Combinatorial algorithms for the affected problems

Determine whether practical or combinatorial algorithms exist for the affected problems, including the sparse triangle and related problems, that achieve polynomial savings analogous to the algebraic algorithms presented in the paper.

Background

The paper’s new algorithms are algebraic and may be impractical because of large hidden constants. The authors contrast them with combinatorial Boolean matrix multiplication, for which only subpolynomial improvements over cubic time are known, and explicitly leave the existence of truly subcubic combinatorial algorithms unresolved.

References

We do not know whether such algorithms exist.

— Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs  (2610.06783 - Alman et al., 5 Oct 2026) in Section 1, paragraph “Combinatorial algorithms”

Is improving upon the known algorithms for this problem hard, or is there another technique that could help here?

— Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs  (2610.06783 - Alman et al., 5 Oct 2026) in Section 8, Section “Conclusion,” paragraph “Fine-grained complexity”

What are the true exponents of $3$SUM, APSP, and the many other problems of Figure~\ref{fig:web} that now have faster algorithms? Our exponents can certainly be improved, and one place to gain is the reductions.

— Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs  (2610.06783 - Alman et al., 5 Oct 2026) in Section 8, Section “Conclusion,” paragraph “Fine-grained complexity”