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

2025-02-07

AI Summary

Characterizing the expressive power of k-SUM within descriptive complexity theory. Method: Employing fine-grained reductions and semantic modeling via Presburger arithmetic, the paper establishes completeness results for k-SUM with respect to k-fold existential first-order sentences and for 3-SUM with respect to three-quantifier, three-inequality formulas. It introduces the logical class FOP_Z to precisely capture this expressiveness. Contribution/Results: The work identifies the first non-arithmetic pair of FOP_Z-complete problems—Pareto Sum Verification and Hausdorff distance computation—revealing a high-dimensional generalization pathway for 3-SUM. Furthermore, it proves that subquadratic (for 3-SUM) or sub-n^{⌈k/2⌉} (for k-SUM) algorithms would strictly accelerate the model-checking of their respective logical fragments, thereby establishing a tight connection between algorithmic efficiency and logical decidability. This yields the first exact correspondence between fine-grained complexity assumptions and the tractability of syntactically restricted first-order logics over integer structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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-$n^{lceil k/2 ceil pm o(1)}$ 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.

Problem

• Identifying largest logical class captured by k-SUM.
• Establishing k-SUM completeness for specific fragments.
• Investigating general problems complete for FOP_ℤ.

Innovation

• Introduces FOPℤ for Presburger arithmetic
• k-SUM complete for k existential quantifiers
• Pareto Sum Verification generalizes 3-SUM