In the last three decades, the $k$-SUM hypothesis has emerged as a satisfying explanation of long-standing time barriers for a variety of algorithmic problems. Yet to this day, the literature knows of only few proven consequences of a refutation of this hypothesis. Taking a descriptive complexity viewpoint, we ask: What is the largest logically defined class of problems \emph{captured} by the $k$-SUM problem? To this end, we introduce a class $\mathsf{FOP}_{\mathbb{Z}}$ of problems corresponding to deciding sentences in Presburger arithmetic/linear integer arithmetic over finite subsets of integers. We establish two large fragments for which the $k$-SUM problem is complete under fine-grained reductions: 1. The $k$-SUM problem is complete for deciding the sentences with $k$ existential quantifiers. 2. The $3$-SUM problem is complete for all $3$-quantifier sentences of $\mathsf{FOP}_{\mathbb{Z}}$ expressible using at most $3$ linear inequalities. Specifically, a faster-than-$n^{\lceil k/2 \rceil \pm o(1)}$ algorithm for $k$-SUM (or faster-than-$n^{2 \pm o(1)}$ algorithm for $3$-SUM, respectively) directly translate to polynomial speedups of a general algorithm for \emph{all} sentences in the respective fragment. Observing a barrier for proving completeness of $3$-SUM for the entire class $\mathsf{FOP}_{\mathbb{Z}}$, we turn to the question which other -- seemingly more general -- problems are complete for $\mathsf{FOP}_{\mathbb{Z}}$. In this direction, we establish $\mathsf{FOP}_{\mathbb{Z}}$-completeness of the \emph{problem pair} of Pareto Sum Verification and Hausdorff Distance under $n$ Translations under the $L_\infty$/$L_1$ norm in $\mathbb{Z}^d$. In particular, our results invite to investigate Pareto Sum Verification as a high-dimensional generalization of 3-SUM.

翻译：在过去的三十年中，$k$-SUM假设已成为解释一系列算法问题长期存在的时间障碍的令人满意的理论框架。然而迄今为止，文献中仅知少数被证实的、由该假设被证伪所导致的推论。从描述复杂性视角出发，我们提出：$k$-SUM问题所能刻画的、具有逻辑定义的最大问题类是什么？为此，我们引入问题类$\mathsf{FOP}_{\mathbb{Z}}$，其对应判定定义在整数有限子集上的Presburger算术/线性整数算术语句。我们建立了两个大的片段，证明$k$-SUM问题在细粒度归约下对于这些片段是完备的：1. $k$-SUM问题对于判定具有$k$个存在量词的语句是完备的。2. $3$-SUM问题对于$\mathsf{FOP}_{\mathbb{Z}}$中所有可使用至多$3$个线性不等式表达的$3$量词语句是完备的。具体而言，针对$k$-SUM的优于$n^{\lceil k/2 \rceil \pm o(1)}$算法（或分别针对$3$-SUM的优于$n^{2 \pm o(1)}$算法）将直接转化为针对相应片段中所有语句的通用算法的多项式级加速。观察到证明$3$-SUM对整个类$\mathsf{FOP}_{\mathbb{Z}}$完备性存在障碍，我们转而探究哪些其他——看似更一般的——问题对$\mathsf{FOP}_{\mathbb{Z}}$是完备的。在此方向上，我们证明了在$\mathbb{Z}^d$中$L_\infty$/$L_1$范数下，帕累托和验证与$n$个平移下的豪斯多夫距离这一对问题对$\mathsf{FOP}_{\mathbb{Z}}$是完备的。特别地，我们的结果促使将帕累托和验证作为3-SUM的高维推广进行研究。