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$\mathcal{A}$和$\mathcal{B}$之间的同态是一个被广泛研究的问题，具有众多应用。由于所有同态构成的集合$\operatorname{Hom}(\mathcal{A},\mathcal{B})$可能非常庞大，因此找到一种能够简洁表示该集合的方法（尤其是支持高效枚举与计数的方法）极具价值。一种简单而有效的实现方式是通过并集和笛卡尔积分解$\operatorname{Hom}(\mathcal{A},\mathcal{B})$。这类数据结构被称为d-表示，由Olteanu和Zavodny在数据库理论背景下提出。他们的研究同时表明：若左侧结构$\mathcal{A}$的树宽有界，则可在多项式时间内找到规模为多项式的d-表示。我们证明，对于有界元数的结构而言，这一结论是最优的：当树宽无界时，存在实例使得任何d-表示的规模都是超多项式的。在此过程中，我们发展了证明d-表示规模下界的工具，特别地，定义了适用于该上下文的归约概念，并证明了图中所有$k$-团（$k$-cliques）的d-表示规模的几乎紧致下界。