On the relative strength of different generalizations of split cuts

Split cuts are among the most important and well-understood cuts for general mixed-integer programs. In this paper we consider some recent generalizations of split cuts and compare their relative strength. More precisely, we compare the elementary closures of split, cross, crooked cross and general multi-branch split cuts as well as cuts obtained from multi-row and basic relaxations. We present a complete containment relationship between the closures of split, rank-2 split, cross, crooked cross and general multi-branch split cuts. More specifically, we show that 3-branch split cuts strictly dominate crooked cross cuts, which in turn strictly dominate cross cuts. We also show that multi-branch split cuts are incomparable to rank-2 split cuts. In addition, we also show that cross cuts, and hence crooked cross cuts, cannot always be obtained from 2-row relaxations or from basic relaxations. Together, these results settle some open questions raised in earlier papers.