Exact dynamic programming for tropical profile partitions

Determine whether an exact dynamic program can compute the g-block optimum for log-magnitude profiles admitting a max-plus approximation with at least two templates.

Background

The paper extends profile-clustering guarantees from rank-one profiles to approximate max-plus, or tropical, representations with multiple templates. Although approximate factorizations yield useful spread bounds, crossing templates may induce different row orderings across coordinates, preventing the straightforward interval dynamic program available in the rank-one case.

References

Because crossing templates can order rows differently across coordinate ranges, whether an exact dynamic program can compute the $g$-block optimum in this setting remains open.

— Contraction-Gauge Preconditioning for Quantized Matrix Multiplication  (2607.18745 - Sao et al., 21 Jul 2026) in Appendix, subsection “Low-dimensional and tropical profiles”