The complexity of tropical graph homomorphisms

TL;DR

This paper explores the computational complexity of tropical graph homomorphism problems, establishing a connection with the Feder-Vardi Dichotomy Conjecture and classifying certain problem subclasses.

Contribution

It introduces the study of tropical graph homomorphism problems, linking their complexity to the Feder-Vardi Dichotomy Conjecture and providing classifications for specific cases.

Findings

$(H,c)$-COLOURING problems exhibit a complexity dichotomy tied to the CSP Dichotomy Conjecture.

Certain classes of $H$-TROPICAL-COLOURING problems have been classified in terms of computational complexity.

The study reveals the richness of the class of tropical graph homomorphism problems.

Abstract

A tropical graph $(H,c)$ consists of a graph $H$ and a (not necessarily proper) vertex-colouring $c$ of $H$. Given two tropical graphs $(G,c_1)$ and $(H,c)$, a homomorphism of $(G,c_1)$ to $(H,c)$ is a standard graph homomorphism of $G$ to $H$ that also preserves the vertex-colours. We initiate the study of the computational complexity of tropical graph homomorphism problems. We consider two settings. First, when the tropical graph $(H,c)$ is fixed; this is a problem called $(H,c)$-COLOURING. Second, when the colouring of $H$ is part of the input; the associated decision problem is called $H$-TROPICAL-COLOURING. Each $(H,c)$-COLOURING problem is a constraint satisfaction problem (CSP), and we show that a complexity dichotomy for the class of $(H,c)$-COLOURING problems holds if and only if the Feder-Vardi Dichotomy Conjecture for CSPs is true. This implies that $(H,c)$-COLOURING problems form a rich class of decision problems. On the other hand, we were successful in classifying the complexity of at least certain classes of $H$-TROPICAL-COLOURING problems.