Enumerating Homomorphisms

TL;DR

This paper investigates the enumeration of all homomorphism solutions in relational structures, revealing that enumeration behaves differently from decision problems and identifying cases where polynomial delay enumeration is feasible.

Contribution

It provides the first analysis of enumeration complexity for homomorphisms, showing the problem's distinct behavior from decision versions and identifying specific tractable cases.

Findings

Enumeration can be done with polynomial delay in certain cases.

The structure of the problem differs significantly from the decision version.

A characterization similar to the decision problem is unlikely for enumeration.

Abstract

The homomorphism problem for relational structures is an abstract way of formulating constraint satisfaction problems (CSP) and various problems in database theory. The decision version of the homomorphism problem received a lot of attention in literature; in particular, the way the graph-theoretical structure of the variables and constraints influences the complexity of the problem is intensively studied. Here we study the problem of enumerating all the solutions with polynomial delay from a similar point of view. It turns out that the enumeration problem behaves very differently from the decision version. We give evidence that it is unlikely that a characterization result similar to the decision version can be obtained. Nevertheless, we show nontrivial cases where enumeration can be done with polynomial delay.