# Subexponential Algorithms for dto-1 Two-Prover Games and for Certifying Almost Perfect Expansion

@inproceedings{Steurer2010SubexponentialAF, title={Subexponential Algorithms for dto-1 Two-Prover Games and for Certifying Almost Perfect Expansion}, author={David Steurer}, year={2010} }

A question raised by the recent subexponential algorithm for Unique Games (Arora, Barak, Steurer, FOCS 2010) is what other “hard-looking” problems admit good approximation algorithms with subexponential complexity. In this work, we give such an algorithm for d-to-1 two-prover games, a broad class of constraint satisfaction problems. Our algorithm has several consequences for Khot’s d-to-1 Conjectures. We also give a related subexponential algorithm for certifying that small sets in a graph have…

## 18 Citations

### Subexponential Algorithms for Unique Games and Related Problems

- Computer Science, Mathematics2010 IEEE 51st Annual Symposium on Foundations of Computer Science
- 2010

A sub exponential time approximation algorithm for the Unique Games problem that is exponential in an arbitrarily small polynomial of the input size, n, and shows that for every $\epsilon>0$ and every regular $n$-vertex graph~$G, one can break into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most n eigenvalues larger than $1-\eta.

### Towards a proof of the 2-to-1 games conjecture?

- MathematicsElectron. Colloquium Comput. Complex.
- 2016

A polynomial time reduction from gap-3LIN to label cover with 2-to-1 constraints is presented and an NP-hardness gap of 1/2−ε vs. ε for unique games is implied, assuming a certain combinatorial hypothesis on the Grassmann graph.

### On the proof of the 2-to-2 Games Conjecture

- MathematicsCurrent Developments in Mathematics
- 2019

This article gives an overview of the recent proof of the 2-to-2 Games Conjecture in [68, 39, 38, 69] (with additional contributions from [75, 18, 67]). The proof requires an understanding of…

### An Exposition of Dinur-Khot-Kindler-Minzer-Safra’s Proof for the 2-to-2 Games Conjecture∗

- Mathematics
- 2018

In a recent paper [KMS18] proved a certain combinatorial hypothesis, which completed the proof of (the imperfect completeness variant of) Khot’s 2-to-2 games conjecture (based on an approach…

### A new point of NP-hardness for unique games

- Computer ScienceSTOC '12
- 2012

For these c, this is the first hardness result showing that it helps to take the alphabet size larger than 2 and the NP-hardness reductions are quasilinear-size and thus show nearly full exponential time is required, assuming the ETH.

### Hypercontractivity, sum-of-squares proofs, and their applications

- Mathematics, Computer ScienceSTOC '12
- 2012

Reductions between computing the 2->4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory and the study of Khot's Unique Games Conjecture are shown.

### New Tools for Graph Coloring

- Computer ScienceAPPROX-RANDOM
- 2011

This algorithm is inspired by recent work of Barak, Raghavendra, and Steurer on using Lasserre Hierarchy for unique games and can be used to show that known integrality gap instances for SDP relaxations like strict vector chromatic number cannot survive a few rounds of Lasserr lifting.

### Label Cover Reductions for Unconditional Approximation Hardness of Constraint Satisfaction

- Computer Science, Mathematics
- 2014

It is shown that all parities except one are positively useless and two protocols do the trick, and the techniques may prove useful in further analyzing the approximability of CSPs without negations.

### Analytical approach to parallel repetition

- Computer ScienceSTOC
- 2014

Improved bounds for few parallel repetitions of projection games are shown, showing that Raz's counterexample to strong parallel repetition is tight even for a small number of repetitions, and a short proof for the NP-hardness of label cover(1, δ) for all δ > 0, starting from the basic PCP theorem.

### C C ] 21 M ay 2 01 2 Hypercontractivity , Sum-of-Squares Proofs , and their Applications

- Computer Science
- 2018

Reductions between computing the 2 → 4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory and known algorithms for the quantum separability problem imply a non-trivial additive approxi mation for the 2→ 4 norm.

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