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Computational hardness of detecting graph lifts and certifying lift-monotone properties of random regular graphs (2404.17012v1)

Published 25 Apr 2024 in cs.CC, cs.DS, math.CO, and math.PR

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.

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Authors (2)
  1. Dmitriy Kunisky (33 papers)
  2. Xifan Yu (5 papers)
Citations (1)
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Summary

  • The paper conjectures that polynomial-time algorithms cannot distinguish random regular graphs from noisy random lifts of other graphs, a core computational hardness assumption.
  • To support the conjecture, the authors use the local statistics hierarchy of semidefinite programs to establish lower bounds, implying polynomial-time algorithms based on this method cannot make the distinction.
  • The conjecture implies that polynomial-time algorithms cannot efficiently certify optimal or improved bounds for properties like the maximum cut or independent set size of random regular graphs.

Computational Hardness of Detecting Graph Lifts and Certifying Lift-Monotone Properties of Random Regular Graphs

The paper explores the computational hardness of distinguishing random lifts of graphs from large random dd-regular graphs under certain noise models. The authors present a conjecture related to the stochastic properties of graph lifts and provide supporting evidence through lower bounds on the local statistics hierarchy of hypothesis testing semidefinite programs. The significance of this work lies in its implications for various optimization problems on random regular graphs, including maximum cuts, independent sets, and chromatic numbers.

Key Contributions and Results

  1. Detection Hardness Conjecture: The authors conjecture that it is infeasible for any polynomial-time algorithm to differentiate between a random dd-regular graph and a noisy random lift of a Ramanujan dd-regular base graph. This premise holds for both bipartite and non-bipartite cases, provided sufficient small noise corruptions are added to the lifts.
  2. Local Statistics Bound: To support their conjecture, the authors establish lower bounds using the local statistics hierarchy of semidefinite programs, drawing parallels with sum-of-squares hierarchies. They argue that polynomial-time algorithms using these methodologies cannot distinguish the graph distributions in question, reinforcing the proposed conjecture's validity.
  3. Implications for Graph Properties: The conjecture's acceptance would imply that no polynomial-time algorithm can efficiently certify tight bounds for several graph properties, specifically:
    • Maximum Cut: It is shown that no polynomial-time method can certify maximum cuts of random 3- or 4-regular graphs more accurately than naive spectral bounds.
    • Independent Set: The maximum independent set size of random 3- or 4-regular graphs similarly resists certification improvements.
    • Chromatic Number: For random 7-regular graphs, the conjecture suggests that the true chromatic number remains elusive to polynomial-time verification, beyond simple spectral techniques.
  4. Robust Certification Techniques: The authors provide a theoretical framework for quiet planting in random graph lifts, guiding practitioners on how subtle modifications can undermine seemingly robust certification strategies. The presented methodology extends beyond mere lift-monotonicity, proposing applicability towards broader certification challenges.

Implications and Future Directions

This paper's implications and methodologies challenge standard expectations in graph theory and computational complexity. By hypothesizing intrinsic hardness in graph detection and property certification tasks, it encourages a reevaluation of existing polynomial-time algorithms and their applicability to random graph instances. Notably, this work prompts further investigation into the theoretical underpinnings of the local statistics hierarchy, potentially sparking advancements in recognizing fundamental limitations of algorithmic approaches.

The intriguing interplay between graph lifting, randomness, and computational challenges opens doors to future exploration. These might include:

  • Extending the dynamics of graph lifts to explore analogous phenomena in hypergraphs or directed graphs.
  • Investigating similar conjectural frameworks within other stochastic models or graph ensembles, potentially revealing widespread structural properties influencing computation.
  • Developing alternative tractable approaches, such as approximation algorithms or heuristic methods, to address tasks deemed difficult under this conjecture.

Conclusion

This paper's foundational conjecture and ensuing evidence posit significant challenges for polynomial-time algorithms in graph theory, influencing how researchers approach the paper of random regular graphs and their certifiable properties. By addressing these theoretical constraints and navigating their implications, researchers can both refine existing methods and chart new paths in computational graph theory.

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