Constant-factor approximation for restricted assignment maximum flow-time

Develop a constant-factor approximation algorithm for maximum flow-time minimization in the general restricted assignment setting where each job j has a machine-dependent processing time d_{ij} on machine i and can be scheduled only on a specified subset M_j ⊆ [m].

Background

The general restricted assignment formulation (jobs with machine-dependent processing times d_{ij} and eligibility sets M_j) is a central scheduling problem. Bansal, Rohwedder, and Svensson proved that a natural LP relaxation has integrality gap O(√log n) and conjectured the existence of an O(1)-approximation.

This paper confirms a constant-factor (3 − 1/(m−1)) approximation for the special case with machine closing times, but the general restricted assignment case remains open.

References

Bansal, Rohwedder and Svensson proved that a natural linear programming relaxation has integrality gap O(\sqrt{\log n}), and conjectured that an O(1)-approximation should be possible.

— Weighted Chairman Assignment and Flow-Time Scheduling (2511.18546 - Liu et al., 23 Nov 2025) in Introduction