The task of computing homomorphisms between two finite relational structures $\mathcal{A}$ and $\mathcal{B}$ is a well-studied question with numerous applications. Since the set $\operatorname{Hom}(\mathcal{A},\mathcal{B})$ of all homomorphisms may be very large having a method of representing it in a succinct way, especially one which enables us to perform efficient enumeration and counting, could be extremely useful. One simple yet powerful way of doing so is to decompose $\operatorname{Hom}(\mathcal{A},\mathcal{B})$ using union and Cartesian product. Such data structures, called d-representations, have been introduced by Olteanu and Zavodny in the context of database theory. Their results also imply that if the treewidth of the left-hand side structure $\mathcal{A}$ is bounded, then a d-representation of polynomial size can be found in polynomial time. We show that for structures of bounded arity this is optimal: if the treewidth is unbounded then there are instances where the size of any d-representation is superpolynomial. Along the way we develop tools for proving lower bounds on the size of d-representations, in particular we define a notion of reduction suitable for this context and prove an almost tight lower bound on the size of d-representations of all $k$-cliques in a graph.
翻译:计算两个有限关系结构之间的同质性任务 $\ mathcal{A} $ 和 $\ mathcal{B} $\ mathcal{B} 是一个有很多应用程序的很好研究的问题。 自从设定 $\ operallname{Hom} (\ mathcal{A},\ mathcal{B}) 以来, 所有同质性结构中的美元( mathcal{A},\ mathcal{B}) 任务可能非常大, 可以用简洁的方法来代表它, 特别是使我们能够高效查点和计数的方法, 可能非常有用。 一个简单但有力的方法就是使用联盟和卡通制产品来解分解 $\ operatorname{Hom} (macal{A},\ mathcal{B} $( macal{B} $) 问题。 自从设定了 $ colternal 的大小 范围以来, 将数据结构, 称为 davadmillaltium 的缩缩缩缩缩缩缩缩缩 结构, 如果我们的缩缩缩缩缩缩缩缩缩的图,, 我们的缩缩的缩的缩的缩的缩缩缩的缩的缩的缩在了。