Homomorphisms and Embeddings

Abstract

The homomorphism problem for relational structures is a fundamental algorithmic problem playing an important role in different areas of computer science. For example, constraint satisfaction problems in artificial intelligence and the containment problem for conjunctive queries in database theory can be viewed as homomorphism problems. We briefly covered the homomorphism problem in Chap. 7, where we proved that its parameterization by the size of the left-hand side structure is Wi$[1]$-complete. In the first two sections of this chapter we study restrictions of the homomorphism problem obtained by requiring the left-hand side structure to be from a certain class of structures. We give a complete classification of the complexity of such restrictions, both in the sense of polynomial time solvability and fixed-parameter tractability. Once again, tree width plays a central role. As a by-product, we obtain a new characterization of the question FPT $\genfrac{}{}{0pt}{}{?}{\displaystyle =}$ W$[1]$ entirely in terms of classical complexity. In the third section we study the related embedding problem. We introduce a powerful new technique for the design of fpt-algorithms, which is known as color coding. In its basic form, color coding yields randomized fpt-algorithms. These can be derandomized by sophisticated hashing techniques. We apply these techniques to prove that the embedding problem, restricted to left-hand side structures of bounded tree width, is fixed-parameter tractable.