We revisit the algorithmic problem of reconstructing a graph from homomorphism counts that has first been studied in (Böker et al., STACS 2024): given graphs $F_1,\ldots,F_k$ and counts $m_1,\ldots,m_k$, decide if there is a graph $G$ such that the number of homomorphisms from $F_i$ to $G$ is $m_i$, for all $i$. We prove that the problem is NEXP-hard if the counts $m_i$ are specified in binary and $Σ_2^p$-complete if they are in unary. Furthermore, as a positive result, we show that the unary version can be solved in polynomial time if the constraint graphs are stars of bounded size.

翻译：我们重新审视了从同态计数重构图的算法问题，该问题最初在（Böker等人，STACS 2024）中被研究：给定图$F_1,\ldots,F_k$及计数$m_1,\ldots,m_k$，判断是否存在图$G$，使得对所有$i$，从$F_i$到$G$的同态数量均为$m_i$。我们证明了当计数$m_i$以二进制形式给出时该问题是NEXP难的，而以一元形式给出时是$Σ_2^p$完全的。此外，作为一个正面结果，我们证明当约束图是有界大小的星图时，一元版本问题可在多项式时间内求解。