This paper addresses the problem of failing RDF queries. Query relaxation is one of the cooperative techniques that allows providing users with alternative answers instead of an empty result. While previous works on query relaxation over RDF data have focused on defining new relaxation operators, we investigate in this paper techniques to find the parts of an RDF query that are responsible of its failure. Finding such subqueries, named Minimal Failing Subqueries (MFSs), is of great interest to efficiently perform the relaxation process. We propose two algorithmic approaches for computing MFSs. The first approach (LBA) intelligently leverages the subquery lattice of the initial RDF query while the second approach (MBA) is based on a particular matrix that improves the performance of LBA. Our approaches also compute a particular kind of relaxed RDF queries, called Maximal Succeeding Subqueries (XSSs). XSSs are subqueries with a maximal number of triple patterns of the initial query. To validate our approaches, a set of thorough experiments is conducted on the LUBM benchmark and a comparative study with other approaches is done.