The Complexity of Homomorphism Reconstruction Revisited

TL;DR

This paper investigates the computational complexity of reconstructing graphs from homomorphism counts, establishing hardness results and identifying tractable cases based on input encoding and graph structure.

Contribution

It proves NEXP-hardness and -completeness for the problem depending on input encoding, and shows polynomial-time solvability for unary counts with bounded star graphs.

Findings

NEXP-hardness when counts are binary

-completeness when counts are unary

Polynomial-time solution for unary counts with bounded star graphs

Abstract

We revisit the algorithmic problem of reconstructing a graph from homomorphism counts that has first been studied in (B\"oker 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 $\Sigma_2^p$-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.