The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into $\ell_1$ (1305.4581v1)

Abstract: In this paper, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into $\ell_1$ with constant distortion. We show that for an arbitrarily small constant $\delta> 0$, for all large enough $n$, there is an $n$-point negative type metric which requires distortion at least $(\log\log n)^{1/6-\delta}$ to embed into $\ell_1.$ Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot, establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (non-uniform) Sparsest Cut problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for Sparsest Cut. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of Unique Games to an integrality gap instance of Sparsest Cut. This enables us to prove a $(\log \log n)^{1/6-\delta}$ integrality gap for Sparsest Cut, which is known to be equivalent to the metric embedding lower bound.

• The paper disproves the ℓ₂ to ℓ₁ embeddability conjecture by constructing n-point negative type metrics with distortion at least (log log n)^(1/6−δ).
• It establishes strong integrality gap instances for Balanced Edge-Separator and Sparsest Cut, highlighting the limitations of SDP relaxations.
• The study leverages PCP reductions and the Unique Games Conjecture to connect computational complexity with metric embeddings, setting new hardness thresholds.

An Overview of "The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into ℓ₁"

This paper by Subhash A. Khot and Nisheeth K. Vishnoi offers a comprehensive analysis that bridges various theoretical aspects of computational complexity, particularly focusing on the Unique Games Conjecture (UGC), integrality gaps for certain cut problems, and the embeddability of negative type metrics into ℓ₁ spaces.

Key Contributions and Findings

1. Disproval of the (ℓ₂, ℓ₁, O(1))-Conjecture: A significant portion of the paper is dedicated to disproving the conjecture proposed by Goemans and Linial regarding the embeddability of negative type metrics into ℓ₁ spaces with a constant distortion. The authors achieve this by constructing an n-point negative type metric that exhibits a distortion of at least (log log n) 1/6−δ, challenging previous understandings.
2. Integrality Gap for Balanced Edge-Separator and Sparsest Cut: The paper constructs an integrality gap instance for the non-uniform Balanced Edge-Separator problem, which directly translates into a similar result for the Sparsest Cut problem. This result underscores the limitations of semi-definite programming (SDP) relaxations in approximating these problems.
3. Relationship between UGC and Metric Embeddings: The authors draw an innovative connection between the UGC and metric embeddings by showing that the conjecture can be leveraged to construct super-constant hardness results and integrality gap instances. This connection is pivotal in proving the non-constant distortion lower bound for embedding negative type metrics into ℓ₁.
4. PCP Reduction and UGC-Hardness: The authors present a PCP (probabilistically checkable proof) reduction from Unique Games to the Balanced Edge-Separator, showcasing that such cut problems are NP-hard to approximate within any constant factor assuming UGC. The reduction also provides a framework to generate powerful integrality gap instances.

Technical Insights

• Metric Space Analysis: Critical to the paper's results is the innovative use of metric space theories. They cleverly select n-point metrics that resist embedding into ℓ₁ with small distortion, leveraging Fourier analysis and group norms.
• Fourier Analytical Techniques: The paper applies advanced Fourier analytical tools to establish quantitative bounds on how functions with specific Fourier spectra behave, crucial for proving non-embeddability results.
• Complexity Theory and SDP: By weaving concepts from complexity theory, such as the UGC and integrations of PCP theorems with SDP constraints, the authors demonstrate significant limitations in approximating certain optimization problems.

Implications and Future Work

The implications of this research extend beyond disproving conjectures. It sets a foundation for further inquiry into:

• Algorithmic Limitations: Understanding the limitations of current algorithmic approaches, especially those based on SDP relaxations.
• UGC's Role in Complexity Theory: Continued exploration of how the UGC can inform and perhaps unify different complexity hardness results.
• Embedding Theory: Advancing the paper of metric embeddings, particularly in relation to other spaces beyond ℓ₁, which could unlock new directions in computational geometry and optimization problem design.

Conclusion

By bridging the UGC to key optimization problems and metric geometry, this paper pushes the boundary of what is known in complexity theory and combinatorial optimization. It challenges long-held assumptions, opening paths for future theoretical advancements and deeper insights into algorithmic approximations and their limits.