The Complexity of Homomorphism Reconstruction Revisited

TL;DR

This work precisely maps the complexity of reconstructing a graph from homomorphism counts, showing the binary-encoded instance is $NEXP$-complete and the unary-encoded instance is $oldsymbol{ ext{Σ}_2^p}$-complete. It develops a circuit-to-graph gadget framework to establish hardness, including a SuccinctClique-based reduction for the coloured version and a gadget-based uncolouring reduction to the standard HomRec problem. On the tractable side, it proves that StarHomRec (and related StarSubRec) is solvable in time $m^{O( ext{ℓ}^2)}$ by exploiting star-count dependencies on degree sequences and a dynamic-programming/Havel–Hakimi construction to realize feasible graphs. Overall, the paper delineates a sharp boundary: general instances are intractable under standard encodings, while structured star-count instances admit efficient, constructive solutions with clear algorithmic pipelines. The results have implications for graph embeddings, reconstruction questions, and the design of latent-space methods based on homomorphism counts.

Abstract

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.

Figures (7)

• Figure 1: The graph $F_{\text{str}_1}$.
• Figure 2: The graph $F_{\text{str}_2}$.
• Figure 4: The graph $G_S$. If the gadgets for $S$ encode a $k$-clique of $G$, all vertices coloured $\mathsf{C}_o$ are adjacent to the "true" value gadget vertex next to $\top$.
• Figure 5: $F$
• Figure 6: $G$

Theorems & Definitions (41)

• theorem 1
• theorem 2
• theorem 3
• corollary 1
• theorem 4
• lemma 1
• proof
• proof : Proof of \ref{['thm:circuitsat']}.
• definition 1: GalperinW83
• lemma 2