Towards Bridging the Gap Between SAT and Max-SAT Refutations

Adapting a resolution proof for SAT to a Max-SAT resolution proof without increasing considerably the size of the proof is an open question. This paper contributes to this topic by exhibiting linear adaptations, in terms of the input SAT proof size, in restricted cases which are regular tree resolution refutations, tree resolution refutations and a new introduced class of refutations that we refer to as semi-tree resolution refutations. We also extend these results by proposing a complete adaptation for any unrestricted SAT refutation to a Max-SAT refutation, which is exponential in the worst case.