Complexity classification of balanced slice design

Determine whether the balanced slice-design optimization problem is weakly or strongly NP-hard beyond the established weak NP-completeness of its restricted two-slice positive-integer decision version.

Background

The paper reduces Partition to the balanced slice-design problem and proves weak NP-completeness for two slices with positive integer profiles satisfying a_k=b_k. A pseudo-polynomial dynamic program exists for that restricted case, but the broader complexity classification of the unrestricted minimization problem is not resolved.

References

The reduction establishes NP-hardness of the unrestricted minimization problem but leaves its classification as weakly or strongly NP-hard open.

— Contraction-Gauge Preconditioning for Quantized Matrix Multiplication  (2607.18745 - Sao et al., 21 Jul 2026) in Appendix, proof of Proposition 12.1 (Hardness of balanced slice design)