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Relations among SETH, 3SUM, and APSP hypotheses

Determine the precise implication relations among the three main fine-grained hypotheses—SETH, the 3SUM Hypothesis, and the APSP Hypothesis—by establishing which implications (if any) hold between them.

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Background

The introduction reviews the three most prominent fine-grained hypotheses—SETH, 3SUM, and APSP—and notes that none is currently known to imply the others.

The authors underscore that clarifying the relationships between these hypotheses has been regarded as a central challenge in the field.

References

Indeed, determining the relations between these hypotheses is sometimes stated as a major open problem in the field , even though there is some tentative evidence that some potential reductions may be impossible .

— k-SUM Hardness Implies Treewidth-SETH (2510.08185 - Lampis, 9 Oct 2025) in Section 1.2 (Background)