On the complexity of some restricted variants of Quotient Pigeon and a weak variant of Kőnig

Abstract

One of the most famous TFNP subclasses is PPP, which is the set of all search problems whose totality is guaranteed by the pigeonhole principle. The author's recent preprint [1] has introduced a TFNP problem related to the pigeonhole principle over a quotient set, called Quotient Pigeon, and shown that the problem Quotient Pigeon is not only PPP-hard but also PLS-hard. In this paper, we formulate other computational problems related to the pigeonhole principle over a quotient set via an explicit representation of the equivalence classes. Our new formulation introduces a non-trivial $\text{PPP}\cap {\text{PPA}}_k$-complete problem for every $k\ge 2$. Furthermore, we consider the computational complexity of a computational problem related to Kőnig's lemma, which is a weaker variant of the problem formulated by Pasarkar et al. [2]. We show that our weaker variant is PPAD-hard and is in $\text{PPP}\cap \text{PPA}$.