Rounding Semidefinite Programming Hierarchies via Global Correlation

“…The idea of speeding up LP and SDP hierarchies for specific problems has been investigated in a series of previous works [dlVK07,BRS11,GS12], which shows that with respect to local analyses of the sum-of-squares algorithm it is sometimes possible to improve the running time from n O(d) to 2 O(d) · n O(1) . However, the scopes and strategies of these works are completely different from ours.…”

Section: Related Workmentioning

confidence: 99%

Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors

Hopkins

Schramm

Shi

et al. 2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems, the best known guarantees are based on the sum-of-squares method. We develop new algorithms inspired by analyses of the sum-of-squares method. Our algorithms achieve the same or similar guarantees as sum-ofsquares for these problems but the running time is significantly faster.For the planted sparse vector problem, we give an algorithm with running time nearly linear in the input size that approximately recovers a planted sparse vector with up to constant relative sparsity in a random subspace of R n of dimension up toΩ( √ n). These recovery guarantees match the best known ones of Barak, Kelner, and Steurer (STOC 2014) up to logarithmic factors.For tensor decomposition, we give an algorithm with running time close to linear in the input size (with exponent ≈ 1.086) that approximately recovers a component of a random 3-tensor over R n of rank up toΩ(n 4/3 ). The best previous algorithm for this problem due to Ge and Ma (RANDOM 2015) works up to rankΩ(n 3/2 ) but requires quasipolynomial time.

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“…In particular, recent results (Arora, Barak, & Steurer, 2010;Barak, Raghavendra, & Steurer, 2011;Guruswami & Sinop, 2011) give subexponential-time algorithms for small set expansion. Still despite this recent progress providing evidence against the SSE conjecture, it remains open.…”

Section: The Small Set Expansion Conjecturementioning

confidence: 99%

Inapproximability of Treewidth and Related Problems

Austrin

Pitassi

et al. 2014

jair

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Graphical models, such as Bayesian Networks and Markov networks play an important role in artificial intelligence and machine learning. Inference is a central problem to be solved on these networks. This, and other problems on these graph models are often known to be hard to solve in general, but tractable on graphs with bounded Treewidth. Therefore, finding or approximating the Treewidth of a graph is a fundamental problem related to inference in graphical models. In this paper, we study the approximability of a number of graph problems: Treewidth and Pathwidth of graphs, Minimum Fill-In, OneShot Black (and Black-White) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as Minimum Cut Linear Arrangement and Interval Graph Completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are NP-hard to approximate to within any constant factor in polynomial time.

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Rounding Semidefinite Programming Hierarchies via Global Correlation

Cited by 114 publications

References 35 publications

Approximating independent sets in sparse graphs

Approximating independent sets in sparse graphs

Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors

Inapproximability of Treewidth and Related Problems

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