k-SUM Hardness Implies Treewidth-SETH

2025-10-09

AI Summary

This paper investigates the computational hardness of the k-SUM and k-XOR problems as evidence for the validity of Primal Treewidth SETH—a variant of the Strong Exponential Time Hypothesis parameterized by the primal graph’s treewidth. Method: Leveraging fine-grained reductions, randomized algorithm constructions, and structural analysis of tree decompositions, the authors establish the first tight connections between k-SUM/k-XOR hardness assumptions and treewidth-parameterized complexity. Contribution/Results: They prove that if no significantly faster algorithms exist for k-SUM (or k-XOR), then Primal Treewidth SETH holds; conversely, a truly subexponential-time treewidth-parameterized SAT algorithm implies breakthrough algorithms for k-SUM/k-XOR. This yields tight lower bounds for numerous classical problems under an assumption plausibly stronger than standard SETH, and introduces a novel framework for deriving graph-structural complexity hypotheses from algebraic problem hardness.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• 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

Abstract

We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an $varepsilon>0$ and an algorithm which solves SAT in time $(2-varepsilon)^{tw}|φ|^{O(1)}$, where $tw$ is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time $n^{(1-δ)frac{k}{2}}$ for some $δ>0$ and all sufficiently large $k$. We also establish an analogous result for the k-XOR problem, where integer addition is replaced by component-wise addition modulo $2$. As an application of our reduction we are able to revisit tight lower bounds on the complexity of several fundamental problems parameterized by treewidth (Independent Set, Max Cut, $k$-Coloring). Our results imply that these bounds, which were initially shown under the SETH, also hold if one assumes the k-SUM or k-XOR Hypotheses, arguably increasing our confidence in their validity.

Problem

• Proving Treewidth-SETH hardness from k-SUM complexity assumptions
• Establishing analogous hardness results for k-XOR problem variants
• Extending treewidth-based lower bounds for fundamental graph problems

Innovation

• Links k-SUM hardness to Treewidth-SETH
• Establishes analogous result for k-XOR problem
• Revisits tight treewidth bounds for fundamental problems