Constant integrality gap for a combined relaxation

Establish whether a relaxation for Bernoulli-job scheduling that combines expected-rank constraints for deadlines with expected-processing constraints on time intervals has a constant integrality gap.

Background

The paper considers a relaxation over start-time distributions. For every deadline, the relaxation can impose expected-rank constraints derived from a laminar matroid limiting the number of positive jobs of sufficiently large size that may start by that deadline. Constraints imposed independently at individual deadlines are insufficient: the paper gives a one-machine construction satisfying all such rank constraints and maintaining expected processing below the length of every interval, while achieving an objective far below the true optimum. A known one-machine inequality closes this gap, but its multi-machine analogue introduces a term proportional to the squared coefficient of variation, which is problematic for Bernoulli jobs. The unresolved issue is whether combining the rank and interval-processing constraints yields a relaxation with a constant integrality gap.