The connection between the bump number problem and flow-shop scheduling with precedence constraints

Abstract

The bump number b( P) of a partial order P is the minimum number of comparable, consecutive pairs of elements in a linear extension of P. We show the strong connection between the bump number problem and flow-shop scheduling problems: bumps in a permutation schedule “bump” the schedule by a fixed amount of time, and finding b( P) is equivalent to a 2-machine flow-shop problem with precedence constraints. We also show that if jobs have equal processing times on each machine then there is an optimal permutation schedule for the flow-shop with precedence constraints.