Fixed-Parameter Tractability of almost CSP Problem with Decisive Relations

Let I be an instance of binary boolean CSP. Consider the problem of deciding whether one can remove at most k constraints of I such that the remaining constraints are satisfiable. We call it the Almost CSP problem. This problem is NP -complete and we study it from the point of view of parameterized complexity where k is the parameter. Two special cases have been studied: when the constraints are inequality relations (Guo et al., WADS 2005) and when the constraints are OR type relations (Razgon and O'Sullivan, ICALP 2008). Both cases are shown to be fixed-parameter tractable (FPT). In this paper, we define a class of decisive relations and show that when all the relations are in this class, the problem is also fixed-parameter tractable. Note that the inequality relation is decisive, thus our result generalizes the result of the parameterized edge-bipartization problem (Guo et al., WADS 2005). Moreover as a simple corollary, if the set of relations contains no OR type relations, then the problem remains fixed-parameter tractable. However, it is still open whether OR type relations and other relations can be combined together while the fixed-parameter tractability still holds.