Noah Fleming

Noah Fleming, Anna Gal, Deniz Imrek et al.

Unlike in TFNP, for which there is an abundance of problems capturing natural existence principles which are incomparable (in the black-box setting), Kleinberg et al. [KKMP21] observed that many of the natural problems considered so far in the second level of the total function polynomial hierarchy (TF$Σ_2$) reduce to the Strong Avoid problem. In this work, we prove that the Linear Ordering Principle does not reduce to Strong Avoid in the black-box setting, exhibiting the first TF$Σ_2$ problem that lies outside of the class of problems reducible to Strong Avoid. The proof of our separation exploits a connection between total search problems in the polynomial hierarchy and proof complexity, recently developed by Fleming, Imrek, and Marciot [FIM25]. In particular, this implies that to show our separation, it suffices to show that there is no small proof of the Linear Ordering Principle in a $Σ_2$-variant of the Sherali-Adams proof system. To do so, we extend the classical pseudo-expectation method to the $Σ_2$ setting, showing that the existence of a $Σ_2$ pseudo-expectation precludes a $Σ_2$ Sherali-Adams proof. The main technical challenge is in proving the existence of such a pseudo-expectation, we manage to do so by solving a combinatorial covering problem about permutations. We also show that the extended pseudo-expectation bound implies that the Linear Ordering Principle cannot be reduced to any problem admitting a low-degree Sherali-Adams refutation.

Noah Fleming, Max Hopkins, Yuichi Yoshida

Minimum dominating set is a basic local covering problem and a core task in distributed computing. Despite extensive study, in the classic LOCAL model there exist significant gaps between known algorithms and lower bounds. Chang and Li prove an $Ω(\log n)$-locality lower bound for a constant factor approximation, while Kuhn--Moscibroda--Wattenhofer gave an algorithm beating this bound beyond $\log Δ$-approximation, along with a weaker lower bound for this degree-dependent setting scaling roughly with $\min\{\log Δ/\log\log Δ,\sqrt{\log n/\log\log n}\}$. Unfortunately, this latter bound is weak for small $Δ$, and never recovers the Chang--Li bound, leaving central questions: does $O(\log Δ)$-approximation require $Ω(\log n)$ locality, and do such bounds extend beyond LOCAL? In this work, we take a major step toward answering these questions in the non-signaling model, which strictly subsumes the LOCAL, quantum-LOCAL, and bounded-dependence settings. We prove every $O(\logΔ)$-approximate non-signaling distribution for dominating set requires locality $Ω(\log n/(\logΔ\cdot \mathrm{poly}\log\logΔ))$. Further, we show for some $β\in (0,1)$, every $O(\log^βΔ)$-approximate non-signaling distribution requires locality $Ω(\log n/\logΔ)$, which combined with the KMW bound yields a degree-independent $Ω(\sqrt{\log n/\log\log n})$ quantum-LOCAL lower bound for $O(\log^βΔ)$-approximation algorithms. The proof is based on two new low-soundness sensitivity lower bounds for label cover, one via Impagliazzo--Kabanets--Wigderson-style parallel repetition with degree reduction and one from a sensitivity-preserving reworking of the Dinur--Harsha framework, together with the reductions from label cover to set cover to dominating set and the sensitivity-to-locality transfer theorem of Fleming and Yoshida.