriction of this problem, namely a generalization of linear inequations with two variables per constraint, is not approximation resistant, and, secondly, that the Not-All-Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d >= 3. In the Boolean case, Not-All-Equal Sat with three variables per constraint is equivalent to Max 2-SAT and thus has a non-trivial approximation algorithm; for larger domain sizes, Max 2-SAT can be reduced to Not-All-Equal Sat with three variables per constraint. Our approximation algorithm implies that a wide class of 2-CSPs called regular 2-CSPs can all be approximated beyond their random assignment threshold.