Computing the Jump Number on Semi-orders Is Polynomial

1) Introduction and notations In this first section we will give our main definitions and recall different characterizations of interval and semi-orders. In section 2 we shall prove that after a decomposition routine, semi-orders have at most 2 consecutive bumps in a linear extension. We also prove, using a "divide-and-conquer" argument, that computing polynomially the jump number can be done provided we can compute polynomially the jump number s2(P) for linear extensions with at most one consecutive bump. In section 3 we provide an algorithm to calculate the jump number.Finally in section 4 we provide a full example and discuss the complexity issues. Interval orders have been thoroughly studied since the early seventies for their importance in the context of measurement theory (see Fishburn [1970], Monjardet [1978] or Golumbic [1985]). They have come back into fashion lately for their possible applications to parallelism (Habib, Morvan & Rampon [1990]). Semiorders are a proper subclass of interval orders, restrictive but also with potential applications (Rabinovitch [1978]).