Computational hardness of detecting graph lifts and certifying lift-monotone properties of random regular graphs

TL;DR

The paper investigates the computational hardness of distinguishing random regular graphs from lifts of Ramanujan graphs and derives a broad set of conditional certification lower bounds for key graph properties. By leveraging the local statistics SDP hierarchy and quiet planting techniques, it shows a statistical-to-computational gap: information-theoretic separation can exist where polynomial-time certificates cannot, conditional on a Ramanujan lift hardness conjecture. The work yields concrete consequences for maxima cuts, chromatic number, independence and domination numbers, and small-set expansions, supported by explicit Ramanujan examples that realize worst-case certificate gaps. The methodology connects lift-monotone properties with certification theory, providing a framework to transfer lift-based hardness into average-case certifiability bounds with potential broad applicability in graph algorithms and complexity.

Abstract

We introduce a new conjecture on the computational hardness of detecting random lifts of graphs: we claim that there is no polynomial-time algorithm that can distinguish between a large random $d$-regular graph and a large random lift of a Ramanujan $d$-regular base graph (provided that the lift is corrupted by a small amount of extra noise), and likewise for bipartite random graphs and lifts of bipartite Ramanujan graphs. We give evidence for this conjecture by proving lower bounds against the local statistics hierarchy of hypothesis testing semidefinite programs. We then explore the consequences of this conjecture for the hardness of certifying bounds on numerous functions of random regular graphs, expanding on a direction initiated by Bandeira, Banks, Kunisky, Moore, and Wein (2021). Conditional on this conjecture, we show that no polynomial-time algorithm can certify tight bounds on the maximum cut of random 3- or 4-regular graphs, the maximum independent set of random 3- or 4-regular graphs, or the chromatic number of random 7-regular graphs. We show similar gaps asymptotically for large degree for the maximum independent set and for any degree for the minimum dominating set, finding that naive spectral and combinatorial bounds are optimal among all polynomial-time certificates. Likewise, for small-set vertex and edge expansion in the limit of very small sets, we show that the spectral bounds of Kahale (1995) are optimal among all polynomial-time certificates.

Figures (4)

• Figure 1: The 3-regular Ramanujan graph used in the proof of Theorems \ref{['thm:max-cut']} and \ref{['thm:ind-set']} on the maximum cut and maximum independent set of 3-regular graphs, respectively. The graph is formed from a bipartite 3-regular graphs by replacing one edge (shown dashed in gray) with a pair of loops. This graph has $d = 3, |V(H)| = 12$, $\max\{|\lambda_2(H)|, |\lambda_n(H)|\} \approx 2.825 < 2.828 \approx 2\sqrt{2}$, $\mathrm{MC}_2(H) = \frac{17}{18} \approx 0.944$, and $\widehat{\alpha}^{\prime}(H) = \frac{5 + 1/2}{12} \approx 0.458$, where $\widehat{\alpha}^{\prime}$ is the modified independence number discussed in the proof in Section \ref{['sec:ind-set']}.
• Figure 2: The 4-regular Ramanujan graph used in the proof of Theorem \ref{['thm:max-cut']} on the maximum cut of 4-regular graphs. The graph is formed from the complete bipartite graph $K_{4, 4}$ by replacing two edges (shown dashed in gray) with pairs of loops. This graph has $d = 4, |V(H)| = 8, \max\{|\lambda_2(H)|, |\lambda_n(H)|\} \approx 3.236 < 3.464 \approx 2\sqrt{3}$, and $\mathrm{MC}_2(H) = \frac{14}{16} = 0.875$.
• Figure 3: The 4-regular Ramanujan graph used in the proof of Theorem \ref{['thm:ind-set']} on the maximum independent set of 4-regular graphs. This graph appears in Figure 3 of JHSWZ-2024-4RegularGraphsRigidity in a different context. This graph has $d = 4$, $|V(H)| = 7$, $\max\{|\lambda_2(H)|, |\lambda_n(H)|\} = 3 < 3.464 \approx 2\sqrt{3}$, and $\widehat{\alpha}(H) = \frac{3}{7} \approx 0.428$.
• Figure 4: The 7-regular Ramanujan graph used in the proof of Theorem \ref{['thm:chromatic']} on the chromatic number. The graph is formed by modifying the complete multipartite graph $K_{3, 3, 3}$ (a Ramanujan 6-regular graph), which appears as the induced subgraph on the "inner" nine vertices. Note that there are three pairs of repeated edges, forming a triangle on the "outer" three vertices. This graph has $d = 7, |V(H)| = 12, \max\{|\lambda_2(H)|, |\lambda_n(H)|\} \approx 3.791 < 4.898 \approx 2\sqrt{6}$, and $\chi(H) = 3$.

Theorems & Definitions (95)

• Definition 1.1: Random lift ALMR-2001-RandomLifts
• Proposition 1.2: Spectral gap of random regular graphs
• Definition 1.3: Ramanujan graph
• Proposition 1.4: Spectral gap of random lifts BC-2019-EigenvaluesRandomLifts
• Definition 1.5: Strong detection
• Conjecture 1.6: Hardness of detecting lifts; informal
• Definition 1.7: Local statistics; informal
• Remark 1.8: O'Donnell's caveat
• Theorem 1.9: Local statistics hardness; informal
• Definition 1.10: Certification