The purpose of this paper is to outline various results regarding the computational complexity and the algorithms of nonmonotonic entailment in di erent coherence-based approaches. Starting from a (non necessarily consistent) belief base E and a pre-order on E, we rst remind di erent mechanisms for selecting preferred consistent subsets. Then we present di erent entailment principles in order to manage these multiple subsets. The crossing point of each generation mechanism m and each entailment principle p de nes an entailment relation (E;) j p;m which we study from the computational complexity point of view. The results are not very encouraging since the complexity of all these nonmonotonic entailment relations is, in most restricted languages, larger than the complexity of monotonic entail-ment. So, we decided to extend Binary Decision Diagrams techniques, which are well suited to the task of solving NP-hard logic-based problems. Both theoretical and experimental results are described along this line in the last sections. Topic area: Nonmonotonic reasoning, computational complexity, algorithms and binary decision diagram.