Completeness Theorems for k-SUM and Geometric Friends: Deciding Fragments of Integer Linear Arithmetic

TL;DR

The paper develops a fine-grained descriptive framework for studying fragments of integer linear arithmetic using the class $\mathsf{FOP}_{\mathbb{Z}}$. It establishes that $k$-SUM is complete for the existential fragment $\mathsf{FOP}_{\mathbb{Z}}(\exists^k)$ and that $3$-SUM is complete for $\mathsf{FOP}_{\mathbb{Z}}$ formulas with inequality dimension at most $3$, linking algorithmic speedups directly to broader fragments. It further shows that two natural geometric problems, Pareto Sum Verification and Hausdorff Distance under $n$ translations, are complete for $\mathsf{FOP}_{\mathbb{Z}}$, implying that subquadratic improvements for these problems would yield subquadratic improvements for a wide range of quantified linear arithmetic sentences. The work also develops counting-witness results and analyzes general quantifier structures, providing a cohesive picture of how fine-grained hardness propagates through logical fragments and geometric instances. Overall, the results offer both a canonical pair of complete problems and a roadmap for translating improvements in specific problems into broad gains across many $\mathsf{FOP}_{\mathbb{Z}}$ formulas, with concrete implications for lower bounds and potential directions for completing the picture of fragment hardness.

Abstract

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.

Theorems & Definitions (43)

• Theorem 1: $k$-SUM is $\mathsf{FOP}_{\mathbb{Z}}(\exists^k)$-complete
• Theorem 2
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6: Pareto Sum Computation Lower Bound
• Lemma 7: Syntactic Complete problems (Informal Version)
• Lemma 7
• Theorem 8