The computational hardness of counting in two-spin models on d-regular graphs (1203.2602v1)

Abstract: The class of two-spin systems contains several important models, including random independent sets and the Ising model of statistical physics. We show that for both the hard-core (independent set) model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on d-regular graphs when the model has non-uniqueness on the d-regular tree. Together with results of Jerrum--Sinclair, Weitz, and Sinclair--Srivastava--Thurley giving FPRAS's for all other two-spin systems except at the uniqueness threshold, this gives an almost complete classification of the computational complexity of two-spin systems on bounded-degree graphs. Our proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting "free energy density" which coincides with the (non-rigorous) Bethe prediction of statistical physics. We use this result to characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs, which then become the basic gadgets in a randomized reduction to approximate MAX-CUT. Our approach is novel in that it makes no use of the second moment method employed in previous works on these questions.

• The paper establishes a near-complete classification of partition function approximation complexity for two-spin models on d-regular graphs.
• It proves that beyond the uniqueness threshold, no FPTAS exists for models like the hard-core and anti-ferromagnetic Ising unless NP equals RP.
• Novel reduction techniques using expander graph gadgets validate the convergence of the normalized log-partition function to the Bethe free energy density.

Review of "The Computational Hardness of Counting in Two-Spin Models on $d$-Regular Graphs"

This paper by Allan Sly and Nike Sun addresses the computational complexity associated with counting problems for two-spin models on $d$-regular graphs, a topic of significant importance in statistical physics and theoretical computer science. Two-spin systems encompass key models such as the Ising and hard-core models, which have broad applications ranging from magnetism to network theory.

Summary of Main Contributions

The authors achieve a near-complete classification of the computational complexity for approximating the partition function of two-spin systems on bounded-degree graphs. Specifically, they prove that it is hard to approximate this partition function or to sample from such models on $d$-regular graphs when the model exhibits non-uniqueness on the $d$-regular tree. This classification hinges upon the uniqueness threshold, which demarcates regimes where approximation becomes computationally infeasible.

Key Theoretical Insights

1. Normalized Log-Partition Function: The authors prove that for any two-spin system on bipartite, locally tree-like graphs, the normalized log-partition function converges to a limiting "free energy density". This density aligns with the Bethe prediction, a non-rigorous approximation from statistical physics, thus cementing its validity in this context.
2. Hardness Results: By focusing on the uniqueness threshold, they show the limits of viability for Fully Polynomial-Time Approximation Schemes (FPTAS) in two-spin systems. For both the hard-core model and the anti-ferromagnetic Ising model, it is established that beyond certain parameter thresholds, no FPTAS exists unless NP = RP.
3. Reduction Techniques: A novel proof strategy is employed that circumvents the second moment method. Using expander graphs as gadgets in a randomized reduction, the paper provides a scaffold to demonstrate that even roughly approximating the partition function within a factor of $e^{cn}$ is hard in these regimes.

Implications and Future Directions

The implications of this research are multifaceted. Practically, this result indicates where computational efforts might become intractable and highlights where efficient algorithms can be confidently used. Theoretically, it provides a foundation for understanding phase transitions in spin systems and may inspire similar computational complexity analyses for other kinds of graphical models.

One future direction could involve extending these methods to analyze multi-spin or more complex systems, particularly those with non-binary states or those that do not exhibit symmetry such as bipartiteness. Another avenue is exploring algorithms near the uniqueness threshold, where approximation is still feasible, to better understand the limits of current techniques in statistical physics and machine learning.

In conclusion, the paper significantly advances the understanding of the computational landscape of two-spin models, situating the problem within the larger context of phase transitions and computational intractability. It lays a robust groundwork for future research at the intersection of computer science, physics, and applied mathematics, where the paper of spin systems continues to have profound theoretical and practical implications.