Space complexity of list H-coloring revisited: the case of oriented trees

TL;DR

This paper characterizes oriented trees for which the list homomorphism problem is solvable in logspace, providing forbidden subgraph criteria, a simple recognition algorithm, and an algebraic perspective.

Contribution

It refines previous results by offering a complete characterization of oriented trees with logspace solvability for LHOM, including algorithms and algebraic insights.

Findings

Characterization of oriented trees with LHOM in L via forbidden subgraphs

A simple O(|V(T)|^3) recognition algorithm for these trees

An algebraic characterization of the identified trees

Abstract

Digraphs H for which the list homomorphism problem with template H (LHOM(H)) is in logspace (L) was characterized by Egri et al. (SODA 2014): LHOM(H) is in L if and only if H does not contain a circular N (assuming L is different from NL). Undirected graphs for which LHOM(H) is in L can be characterized in terms forbidden induced subgraphs, and also via a simple inductive construction (Egri et al., STACS 2010). As a consequence, the logspace algorithm in the undirected case is simple and easy to understand. No such forbidden subgraph or inductive characterization, and no such simple and easy-to-understand algorithm is known in the case of digraphs. In this paper, in the case of oriented trees, we refine and strengthen the results of Egri et al. (SODA 2014): we give a characterization of oriented trees T for which LHOM(T) is in L both in terms of forbidden induced subgraphs, and also via a simple inductive construction. Using this characterization, we obtain a simple and easy-to-analyze logspace algorithm for LHOM(T). We also show how these oriented trees can be recognized in time O(|V(T)|^3) (the straightforward implementation of the algorithm given in SODA 2014 runs in time O(|V(H)|^8) for oriented trees). An algebraic characterization of these trees is also provided.