On the Structure of $(\min,+)$ Convolution

Abstract: The $(\min,+)$ convolution is a central problem in fine-grained complexity, and whether it admits a truly subquadratic algorithm remains open. We study it through tropical polynomials, where $(\min,+)$ convolution is exactly polynomial multiplication. We introduce tropical decomposition width, a parameter measuring how finely a tropical polynomial can be decomposed into low-degree factors. We prove modular convexity theorems showing that bounded tropical decomposition width forces strong convexity on arithmetic subpolynomials. This yields deterministic algorithms for computing $a\otimes b$ in $$O(n\max(\operatorname{tdw}(a),\operatorname{tdw}(b))<sup>2)$$ time when the width is given, and in $$O(ne<sup>{\min(\operatorname{tdw}(a),\operatorname{tdw}(b))(1+o(1))})$$ time otherwise, without requiring a decomposition. For Multiple-Sequence $(\min,+)$ Convolution, we give a randomized algorithm running in $O(kn<sup>2\sqrt{\min(k,n)}\log<sup>{1.5}(kn))$ time for $k$ sequences of length at most $n$, improving the natural $O(k<sup>2n<sup>2)$ bound. We also obtain conditional lower bounds, a faster single-entry algorithm, and new upper bounds for Multiple-Choice Knapsack. Finally, bounded-decomposition-width classes admit interpolation algebras of finite generating rank, whereas distinguishing all tropical polynomials of degree at most $n$ requires rank exactly $\lfloor n/2\rfloor+1$. We further show that tropical decomposition width cannot decrease under any flat $\mathbb T$-algebra extension. These results connect efficient tropical multiplication with structural rigidity.