Rounding Semidefinite Programming Hierarchies via Global Correlation

TL;DR

This paper introduces a novel rounding method for SDP hierarchies based on graph spectrum, enabling improved algorithms for 2-CSPs and Unique Games that are faster and closer to optimal than previous approaches.

Contribution

It presents a new graph-spectrum-based rounding technique for SDP hierarchies, leading to faster algorithms for 2-CSPs and Unique Games, and provides bounds on the number of rounds needed.

Findings

Achieves near-optimal solutions for 2-CSPs using fewer SDP rounds.

Provides bounds on rounds needed based on eigenvalues of the constraint graph.

Runs faster than traditional SDP hierarchies on certain instances.

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 $2^{O(r)}\poly(n)$.