Rounding Semidefinite Programming Hierarchies via Global Correlation
Boaz Barak, Prasad Raghavendra, David Steurer · 2011
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 in- put 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 3, a rounding algorithm for r rounds of the Lasserre SDP hierarchy for 3 that obtains an integral solution which is at most ε worse than the relaxation's value (normalized to lie in [0, 1]), as long as r >; k·rank≥θ(3)/ poly(ε), where k is the alphabet size of J, θ = poly(ε/k), and rank≥θ(J) denotes the number of eigenvalues larger than θ in the normalized adjacency matrix of the constraint graph of J. In the case that J is a Unique Games instance, the threshold θ is only a polynomial in ε, and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for 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 nolO(r)constraints specified by the rthlevel of the Lasserre hierarchy, and in some cases r rounds of our program can be evaluated in time 2O(r)poly(n).