关于（min，+）-卷积的等价问题
On problems equivalent to (min,+)-convolution
摘要：
近年来，在解释许多基本多项式可解问题的朴素解的改进的明显困难方面取得了重大进展。这一进展是以条件下限的形式出现的--从一个假设很难的问题中减少。困难的问题包括3SUM，所有对最短路径，SAT，正交向量等。在$(\min,+)$ -卷积问题中，目标是计算序列$(c[i])^{n-1}_{i=0}$，其中$c[k] = $ $\min_{i=0,\ldots,k} $ $\{a[i] $ $+$ $b[k-i]\}$，给定序列$(a[i])^{n-1}_{i=0}$和$(b[i])_{i=0}^{n-1}$。这可以很容易地在$O(n^2)$时间内完成，但是对于$\varepsilon > 0$没有已知的$O(n^{2-\varepsilon})$算法。在本文中，我们进行了系统的研究$(\min,+)$ -卷积问题作为一个硬度假设。首先，我们建立了一组其他问题，包括经典的背包问题的变种和次加性序列相关的问题，这个问题的等价性。$(\min,+)$ -卷积问题已被用作许多问题的算法中的构建块，特别是弦学中的问题。它也被认为是一个特别的硬度假设。其次，我们调查了其中的一些联系，并提供了新的减少和其他结果。我们还解释了为什么用SETH代替这个假设对于某些问题可能是不可能的。
英文摘要：
In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower bounds -- reductions from a problem assumed to be hard. The hard problems include 3SUM, All-Pairs Shortest Path, SAT, Orthogonal Vectors, and others. In the $(\min,+)$-convolution problem, the goal is to compute a sequence $(c[i])^{n-1}_{i=0}$, where $c[k] = $ $\min_{i=0,\ldots,k} $ $\{a[i] $ $+$ $b[k-i]\}$, given sequences $(a[i])^{n-1}_{i=0}$ and $(b[i])_{i=0}^{n-1}$. This can easily be done in $O(n^2)$ time, but no $O(n^{2-\varepsilon})$ algorithm is known for $\varepsilon > 0$. In this paper, we undertake a systematic study of the $(\min,+)$-convolution problem as a hardness assumption. First, we establish the equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The $(\min,+)$-convolution problem has been used as a building block in algorithms for many problems, notably problems in stringology. It has also appeared as an ad hoc hardness assumption. Second, we investigate some of these connections and provide new reductions and other results. We also explain why replacing this assumption with the SETH might not be possible for some problems.
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