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Electronic Colloquium on Computational Complexity

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All reports by Author Amey Bhangale:

TR20-130 | 30th August 2020
Amey Bhangale, Subhash Khot

Optimal Inapproximability of Satisfiable k-LIN over Non-Abelian Groups

A seminal result of H\r{a}stad [J. ACM, 48(4):798–859, 2001] shows that it is NP-hard to find an assignment that satisfies $\frac{1}{|G|}+\varepsilon$ fraction of the constraints of a given $k$-LIN instance over an abelian group, even if there is an assignment that satisfies $(1-\varepsilon)$ fraction of the constraints, for any constant ... more >>>

TR20-075 | 6th May 2020
Amey Bhangale, Prahladh Harsha, Orr Paradise, Avishay Tal

Rigid Matrices From Rectangular PCPs

Revisions: 1

We introduce a variant of PCPs, that we refer to as *rectangular* PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the *row* of each query and the other determining the *column*.

We ... more >>>

TR19-148 | 1st November 2019
Amey Bhangale, Subhash Khot

Simultaneous Max-Cut is harder to approximate than Max-Cut

Revisions: 1

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 >>>

TR19-048 | 2nd April 2019
Per Austrin, Amey Bhangale, Aditya Potukuchi

Simplified inpproximability of hypergraph coloring via t-agreeing families

We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99], Ahlswede-Khachatrian [AK98] and Frankl [F76] on the size of such families, we give simple and unified proofs ... more >>>

TR19-004 | 11th January 2019
Amey Bhangale, Subhash Khot

UG-hardness to NP-hardness by Losing Half

Revisions: 1

The $2$-to-$2$ Games Theorem of [KMS-1, DKKMS-1, DKKMS-2, KMS-2] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least $(\frac{1}{2}-\varepsilon)$ fraction of the constraints $vs.$ no assignment satisfying more than $\varepsilon$ fraction of the constraints, for every constant $\varepsilon>0$. We show that the reduction ... more >>>

TR18-073 | 21st April 2018
Amey Bhangale

NP-hardness of coloring $2$-colorable hypergraph with poly-logarithmically many colors

We give very short and simple proofs of the following statements: Given a $2$-colorable $4$-uniform hypergraph on $n$ vertices,

(1) It is NP-hard to color it with $\log^\delta n$ colors for some $\delta>0$.
(2) It is $quasi$-NP-hard to color it with $O\left({\log^{1-o(1)} n}\right)$ colors.

In terms of ... more >>>

TR17-030 | 15th February 2017
Amey Bhangale, Subhash Khot, Devanathan Thiruvenkatachari

An Improved Dictatorship Test with Perfect Completeness

A Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is called a dictator if it depends on exactly one variable i.e $f(x_1, x_2, \ldots, x_n) = x_i$ for some $i\in [n]$. In this work, we study a $k$-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems.

... more >>>

TR16-205 | 22nd December 2016
Amey Bhangale, Irit Dinur, Inbal Livni Navon

Cube vs. Cube Low Degree Test

We revisit the Raz-Safra plane-vs.-plane test and study the closely related cube vs. cube test. In this test the tester has access to a "cubes table" which assigns to every cube a low degree polynomial. The tester randomly selects two cubes (affine sub-spaces of dimension $3$) that intersect on a ... more >>>

TR16-112 | 18th July 2016
Mohammad T. Hajiaghayi, Amey Bhangale, Rajiv Gandhi, Rohit Khandekar, Guy Kortsarz

Bicovering: Covering edges with two small subsets of vertices

We study the following basic problem called Bi-Covering. Given a graph $G(V,E)$, find two (not necessarily disjoint) sets $A\subseteq V$ and $B\subseteq V$ such that $A\cup B = V$ and that every edge $e$ belongs to either the graph induced by $A$ or to the graph induced by $B$. The ... more >>>

TR15-069 | 21st April 2015
Amey Bhangale, Ramprasad Saptharishi, Girish Varma, Rakesh Venkat

On Fortification of General Games

Revisions: 1

A recent result of Moshkovitz~\cite{Moshkovitz14} presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in \cite{Moshkovitz14} to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel ... more >>>

TR14-098 | 30th July 2014
Amey Bhangale, Swastik Kopparty, Sushant Sachdeva

Simultaneous Approximation of Constraint Satisfaction Problems

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design the first nontrivial approximation algorithms in this context.

Our main result is that ... more >>>

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