Dichotomy for tree-structured trigraph list homomorphism problems

TL;DR

This paper establishes a dichotomy for a class of tree-structured trigraph list homomorphism problems, showing they are either polynomially solvable or NP-complete, similar to digraph cases.

Contribution

It identifies a large class of tree-like trigraphs where list homomorphism problems exhibit a dichotomy, linking their complexity to related digraph problems.

Findings

Tree-like trigraph list homomorphism problems are polynomially equivalent to digraph problems.

The paper characterizes conditions under which these problems are polynomial or NP-complete.

Relaxing the structural conditions can lead to computational hardness.

Abstract

Trigraph list homomorphism problems (also known as list matrix partition problems) have generated recent interest, partly because there are concrete problems that are not known to be polynomial time solvable or NP-complete. Thus while digraph list homomorphism problems enjoy dichotomy (each problem is NP-complete or polynomial time solvable), such dichotomy is not necessarily expected for trigraph list homomorphism problems. However, in this paper, we identify a large class of trigraphs for which list homomorphism problems do exhibit a dichotomy. They consist of trigraphs with a tree-like structure, and, in particular, include all trigraphs whose underlying graphs are trees. In fact, we show that for these tree-like trigraphs, the trigraph list homomorphism problem is polynomially equivalent to a related digraph list homomorphism problem. We also describe a few examples illustrating that our conditions defining tree-like trigraphs are not unnatural, as relaxing them may lead to harder problems.