Irit Dinur, Elchanan Mossel, Oded Regev

We study the approximate-coloring(q,Q) problem: Given a graph G, decide

whether \chi(G) \le q or \chi(G)\ge Q. We derive conditional

hardness for this problem for any constant 3\le q < Q. For q \ge

4, our result is based on Khot's 2-to-1 conjecture [Khot'02].

For q=3, we base our hardness ...
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Venkatesan Guruswami, Ali Kemal Sinop

We prove almost tight hardness results for finding independent sets in bounded degree graphs and hypergraphs that admit a good

coloring. Our specific results include the following (where $\Delta$, assumed to be a constant, is a bound on the degree, and

$n$ is the number of vertices):

Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra, Yi Wu

We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with $(1-\epsilon)$ of the examples, it is $\mathrm{NP}$-hard to find a halfspace that is correct on $(1/2+\epsilon)$ of the examples, for arbitrary constants $\epsilon ... more >>>

Venkatesan Guruswami, Euiwoong Lee

Consider a $K$-uniform hypergraph $H = (V,E)$. A coloring $c : V \rightarrow \{1, 2, \dots, k \}$ with $k$ colors is rainbow if every hyperedge $e$ contains at least one vertex from each color, and is called perfectly balanced when each color appears the same number of times. A ... more >>>

Badih Ghazi, Pritish Kamath, Madhu Sudan

We present decidability results for a sub-class of "non-interactive" simulation problems, a well-studied class of problems in information theory. A non-interactive simulation problem is specified by two distributions $P(x,y)$ and $Q(u,v)$: The goal is to determine if two players, Alice and Bob, that observe sequences $X^n$ and $Y^n$ respectively where ... more >>>

Joshua Brakensiek, Venkatesan Guruswami

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 >>>

Amey Bhangale, Subhash Khot

A systematic study of simultaneous optimization of constraint satisfaction problems was initiated in [BKS15]. The simplest such problem is the simultaneous Max-Cut. [BKKST18] gave a $.878$-minimum approximation algorithm for simultaneous Max-Cut which is {\em almost optimal} assuming the Unique Games Conjecture (UGC). For a single instance Max-Cut, [GW95] gave an ... more >>>

Srinivasan Arunachalam, Penghui Yao

In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit pseudorandom generators (PRGs) for arbitrary $m$-facet polytopes in $n$ variables with seed length poly-logarithmic in $m,n$, concluding a sequence of works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, ... more >>>

Amey Bhangale, Subhash Khot, Dor Minzer

We propose a framework of algorithm vs. hardness for all Max-CSPs and demonstrate it for a large class of predicates. This framework extends the work of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable Max-CSPs.

Our framework is based on a new hybrid approximation algorithm, which uses ... more >>>