Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where one would like to answer queries or classify graphs solely based on the representation of a graph $G$ as a finite vector of homomorphism counts from some fixed finite set of graphs to $G$. We study the computational complexity of the arguably most fundamental computational problem associated to these representations, the homomorphism reconstructability problem: given a finite sequence of graphs and a corresponding vector of natural numbers, decide whether there exists a graph $G$ that realises the given vector as the homomorphism counts from the given graphs. We show that this problem yields a natural example of an $\mathsf{NP}^{#\mathsf{P}}$-hard problem, which still can be $\mathsf{NP}$-hard when restricted to a fixed number of input graphs of bounded treewidth and a fixed input vector of natural numbers, or alternatively, when restricted to a finite input set of graphs. We further show that, when restricted to a finite input set of graphs and given an upper bound on the order of the graph $G$ as additional input, the problem cannot be $\mathsf{NP}$-hard unless $\mathsf{P} = \mathsf{NP}$. For this regime, we obtain partial positive results. We also investigate the problem's parameterised complexity and provide fpt-algorithms for the case that a single graph is given and that multiple graphs of the same order with subgraph instead of homomorphism counts are given.

翻译：近年来，通过同态计数表示图催生了同态不可区分性的优美理论。此外，同态计数在数据库理论和机器学习中具有重要应用前景——在这些领域中，人们希望仅基于图$G$的表示（即从某个固定有限图集合到$G$的同态计数构成的有限向量）来回答查询或对图进行分类。我们研究这些表示中最基本的计算问题——同态可重构性问题的计算复杂性：给定一个有限图序列及对应的自然数向量，判定是否存在图$G$能实现该向量作为从给定图到$G$的同态计数。我们证明该问题是$\mathsf{NP}^{#\mathsf{P}}$-难问题的自然实例，当限制输入图为有限集且给定图具有有界树宽时，或当限制输入图为有限集时，该问题仍可能是$\mathsf{NP}$-难的。进一步证明，当限制输入图为有限集并额外给定图$G$的阶数上界时，除非$\mathsf{P} = \mathsf{NP}$，否则该问题不能是$\mathsf{NP}$-难的。在此情形下，我们获得了部分肯定结果。我们还研究了该问题的参数化复杂性，并针对单输入图情形以及给定多个同阶图（以子图计数而非同态计数）的情形给出了固定参数可解算法。