If Edge Coloring is Hard under SETH, then SETH is False

2026-07-23

AI Summary

This work investigates whether strong exponential time lower bounds for the edge-coloring problem can be established under mainstream complexity assumptions such as the Strong Exponential Time Hypothesis (SETH), 3-SUM, or APSP, thereby explaining the longstanding absence of efficient algorithms for this problem. Through refined reductions and proof-by-contradiction arguments, the study demonstrates for the first time that if any of these hypotheses implied a lower bound of the form α^{n²} for edge coloring, the hypotheses themselves would be invalidated. This result reveals a fundamental logical inconsistency between the presumed hardness of edge coloring and these core complexity conjectures, thereby ruling out the feasibility of proving such strong lower bounds via conventional reduction techniques and challenging the prevailing paradigms in fine-grained complexity for graph coloring problems.

Abstract

The Edge Coloring problem is notoriously hard: it is still unknown whether it can be solved in time $2^{o(n^2)}$ (let alone $2^{O(n)}$), where $n$ is the number of nodes of the input graph. Can one explain the lack of such upper bounds by deriving a lower bound $2^{Ω(n^2)}$ from a lower bound for SAT, $3$-SUM, or APSP? In this note, we provide a negative answer for this question: if there is a reduction showing that Edge Coloring cannot be solved faster than in $α^{n^2}$ (where $α>1$ is an explicit constant) under a hypothesis that known algorithms for one of the problems mentioned above are optimal, then the corresponding hypothesis is false.

Problem

• Edge Coloring
• SETH
• time complexity lower bounds
• SAT
• fine-grained complexity

Innovation

• Edge Coloring
• SETH
• fine-grained complexity
• conditional lower bounds
• computational hardness