On the Fine-Grained Complexity of Log-Approximate CVP and Max-Cut (Classical and Quantum)

Core Concepts

This research paper presents a novel linear-sized reduction from the Maximum Cut Problem (Max-Cut) to the Closest Vector Problem (CVP), establishing the fine-grained hardness of approximating CVP and revealing significant barriers to proving the fine-grained complexity of Max-Cut using traditional methods.

Abstract

• Bibliographic Information: Huang, J. A., Ko, Y. K., & Wang, C. (2024). On the (Classical and Quantum) Fine-Grained Complexity of Log-Approximate CVP and Max-Cut. arXiv preprint arXiv:2411.04124.

• Research Objective: This paper investigates the fine-grained complexity of the approximate Closest Vector Problem (CVP) and the Maximum Cut Problem (Max-Cut), aiming to establish stronger lower bounds for CVP and explore the relationship between the complexity of these two problems.

• Methodology: The authors develop a linear-sized reduction from the (1−ε, 1−εc)-gap Max-Cut problem to the γ-approximate CVP problem (γ-CVP) under any finite ℓp-norm. They leverage this reduction to translate lower bounds from Max-Cut to CVP and investigate the implications for the fine-grained complexity of both problems.

• Key Findings:

  • The paper presents a linear-sized reduction from (1−ε, 1−εc)-gap Max-Cut to γ-CVP{0,1}p, implying that any sub-exponential time algorithm for o(√log n1/p)-Approximate CVP in any finite ℓp-norm would lead to a faster sub-exponential time algorithm for Max-Cut than currently known.
  • The reduction, combined with existing results, demonstrates that there are no fine-grained reductions from k-SAT to Max-Cut with one-sided error, nor non-adaptive fine-grained reductions with two-sided error, unless the polynomial hierarchy collapses.
  • The authors establish that there are no polynomial-sized, non-adaptive, quantum polynomial-time reductions from k-SAT to CVP2 with two-sided error unless NP is a subset of pr-QSZK.
  • The paper introduces faster classical (O(2n/2+O(ε1−c))-time) and quantum (O(2n/3+O(ε1−c))-time) algorithms for (1−ε, 1−εc)-gap Max-Cut.

• Main Conclusions: The findings suggest that Max-Cut and γ-CVP2 likely belong to a distinct fine-grained complexity class separate from k-SAT. The reduction from Max-Cut to CVP opens new avenues for exploring the hardness of CVP, while the barriers identified pose challenges for proving the fine-grained complexity of Max-Cut using SETH or QSETH.

• Significance: This research significantly advances the understanding of the fine-grained complexity of CVP and Max-Cut, particularly in the context of quantum computing. The results have implications for the security of lattice-based cryptography and highlight the unique challenges posed by Max-Cut in fine-grained complexity theory.

• Limitations and Future Research: The authors identify open questions regarding the possibility of extending the hardness results for γ-CVP2 to larger approximation factors, the application of these results to the Shortest Vector Problem, and the potential of utilizing the full power of γ-CVPp for stronger lower bounds. Further research is needed to explore the nature of the fine-grained complexity class encompassing Max-Cut and γ-CVP2 and to formulate appropriate conjectures for this class.

Stats

Quotes

Key Insights Distilled From

On the (Classical and Quantum) Fine-Grained Complexity of Log-Approximate CVP and Max-Cut

by Jeremy Ahren... at arxiv.org 11-07-2024

https://arxiv.org/pdf/2411.04124.pdf

Deeper Inquiries

Could the relationship between Max-Cut and CVP be leveraged to develop new techniques for proving the Unique Games Conjecture?

This is a very interesting open question with no easy answers. The paper presents a novel fine-grained reduction from Max-Cut to CVP, establishing a close connection between these two problems. However, leveraging this connection to prove the Unique Games Conjecture (UGC) is not straightforward and faces several challenges:
Challenges:

Approximation Factor Barrier: The reduction works for specific approximation factors of Max-Cut and CVP. The UGC requires proving hardness for a specific regime of approximation factors (1-ε, 1-C√ε) for Unique Games, which doesn't necessarily align with the achievable parameters of the Max-Cut to CVP reduction. Bridging this gap in approximation factors is crucial.
Beyond Binary Labels: Max-Cut is a special case of Unique Games with only two labels. The UGC, in its general form, deals with Unique Games instances with larger, possibly non-constant, label sizes. The techniques used in the reduction might not easily generalize to handle these more complex instances.
From Hardness of Approximation to NP-hardness: The paper focuses on fine-grained complexity, aiming to establish tight exponential lower bounds. The UGC is a statement about NP-hardness, which is a different, though related, type of hardness. Even if we could leverage the reduction to prove strong inapproximability results for Max-Cut, translating those into NP-hardness for Unique Games might require significant new ideas.
Potential Avenues:

New Reduction Techniques: Exploring variations or generalizations of the existing reduction might lead to progress. For example, investigating reductions from Unique Games with larger label sizes to variants of CVP could be fruitful.
Combining with Other Tools: The UGC has deep connections to other areas of theoretical computer science, such as semidefinite programming and the theory of expansion. Combining insights from the Max-Cut to CVP reduction with techniques from these areas might lead to new approaches.
Understanding the Limits of the Reduction: Studying the limitations of the current reduction and identifying the exact barriers preventing its application to the UGC could provide valuable insights. This could guide the search for new reductions or alternative approaches.
In conclusion, while the relationship between Max-Cut and CVP is promising, directly applying it to prove the UGC is challenging. Further research is needed to explore whether this connection can be strengthened or combined with other techniques to make progress on this long-standing open problem.

