Extremal combinatorics, iterated pigeonhole arguments, and generalizations of PPP

TL;DR

This paper introduces a new complexity class PLC within TFNP, based on iterated pigeonhole arguments, and explores its relationship with existing classes like PPP, analyzing problems from extremal combinatorics.

Contribution

It defines the PLC class, relates it to PPP, and classifies several combinatorial problems within these classes, including open questions about their completeness.

Findings

PLC includes PPP and many unclassified problems.

The paper classifies some problems as PPP-complete.

It reframes PPP as an optimization problem and explores related hierarchies.

Abstract

We study the complexity of computational problems arising from existence theorems in extremal combinatorics. For some of these problems, a solution is guaranteed to exist based on an iterated application of the Pigeonhole Principle. This results in the definition of a new complexity class within TFNP, which we call PLC (for "polynomial long choice"). PLC includes all of PPP, as well as numerous previously unclassified total problems, including search problems related to Ramsey's theorem, the Sunflower theorem, the Erd\H{o}s-Ko-Rado lemma, and K\"onig's lemma. Whether the first two of these four problems are PLC-complete is an important open question which we pursue; in contrast, we show that the latter two are PPP-complete. Finally, we reframe PPP as an optimization problem, and define a hierarchy of such problems related to Tur\'an's theorem.