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Electronic Colloquium on Computational Complexity

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Reports tagged with Unique Games Conjecture:
TR06-088 | 9th July 2006
Per Austrin

Balanced Max 2-Sat might not be the hardest

We show that, assuming the Unique Games Conjecture, it is NP-hard to approximate Max 2-Sat within $\alpha_{LLZ}^{-}+\epsilon$, where $0.9401 < \alpha_{LLZ}^{-} < 0.9402$ is the believed approximation ratio of the algorithm of Lewin, Livnat and Zwick.

This result is surprising considering the fact that balanced instances of Max 2-Sat, i.e. ... more >>>

TR08-008 | 8th February 2008
Venkatesan Guruswami, Prasad Raghavendra

Constraint Satisfaction over a Non-Boolean Domain: Approximation algorithms and Unique-Games hardness

Revisions: 1

We study the approximability of the \maxcsp problem over non-boolean domains, more specifically over $\{0,1,\ldots,q-1\}$ for some integer $q$. We obtain a approximation algorithm that achieves a ratio of $C(q) \cdot k/q^k$ for some constant $C(q)$ depending only on $q$. Further, we extend the techniques of Samorodnitsky and Trevisan to ... more >>>

TR08-009 | 7th December 2007
Per Austrin, Elchanan Mossel

Approximation Resistant Predicates From Pairwise Independence

We study the approximability of predicates on $k$ variables from a
domain $[q]$, and give a new sufficient condition for such predicates
to be approximation resistant under the Unique Games Conjecture.
Specifically, we show that a predicate $P$ is approximation resistant
if there exists a balanced pairwise independent distribution over
more >>>

TR08-018 | 28th February 2008
Ran Raz

A Counterexample to Strong Parallel Repetition

The parallel repetition theorem states that for any two-prover game,
with value $1- \epsilon$ (for, say, $\epsilon \leq 1/2$), the value of
the game repeated in parallel $n$ times is at most
$(1- \epsilon^c)^{\Omega(n/s)}$, where $s$ is the answers' length
(of the original game) and $c$ is a universal ... more >>>

TR09-128 | 29th November 2009
Gillat Kol, Ran Raz

Locally Testable Codes Analogues to the Unique Games Conjecture Do Not Exist

The Unique Games Conjecture (UGC) is possibly the most important open problem in the research of PCPs and hardness of approximation. The conjecture is a strengthening of the PCP Theorem, predicting the existence of a special type of PCP verifiers: 2-query verifiers that only make unique tests. Moreover, the UGC ... more >>>

TR09-138 | 14th December 2009
Gillat Kol, Ran Raz

Bounds on 2-Query Locally Testable Codes with Affine Tests

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an affine test. That is, we consider LTCs over a finite field F, with codeword testers that only use tests of the form $av_i + bv_j = c$, where v is ... more >>>

TR10-063 | 12th April 2010
Venkatesan Guruswami, Yuan Zhou

Tight Bounds on the Approximability of Almost-satisfiable Horn SAT and Exact Hitting Set}

Revisions: 1

We study the approximability of two natural Boolean constraint satisfaction problems: Horn satisfiability and exact hitting set. Under the Unique Games conjecture, we prove the following optimal inapproximability and approximability results for finding an assignment satisfying as many constraints as possible given a {\em
near-satisfiable} instance.

\item ... more >>>

TR10-112 | 15th July 2010
Subhash Khot, Dana Moshkovitz

NP-Hardness of Approximately Solving Linear Equations Over Reals

In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each
equation contains at most three variables.

Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be ``non-trivial". Here is
an informal statement of our ... more >>>

TR10-177 | 16th November 2010
Venkatesan Guruswami, Prasad Raghavendra, Rishi Saket, Yi Wu

Bypassing UGC from some optimal geometric inapproximability results

Revisions: 1

The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this ... more >>>

TR11-066 | 25th April 2011
Venkatesan Guruswami, Ali Kemal Sinop

Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Quadratic Integer Programming with PSD Objectives

Revisions: 1

We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These ... more >>>

TR11-098 | 11th July 2011
Marek Karpinski, Richard Schmied, Claus Viehmann

Tight Approximation Bounds for Vertex Cover on Dense k-Partite Hypergraphs

We establish almost tight upper and lower approximation bounds for the Vertex Cover problem on dense k-partite hypergraphs.

more >>>

TR12-120 | 24th September 2012
Boaz Barak

Proof vs. Truth in Computational Complexity

Revisions: 1

In this survey, I discuss the general question of what evidence can we use to predict the answer for open questions in computational complexity, as well as the concrete evidence currently known for two conjectures: Khot's Unique Games Conjecture and Feige's Random 3SAT Hypothesis.

more >>>

TR13-125 | 11th September 2013
Venkatesan Guruswami, Euiwoong Lee

Complexity of approximating CSP with Balance / Hard Constraints

We study two natural extensions of Constraint Satisfaction Problems (CSPs). {\em Balance}-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of {\em Hard}-Max-CSP consists of {\em soft constraints} and {\em hard constraints}, and the goal is to maximize ... more >>>

TR13-184 | 23rd December 2013
Boaz Barak, Jonathan Kelner, David Steurer

Rounding Sum-of-Squares Relaxations

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a *combining algorithm* -- an algorithm that maps a distribution over solutions into a (possibly ... more >>>

TR14-059 | 21st April 2014
Boaz Barak, David Steurer

Sum-of-squares proofs and the quest toward optimal algorithms

Revisions: 2

In order to obtain the best-known guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method, surprisingly suggest that this tailoring is not necessary and that a single efficient algorithm could achieve best possible ... more >>>

TR14-142 | 1st November 2014
Subhash Khot, Dana Moshkovitz

Candidate Lasserre Integrality Gap For Unique Games

We propose a candidate Lasserre integrality gap construction for the Unique Games problem.
Our construction is based on a suggestion in [KM STOC'11] wherein the authors study the complexity of approximately solving a system of linear equations over reals and suggest it as an avenue towards a (positive) resolution ... more >>>

TR15-105 | 21st June 2015
Venkatesan Guruswami, Euiwoong Lee

Towards a Characterization of Approximation Resistance for Symmetric CSPs

A Boolean constraint satisfaction problem (CSP) is called approximation resistant if independently setting variables to $1$ with some probability $\alpha$ achieves the best possible approximation ratio for the fraction of constraints satisfied. We study approximation resistance of a natural subclass of CSPs that we call Symmetric Constraint Satisfaction Problems (SCSPs), ... more >>>

TR16-116 | 26th July 2016
Subhash Khot, Rishi Saket

Approximating CSPs using LP Relaxation

This paper studies how well the standard LP relaxation approximates a $k$-ary constraint satisfaction problem (CSP) on label set $[L]$. We show that, assuming the Unique Games Conjecture, it achieves an approximation within $O(k^3\cdot \log L)$ of the optimal approximation factor. In particular we prove the following hardness result: let ... more >>>

TR16-124 | 12th August 2016
Subhash Khot

On Independent Sets, $2$-to-$2$ Games and Grassmann Graphs

Revisions: 1 , Comments: 1

We present a candidate reduction from the $3$-Lin problem to the $2$-to-$2$ Games problem and present a combinatorial hypothesis about
Grassmann graphs which, if correct, is sufficient to show the soundness of the reduction in
a certain non-standard sense. A reduction that is sound in this non-standard sense
implies that ... more >>>

TR17-080 | 1st May 2017
Joshua Brakensiek, Venkatesan Guruswami

The Quest for Strong Inapproximability Results with Perfect Completeness

The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect ... more >>>

TR17-141 | 19th September 2017
Joshua Brakensiek, Venkatesan Guruswami

A Family of Dictatorship Tests with Perfect Completeness for 2-to-2 Label Cover

We give a family of dictatorship tests with perfect completeness and low-soundness for 2-to-2 constraints. The associated 2-to-2 conjecture has been the basis of some previous inapproximability results with perfect completeness. However, evidence towards the conjecture in the form of integrality gaps even against weak semidefinite programs has been elusive. ... more >>>

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