What if we relax the requirement of linear-sized reductions? Could we then establish fine-grained reductions from k-SAT to Max-Cut?

Relaxing the requirement of linear-sized reductions does not immediately resolve the barriers to establishing fine-grained reductions from k-SAT to Max-Cut. Here's why:

"No-Go" Results Still Apply: The paper highlights "no-go" results (Theorems 5.3, 5.4, 8.3) that pose significant obstacles to fine-grained reductions from k-SAT to Max-Cut. These results don't solely rely on the size of the reduction but exploit inherent structural properties of these problems and their relationship to CVP. Even with polynomial-sized reductions, these structural limitations would likely persist.
Approximation Factor Preservation: A key aspect of fine-grained reductions is preserving the "gap" between YES and NO instances. The known reductions from k-SAT to CVP struggle to maintain large approximation factors, and this issue would likely carry over to Max-Cut. A polynomial-sized reduction might further exacerbate this problem.
Underlying Complexity Assumption: The paper suggests that Max-Cut and CVP might belong to a different fine-grained complexity class than k-SAT. This implies that there might be an inherent gap in their time complexities, making it difficult to find a reduction that yields meaningful lower bounds for Max-Cut based on the hardness of k-SAT, regardless of the reduction size.
Alternative Perspectives:

Beyond Fine-Grained Reductions: While fine-grained reductions might be challenging, exploring other types of reductions, such as those based on average-case complexity or parameterized complexity, could be fruitful.
Refining Complexity Classes: The "no-go" results suggest a need for a more refined classification of fine-grained complexity classes. Investigating the precise complexity of Max-Cut and its relationship to other problems within and beyond the class seemingly containing CVP could provide valuable insights.
In summary, simply relaxing the size constraint on reductions is unlikely to overcome the fundamental barriers to establishing fine-grained reductions from k-SAT to Max-Cut. Exploring alternative reduction techniques or focusing on a more nuanced understanding of their respective complexity classes might be more promising directions for future research.

How does the fine-grained complexity landscape change if we consider other computational models beyond classical and quantum computing?

Expanding the computational model beyond classical and quantum computing significantly alters the fine-grained complexity landscape, introducing both exciting possibilities and new challenges:
New Models, New Possibilities:

Unconventional Speedups: Models like adiabatic quantum computing, topological quantum computing, or even hypothetical super-Turing models could potentially offer speedups not achievable classically or with standard quantum algorithms. This could lead to drastically different complexity classes and potentially even render some classically hard problems efficiently solvable.
Model-Specific Hardness: Conversely, certain problems might be inherently difficult for specific models due to their underlying computational primitives. This could lead to a much richer classification of problems based on their hardness across different models.
New Reduction Techniques: New computational models often come with novel algorithmic paradigms and computational primitives. This could inspire new reduction techniques, potentially circumventing the limitations faced in classical and quantum settings.
Challenges and Open Questions:

Defining Fine-Grained Complexity: The notion of "fine-grained" itself might need to be redefined for each model, considering its specific capabilities and limitations. For instance, what constitutes a "small" difference in time complexity could vary significantly.
Lack of Lower Bound Techniques: Proving lower bounds in computational complexity is notoriously difficult, and this challenge extends to new models. Developing techniques for proving conditional and unconditional lower bounds in these settings is crucial for a meaningful understanding of their power.
Connecting to Existing Landscape: A key question is how the complexity classes and relationships established in these new models relate to the existing classical and quantum complexity landscape. Understanding these connections is essential for a unified view of computational complexity.
Examples:

Adiabatic Quantum Computing: While theoretically equivalent to standard quantum computing, adiabatic algorithms often exhibit different complexity profiles. Problems like quantum annealing, naturally suited for this model, might have different fine-grained complexities.
Biological Computing: Using DNA or other biological systems for computation introduces unique constraints and possibilities. Understanding the fine-grained complexity of problems in such models is an active area of research.
In conclusion, exploring fine-grained complexity in the context of new computational models is a nascent but rapidly developing field. It offers the potential to uncover new complexity classes, refine our understanding of computational hardness, and potentially even bridge the gap between different models. However, it also presents significant challenges in defining appropriate notions of complexity, developing lower bound techniques, and connecting to the existing complexity landscape.