Sum-of-Squares Lower Bounds for Sparse Independent Set

Abstract

Given a graph and an integer k, Densest k-Subgraph is the algorithmic task of finding the subgraph on k vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an O(n 1/4+ε )-factor approximation (for any ε > 0) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an O(n 1/4−ε )-factor approximation. In the average case, Densest k-Subgraph is a prototypical noisy inference task which is conjectured to exhibit a statistical-computational gap.In this work, we provide the strongest evidence yet of hardness for Densest k-Subgraph by showing matching lower bounds against the powerful Sum-of-Squares (SoS) algorithm, a meta-algorithm based on convex programming that achieves stateof-art algorithmic guarantees for many optimization and inference problems. For k ≤ n

On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices

“…Graph matrices are a type of matrix which plays a key role in analyzing the sum of squares hierarchy on average-case problems [4,6,7,9,11] and is also useful for analyzing other methods involving higher moments [3] (for background on graph matrices, see [3]). However, the behavior of graph matrices is only partially understood.…”

“…However, the behavior of graph matrices is only partially understood. At present, we have rough norm bounds on graph matrices [3,7,12] which are sufficient for many applications, but we can hope to analyze graph matrices much more precisely.…”