arXiv subjects

Aditya Potukuchi

Publications and source records attributed to Aditya Potukuchi.

24 records · Page 2Linked to original sources

arXiv2018

Discrepancy in random hypergraph models

We study hypergraph discrepancy in two closely related random models of hypergraphs on $n$ vertices and $m$ hyperedges. The first model, $\mathcal{H}_1$, is when every vertex is present in exactly $t$ randomly chosen hyperedges. The premise of this is closely tied to, and motivated by the Beck-Fiala conjecture. The second, perhaps more natural model, $\mathcal{H}_2$, is when the entries of the $m \times n$ incidence matrix is sampled in an i.i.d. fashion, each with probability $p$. We prove the following: 1. In $\mathcal{H}_1$, when $\log^{10}n \ll t \ll \sqrt{n}$, and $m = n$, we show that the discrepancy of the hypergraph is almost surely at most $O(\sqrt{t})$. This improves upon a result of Ezra and Lovett for this range of parameters. 2. In $\mathcal{H}_2$, when $p= \frac{1}{2}$, and $n = Ω(m \log m)$, we show that the discrepancy is almost surely at most $1$. This answers an open problem of Hoberg and Rothvoss.

math.CO↗

arXiv2018

Improved Inapproximability of Rainbow Coloring

A rainbow $q$-coloring of a $k$-uniform hypergraph is a $q$-coloring of the vertex set such that every hyperedge contains all $q$ colors. We prove that given a rainbow $(k - 2\lfloor \sqrt{k}\rfloor)$-colorable $k$-uniform hypergraph, it is NP-hard to find a normal $2$-coloring. Previously, this was only known for rainbow $\lfloor k/2 \rfloor$-colorable hypergraphs (Guruswami and Lee, SODA 2015). We also study a generalization which we call rainbow $(q, p)$-coloring, defined as a coloring using $q$ colors such that every hyperedge contains at least $p$ colors. We prove that given a rainbow $(k - \lfloor \sqrt{kc} \rfloor, k- \lfloor3\sqrt{kc} \rfloor)$-colorable $k$ uniform hypergraph, it is NP-hard to find a normal $c$-coloring for any $c = o(k)$. The proof of our second result relies on two combinatorial theorems. One of the theorems was proved by Sarkaria (J. Comb. Theory. 1990) using topological methods and the other theorem we prove using a generalized Borsuk-Ulam theorem.

cs.CC↗

arXiv2018

A short note on the joint entropy of n/2-wise independence

In this note, we prove a tight lower bound on the joint entropy of $n$ unbiased Bernoulli random variables which are $n/2$-wise independent. For general $k$-wise independence, we give new lower bounds by adapting Navon and Samorodnitsky's Fourier proof of the `LP bound' on error correcting codes. This counts as partial progress on a problem asked by Gavinsky and Pudlák.

cs.DM↗

arXiv2017

Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields

Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk showed that for binary Reed-Muller codes of length $n$ and distance $d = O(1)$, one can correct $\operatorname{polylog}(n)$ random errors in $\operatorname{poly}(n)$ time (which is well beyond the worst-case error tolerance of $O(1)$). In this paper, we consider the problem of `syndrome decoding' Reed-Muller codes from random errors. More specifically, given the $\operatorname{polylog}(n)$-bit long syndrome vector of a codeword corrupted in $\operatorname{polylog}(n)$ random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in $\operatorname{polylog}(n)$ time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of Saptharishi et al. 2. The second algorithm is obtained by implementing the steps of the Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.

cs.IT↗

arXiv2017

On zeros of a polynomial in a finite grid

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain "Condition (D)" on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further Generalized Alon-Füredi Theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon-Füredi. We then discuss the relationship between Alon-Füredi and results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon-Füredi Theorem and its generalization in terms of Reed--Muller type affine variety codes is shown which gives us the minimum Hamming distance of these codes. Then we apply the Alon-Füredi Theorem to quickly recover (and sometimes strengthen) old and new results in finite geometry, including the Jamison/Brouwer-Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

math.CO↗

arXiv2016

Cauchy-Davenport Theorem for linear maps: Simplification and Extension

We give a new proof of the Cauchy-Davenport Theorem for linear maps given by Herdade et al., (2015). This theorem gives a lower bound on the size of the image of a linear map on a grid. Our proof is purely combinatorial and offers a partial insight into the range of parameters not handled previously.

math.CO↗