The PPP-completeness of the Ward-Szabo theorem

TL;DR

This paper proves that the Ward-Szabo theorem's associated total search problem is complete for the complexity class PPP, linking it to the pigeonhole principle.

Contribution

It establishes the Ward-Szabo problem as PPP-complete, clarifying its precise position within the TFNP complexity classes.

Findings

Ward-Szabo problem is PWPP-hard and in TFNP.

The problem is shown to be PPP-complete.

This clarifies the computational complexity of Ward-Szabo.

Abstract

Ward and Szab\'o [WS94] have shown that a complete graph with $N^2$ nodes whose edges are colored by $N$ colors and that has at least two colors contains a bichromatic triangle. This fact leads us to a total search problem: Given an edge-coloring on a complete graph with $N^2$ nodes using at least two colors and at most $N$ colors, find a bichromatic triangle. Bourneuf, Folwarczn\'y, Hub\'acek, Rosen, and Schwartzbach [Bou+23] have proven that such a total search problem, called Ward-Szab\'o, is PWPP-hard and belongs to the class TFNP, a class for total search problems in which the correctness of every candidate solution is efficiently verifiable. However, it is open which TFNP subclass contains Ward-Szab\'o. This paper will improve the computational complexity of Ward-Szab\'o. We prove that Ward-Szab\'o is a complete problem for the complexity class PPP, a TFNP subclass of problems in which the existence of solutions is guaranteed by the pigeonhole principle.