Any results on binary boolean CSP beyond the fixed-parameter tractability of almost 2SAT problem? Let $\varphi$ be a 2CNF formula and $k$ a nonnegative integer. It is proved in this paper that the problem of deciding whether one can delete at most $k$ clauses to make $\varphi$ satisfable, is fixed-parameter tractable, where $k$ is the parameter. My question is whether there are some work that generalize this result to other binary boolean CSP? (That is, to decide whether one can delete at most $k$ constraints to make some CSP instance satisfiable, parameterized by $k$) Or any negative results?
To my knowledge classifying this CSP variant is wide open. You can express a few fixed-parameter tractable problems in the setting (e.g. d-Hitting Set is roughly the case where you have positive clauses of size at most d plus negative assignments; roughly means that the CSP problem is slightly more general but easily reduces back to d-HS, or at least weighted d-HS). Even for constraints that you can implement via existentially quantified 2-CNF formulas it is open what the complexity is. The problem is that when implementing constraints in this way, while they are 2-CNF, you pay only one to delete the whole thing. Hence even simple constraints which are just the conjunction of two others can be hard (I may have example + reference later).