Improving alphabet-size bounds in low-soundness PCPs (sliding scale)

Improve the alphabet-size bounds in low-soundness PCP constructions used to produce label cover instances—specifically, obtain significantly smaller alphabets for a given soundness level—thereby advancing toward the sliding scale conjecture and enabling sharper downstream hardness and sensitivity lower bounds in reductions to set cover and dominating set.

Background

The construction underlying the paper’s Dinur–Harsha-based label cover lower bound yields low-soundness instances with alphabet sizes that are exponential in a polynomial of the soundness parameter, which limits the strength of the resulting dominating-set lower bounds. The authors explicitly point out that shrinking these alphabets is a central objective in PCP theory.

This problem is tightly connected to the sliding scale conjecture, which predicts optimal trade-offs between proof length, query complexity, and soundness; progress on alphabet-size reductions would have broad consequences beyond the scope of this paper.

References

Unfortunately, \Cref{thm:intro-label-cover} cannot be used to prove a lower bound against $O(\log \Delta)$-approximation for dominating set due to the large left-alphabet size, which restricts us to $O(\log\beta \Delta)$-approximation for some $\beta>0$. More generally, the degree $\Delta$ of the final dominating set instance in our reduction is at least the degree and alphabet size of the starting low soundness label cover, so it is critical we keep these parameters as small as possible. Improving the alphabet-size bound in the PCP underlying \Cref{thm:intro-label-cover} is a central open problem in PCP theory, related to the so-called `sliding scale conjecture', and doing so would have consequences far beyond just improving our dominating set lower bound [citation].

— Non-Signaling Locality Lower Bounds for Dominating Set  (2604.02582 - Fleming et al., 2 Apr 2026) in Section 1.1: Proof Overview (Key Ingredients)