On the Unique Games Conjecture

Khot, Subhash

doi:10.1109/sfcs.2005.61

Cited by 50 publications

(50 citation statements)

References 102 publications

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“…From the point of view of worst-case approximation in theoretical computer science, semidefinite programming [and the so-called sum-of-squares hierarchy (15-18)] seems to provide an "off the shelf" hierarchy of optimal algorithms for a large class of problems (19) [this is related to the famous unique games conjecture (20)]. In fact, semidefinite programming has been used to highlight another phase transition (10) in the binary stochastic block model, below which it is information theoretically impossible to exactly recover the community memberships of all of the nodes (although it may be possible to outperform a random guess).…”

Section: Semidefinite Programmingmentioning

confidence: 99%

Inference on graphs via semidefinite programming

Bandeira¹

2016

Proc. Natl. Acad. Sci. U.S.A.

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Section: Semidefinite Programmingmentioning

confidence: 99%

Inference on graphs via semidefinite programming

Bandeira¹

2016

Proc. Natl. Acad. Sci. U.S.A.

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“…The conjecture has inspired a remarkable body of work since its formulation. Under UGC, many of the known algorithms in approximation are proven to be tight (for an excellent survey on this topic, see Khot & Vishnoi, 2005). For instance, under the UGC, the Vertex Cover problem is NP-hard to approximate within a factor of 2 − , for any > 0 (Khot & Regev, 2008).…”

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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“…In addition, to obtain the previous order, Alice and Bob can share the maximally entangled state and Alice can perform some simple von Neumann measurements on her party. Our approach is based on a modification of the Khot-Vishnoi (KV) game [23] and the results in [21] to obtain an asymmetric version of it, which will work in our setting. Moreover, our result is near optimal (up to a logarithm factor) in terms of Alice's outputs and the Hilbert space dimension (see Propositions 1 and 2).…”

Section: Introductionmentioning

confidence: 99%

Large bipartite Bell violations with dichotomic measurements

Palazuelos

Yin

2015

Phys. Rev. A

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In this paper, we introduce a simple and natural bipartite Bell scenario by considering the correlations between two parties defined by general measurements in one party and dichotomic ones in the other. We show that unbounded Bell violations can be obtained in this context. Since such violations cannot occur when both parties use dichotomic measurements, so far our setting can be considered as the simplest one where this phenomenon can be observed. Our example is essentially optimal in terms of the outputs and the Hilbert space dimension.

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On the Unique Games Conjecture

Abstract: This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry.

Cited by 50 publications

References 102 publications

Inference on graphs via semidefinite programming

Inference on graphs via semidefinite programming

Inapproximability of Treewidth and Related Problems

Large bipartite Bell violations with dichotomic measurements

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