$(\text{min},+)$卷积的结构
On the Structure of $(\min,+)$ Convolution
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中文总结 AI 辅助
本文研究细粒度复杂度核心问题$(\text{min},+)$卷积，引入热带分解宽度参数，给出相关确定性、随机算法及下界，揭示其与结构刚性的联系
中文摘要 AI 辅助
$(\text{min},+)$卷积是细粒度复杂度领域的核心问题，其是否存在真正的次二次算法仍是开放问题。本文通过热带多项式研究该问题，其中$(\text{min},+)$卷积等价于多项式乘法。我们引入热带分解宽度（tropical decomposition width）这一参数，用于衡量热带多项式分解为低次因子的精细程度。我们证明了模凸性定理，表明有界热带分解宽度会迫使算术子多项式呈现强凸性。这给出了计算$a\boxtimes b$的确定性算法：当给定宽度时，时间复杂度为$O(n\boldsymbol{\text{max}}(\text{tdw}(a),\text{tdw}(b))^2)$；未给定宽度时，时间复杂度为$O(n e^{\boldsymbol{\text{min}}(\text{tdw}(a),\text{tdw}(b))(1+o(1))})$，且无需进行分解。对于多序列$(\text{min},+)$卷积，我们给出了一种随机算法，对于$k$条长度不超过$n$的序列，其时间复杂度为$O(kn^2\boldsymbol{\text{min}}(k,n)^{0.5}\boldsymbol{\text{log}}^{1.5}(kn))$，改进了自然的$O(k^2n^2)$界。我们还得到了条件下界、更快的单元素算法以及多选背包问题的新上界。最后，有界分解宽度类具有有限生成秩的插值代数，而区分所有次数不超过$n$的热带多项式恰好需要秩$\boldsymbol{\text{floor}}(n/2)+1$。我们进一步证明，在任何平坦$\boldsymbol{\text{T}}$-代数扩张下，热带分解宽度不会减小。这些结果将高效热带乘法与结构刚性联系起来。
英文摘要
The $(\min,+)$ convolution is a central problem in fine-grained complexity, and it remains open whether it can be computed in truly subquadratic time. We study it through tropical polynomials, where $(\min,+)$ convolution is exactly tropical polynomial multiplication. We introduce the tropical decomposition width, $\operatorname{tdw}(A)$, which measures how finely a tropical polynomial can be decomposed into factors of small degree. We prove two modular convexity theorems showing that bounded tropical decomposition width forces convexity on arithmetic progression subpolynomials. This yields deterministic algorithms for computing $a\otimes b$ in $$O\left(n\max(\operatorname{tdw}(a),\operatorname{tdw}(b))^2\right)$$ when $\max(\operatorname{tdw}(a),\operatorname{tdw}(b))$ is given, and in $$O\left(ne^{\min(\operatorname{tdw}(a),\operatorname{tdw}(b))(1+o(1))}\right)$$ without prior knowledge of the width. Neither algorithm requires a decomposition of the input sequences. The same structural ideas give a randomized algorithm for Multiple-Sequence $(\min,+)$ Convolution: given $k$ sequences of length at most $n$, their convolution can be computed in $$O\left(kn^2\sqrt{\min(k,n)}\log^{1.5}(kn)\right)$$ time, improving the natural $O(k^2n^2)$ bound. Finally, we introduce interpolation algebras for tropical polynomials and show that classes with bounded tropical decomposition width admit interpolation algebras of finite generating rank, whereas distinguishing all tropical polynomials of degree at most $n$ requires generating rank $\lfloor n/2\rfloor+1$. We also prove that tropical decomposition width cannot decrease under any flat $\mathbb T$-algebra extension. Together, these results connect the tractability of $(\min,+)$ convolution with structural rigidity in tropical polynomial multiplication.
发表机构
- Tsinghua University（清华大学）
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