CS 264 : Beyond Worst - Case Analysis Lectures # 11 and 12 : SDP Algorithms for Semi - Random Bisection and Clique ∗
نویسنده
چکیده
Lectures #9 and #10 studied the planted bisection (a.k.a. community detection) and planted clique models, where we posited specific parameterized input distributions (parameterized by edge densities p and q within and between clusters, or by planted the clique size k), in which a “clearly optimal” solution is planted in an otherwise random graph. The goal was to identify necessary and sufficient conditions on the parameters (p − q, or k) of the distribution such that the planted solution can be recovered in polynomial time. We obtained state-of-the-art positive results for these problems, for p − q = Ω( √ logn n ) and k = Ω( √ n). We obtained these results using spectral algorithms, which compute the second eigenvector of the adjacency matrix followed by some problem-specific postprocessing. How should we feel about these results? Technically, the results are quite interesting. And in the end, the theory ends up advocating natural algorithms that are not overly tailored to the assumed input distributions, so there is hope that these algorithms could perform well much more generally. Indeed, at least for graph partitioning, spectral algorithms constitute one of the dominant paradigms in practice. Still, one can’t help but notice that our analysis of these spectral algorithms strongly exploited the assumed input distribution (e.g., through bounds on the eigenvalues of random mean-zero symmetric matrices). Can we do better? That is, can we obtain more robust versions of our recovery results, that assume much less about the underlying input distribution? The answer is yes, although we will have to up our game—spectral algorithms will no longer be sufficiently powerful, and we’ll need to rely on semidefinite-programming (SDP)-based algorithms.
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