Truly subquadratic min-plus convolution

Determine whether general $(\min,+)$ convolution admits a truly subquadratic-time algorithm, namely an algorithm running in $O(n^{2-\varepsilon})$ time for some constant $\varepsilon>0$.

Background

The paper studies (min⁡,+)(\min,+) convolution, which is equivalent to multiplication of tropical polynomials. Although structured instances admit faster algorithms, the general problem remains unresolved, and the paper does not establish a truly subquadratic algorithm or an unconditional truly superlinear lower bound.

References

The core open question is whether it admits a truly subquadratic-time algorithm, that is, an algorithm running in $O(n{2-\varepsilon})$ time for some constant $\varepsilon>0$.

— On the Structure of $(\min,+)$ Convolution  (2608.13310 - Zhou, 13 Aug 2026) in Section 1, Introduction

Can the time complexities of Multiple-Sequence $(\min,+)$ Convolution and Multiple-Choice Knapsack be improved? Our algorithm does not use the powerful additive-combinatorial techniques that have led to recent improvements for $0$-$1$ Knapsack .

— On the Structure of $(\min,+)$ Convolution  (2608.13310 - Zhou, 13 Aug 2026) in Section titled “Open Problems”

The (\min,+)-Convolution Conjecture states that, given two integer sequences of length n with entries in [-W, W], no algorithm can compute their (\min, +)-convolution in O(n{2-\varepsilon}\polylog W) time for any >0.

— Computing All Optimal Partial $p$-Wasserstein Matchings on the Line  (2608.18875 - Angrick et al., 19 Aug 2026) in Section 6, Lower Bounds