The quadrangle inequality (QI) is a condition on cost functions. For indices : Picture four points on a line.
The left side adds costs of two overlapping intervals and that share . The right side adds the full span plus the overlap . QI says: overlapping intervals cost no more than span-plus-overlap. When cost satisfies QI, optimal splits become monotonic. Without QI, you'd search all possibilities. With QI, you can prune.