The Complexity of Ramsey Arrowing: A Computational Approach for Hardness Proofs

Abstract

In graph Ramsey theory, the arrowing operator is used to describe the appearance of unavoidable substructures within colored graphs; for graphs G, F, and H, we say G → (F,H) (read, G arrows F, H) if every red/blue coloring of G’s edges contains a red F or a blue H. For fixed F and H, the (F,H)-Arrowing problem asks whether G → (F,H) for some given graph G. (F,H)-Arrowing has been shown to be in P or coNP-complete for different pairs (F,H). However, categorizing the complexity for all pairs still remains wide open.
In general, categorizing the complexity of problems whose nature depends on some underlying graph - or, in our case, pair of graphs - is a daunting task, and (F,H)-Arrowing is no exception. Hardness proofs typically rely on ad-hoc, laborious constructions of special graphs known as "gadgets." In this work, we present a simple, computational approach to find these gadgets for small (F,H)-Arrowing problems and show how these can be extended to other (F,H)-Arrowing problems.
Our main focus is on the simplest case for which the complexity remains uncategorized: F = P₃. We showcase the efficacy of our computational approach by presenting hardness proofs for (P₃, H)-Arrowing problems previously not known to be coNP-hard. Moreover, we show how to generalize hardness to other (P₃,H)-Arrowing problems by either: (1) carefully inspecting and modifying our found gadgets, or (2) coming up with intuitive constructions to reduce (P₃, H')-Arrowing to (P₃, H)-Arrowing, where H' is a subgraph of H. We also discuss how our methodology can be extended to work for other (F,H)-Arrowing problems by showing new results for F = P₄ and K_{1,3}.
We see our work as an important step towards categorizing the complexity of (F,H)-Arrowing for all pairs (F,H). (F,H)-Arrowing is thought to be hard when (F',H')-Arrowing is hard where F' and H' are subgraphs of F and H, respectively. Under this assumption, our results narrow down the only uncategorized family of problems for which the problem may lie in P. Beyond the complexity of (F,H)-Arrowing, we believe that the broader impact of our work is showing how computational methodologies can be adopted for proving hardness, and we hope our approach will be adopted to find gadgets for other open graph problems as well.