Restricted CSPs and F-free Digraph Algorithmics

TL;DR

This paper investigates the complexity of directed graph homomorphism problems under restrictions related to forbidden paths, establishing NP-hardness and polynomial cases, and introduces restricted CSPs with a complexity dichotomy.

Contribution

It extends the study of graph homomorphism complexity to directed graphs with forbidden substructures and introduces restricted CSPs with a new complexity dichotomy.

Findings

NP-hardness persists for certain restricted digraph classes.

Polynomial-time algorithms exist for some acyclic restricted digraphs.

A dichotomy theorem classifies restricted CSPs as either polynomial or NP-complete.

Abstract

In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ${\mathbb P}_k$-free and ${\mathbb P}_k$-subgraph-free graphs. We consider the directed version of this research line, by addressing the questions, is it true that digraph homomorphism problems CSP$({\mathbb H})$ have a P versus NP-complete dichotomy when the input is restricted to $\vec{\mathbb P}_k$-free (resp.\ $\vec{\mathbb P}_k$-subgraph-free) digraphs? Our main contribution in this direction shows that if CSP$({\mathbb H})$ is NP-complete, then there is a positive integer $N$ such that CSP$({\mathbb H})$ remains NP-hard even for $\vec{\mathbb P}_N$-subgraph-free digraphs. Moreover, it remains NP-hard for acyclic $\vec{\mathbb P}_N$-subgraph-free digraphs, and becomes polynomial-time solvable for $\vec{\mathbb P}_{N-1}$-subgraph-free acyclic digraphs. We then verify the questions above for digraphs on three vertices and a family of smooth tournaments. We prove these results by establishing a connection between $\mathbb F$-(subgraph)-free algorithmics and constraint satisfaction theory. On the way, we introduce restricted CSPs, i.e., problems of the form CSP$({\mathbb H})$ restricted to yes-instances of CSP$({\mathbb H}')$ -- these were called restricted homomorphism problems by Hell and Ne\v{s}et\v{r}il. Another main result of this paper presents a P versus NP-complete dichotomy for these problems. Moreover, this complexity dichotomy is accompanied by an algebraic dichotomy in the spirit of the finite domain CSP dichotomy.