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Gap Hamming Distance completion to a total problem with constant-cost protocol

Determine whether there exists any completion of the partial Gap Hamming Distance function GapHD_γ to a total communication problem that admits a constant-cost randomized protocol.

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Background

The paper shows that every constant‑cost problem reduces to an appropriate Gap Hamming Distance instance, but emphasizes limitations because GapHD_γ is partial. Whether a total completion of GapHD_γ retains constant-cost remains unknown, and a positive result could strengthen the utility of GapHD in characterizing BPP.

References

This is because $GapHD_\gamma$ is a partial problem, and it is not known whether any of its completions into a total communication problem has a constant-cost protocol.

— Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204 - Fang et al., 29 Jul 2024) in Appendix: Reduction to Gap Hamming Distance