Corrigendum: PLS is contained in PLC
Takashi Ishizuka
TL;DR
This corrigendum examines the relationship between TFNP subclasses $PLS$ and $PLC$, identifying a significant error in the prior proof that aimed to show $PLS \\subseteq PLC$ via a Quotient Pigeon construction. It introduces Quotient Pigeon as a TFNP problem extending Pigeon and proves it is both $PPP$-hard and $PLS$-hard, outlining reductions from LocalOPT to Quotient Pigeon and from Quotient Pigeon to Long Choice to situate $PLS$ within $PLC$ through a transitive reduction framework. Despite the error, the work clarifies the landscape by formalizing a reduction pipeline and highlighting key structural obstacles, while leaving open questions about $PPA$ containment and the precise hardness/complete status of Quotient Pigeon within $PLC$. The discussion advances understanding of how quotient-like constructions can unify TFNP subclasses and raises future directions for identifying $PLC$-complete problems and connections to non-interactive variants like Unary Long Choice. The overall impact lies in refining the boundary of total search problems and proposing concrete avenues to unify classical TFNP subclasses.
Abstract
Recently, Pasarkar, Papadimitriou, and Yannakakis (ITCS 2023) have introduced the new TFNP subclass called PLC that contains the class PPP; they also have proven that several search problems related to extremal combinatorial principles (e.g., Ramsey's theorem and the Sunflower lemma) belong to PLC. This short paper shows that the class PLC also contains PLS, a complexity class for TFNP problems that can be solved by a local search method. However, it is still open whether PLC contains the class PPA.