Approximability Status of Non-Preemptive Throughput Maximization

Determine whether the (Non-Preemptive) Throughput Maximization problem on a single machine is APX-hard, or develop a polynomial-time approximation scheme (PTAS) for this problem, thereby clarifying its approximability status.

Background

The (Non-Preemptive) Throughput Maximization problem on a single machine asks for the maximum number of jobs that can be scheduled non-preemptively within their time windows. The problem is strongly NP-hard, and prior to this work the best known polynomial-time approximation ratio was approximately 1.551.

The paper improves the approximation ratio to 4/3+ in polynomial time and to 5/4+ in pseudo-polynomial time, yet the fundamental complexity-theoretic and approximability classification remains unsettled. In particular, the authors point out that APX-hardness is not established for this problem, leaving open whether a PTAS may exist.

References

However, Throughput Maximization is not known to be APX-hard and, hence, it might still admit a PTAS! We find intriguing that the approximability status of such a basic problem is still rather unclear.

— Improved Approximation Algorithms for Non-Preemptive Throughput Maximization  (2603.29451 - Armbruster et al., 31 Mar 2026) in Introduction (Section 1)