Rounding Semidefinite Programming Hierarchies via Global Correlation (1104.4680v1)

Abstract: We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most \eps worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k\cdot\rank_{\geq \theta}(\Ins)/\poly(\e) \;, where k is the alphabet size of I, $\theta=\poly(\e/k)$, and $\rank_{\geq \theta}(\Ins)$ denotes the number of eigenvalues larger than $\theta$ in the normalized adjacency matrix of the constraint graph of $\Ins$. In the case that $\Ins$ is a \uniquegames instance, the threshold $\theta$ is only a polynomial in $\e$, and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for \emph{every} instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the $n^{O(r)}$ constraints specified by the $r^{th}$ level of the Lasserre hierarchy, and in some cases $r$ rounds of our program can be evaluated in time $2^{O(r)}\poly(n)$.

• The paper presents a novel SDP rounding algorithm for 2-CSPs that leverages graph spectral properties to tightly bound approximation loss.
• It establishes a clear link between the SDP hierarchy’s performance and the threshold rank of the constraint graph’s eigenvalues, enhancing computational insights.
• The approach challenges the Unique Games Conjecture by performing faster on specific instances and offers promising avenues for specialized optimization strategies.

Rounding Semidefinite Programming Hierarchies via Global Correlation

The research presented in this paper introduces novel methodologies for rounding vector solutions from semidefinite programming (SDP) hierarchies into integral solutions. The approach leverages a connection between these hierarchies and the spectrum of the input graph, delivering new algorithms specifically for constraint satisfaction problems with two-variable constraints (2-CSPs).

Algorithmic Advances for 2-CSPs

The paper proposes a rounding algorithm for the Lasserre SDP hierarchy that effectively transforms vector solutions into integral solutions with minimal loss relative to the original SDP relaxation value. For any 2-CSP instance, the authors demonstrate that the difference between the rounded solution's value and the SDP relaxation's value is bounded efficiently if the number of rounds $r$ exceeds a threshold determined by the rank of the constraint graph's adjacency matrix.

The SDP hierarchy-based algorithm offers enhanced performance even on specific Unique Games instances, challenging Khot's Unique Games Conjecture by restricting the set of potential hard instances. The algorithm runs faster on a natural family of instances, providing further support to those working within computational complexity theory, suggesting a nuanced understanding of which instances contribute to the conjecture's validity.

Strong Numerical Results and Implications

The authors provide a rigorous theoretical foundation, demonstrating that the hierarchy's performance is intrinsically linked to the input graph's eigenvalues. Specifically, semidefinite hierarchies are shown to be sensitive to the graph's threshold rank—a measure of the number of eigenvalues exceeding a given threshold. This relationship offers insight into which instances are computationally hard, affecting algorithm design beyond merely improving approximation factors.

Moreover, notable practical implications stem from the observation that graphs with a small number of large eigenvalues enable effective SDP rounding. Surprisingly, traditional instances known for fooling the Goemans-Williamson SDP—such as those inspired by noisy Gaussian graphs—are shown to possess low threshold rank, lending credence to the algorithm's efficiency.

Future Speculations and Theoretical Developments

The authors' approach heralds a promising direction for future research in AI and complex optimization problems. For practitioners, understanding the behavior of SDP hierarchies on graphs with varying spectral properties may drive the development of specialized algorithms tailored to leverage the spectral profiles of real-world problem instances.

Speculatively, the paper opens up avenues to explore potential equivalencies and new hierarchies that might outperform classical methods in computational problem-solving. Investigating these pathways could refine our grasp of NP-hard problems and the landscape of polynomial-time solvability.

In conclusion, while the paper refrains from making sensational claims, its contributions serve as a robust springboard for those entrenched in computational theory seeking to refine understanding and efficiency in solving complex two-variable constraint-based problems through SDP hierarchies.