The bipartite margin shop and maximum red matchings free of blue–red alternating cycles

Abstract

The margin shop arises as a model of margining investment portfolios in a batch, a mandatory end-of-day risk management operation for any prime brokerage firm. The margin-shop scheduling problem is the extension of the preemptive flow-shop scheduling problem where precedence constraints can be introduced between preempted parts of jobs. This paper is devoted to the bipartite case which is equivalent to the problem of finding a maximum red matching that is free of blue–red alternating cycles in a complete bipartite graph with blue and red edges. It is also equivalent to the version of the jump-number problem for bipartite posets where jumps inside only one part should be counted. We show that the unit-time bipartite margin-shop scheduling problem is NP-hard but can be solved in polynomial time if the precedence graph is of degree at most two or a forest.