A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

Given a $k$-uniform hyper-graph, the E$k$-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that E$k$-Vertex-Cover is NP-hard to approximate within factor $(k-1-\epsilon)$ for any $k \geq 3$ and any $\epsilon>0$. The result is essentially tight as this problem can be easily approximated within factor $k$. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of $s$-wise $t$-intersecting families of subsets

On the Combinatorial Version of the Slepian-Wolf Problem

We study the following combinatorial version of the Slepian-Wolf coding scheme. Two isolated Senders are given binary strings $X$ and $Y$ respectively; the length of each string is equal to $n$, and the Hamming distance between the strings is at most $\alpha n$. The Senders compress their strings and communicate the results to the Receiver. Then the Receiver must reconstruct both strings $X$ and $Y$. The aim is to minimise the lengths of the transmitted messages. For an asymmetric variant of this problem (where one of the Senders transmits the input string to the Receiver without compression) with deterministic encoding a nontrivial lower bound was found by A.Orlitsky and K.Viswanathany. In our paper we prove a new lower bound for the schemes with syndrome coding, where at least one of the Senders uses linear encoding of the input string. For the combinatorial Slepian-Wolf problem with randomized encoding the theoretical optimum of communication complexity was recently found by the first author, though effective protocols with optimal lengths of messages remained unknown. We close this gap and present a polynomial time randomized protocol that achieves the optimal communication complexity.Comment: 20 pages, 14 figures. Accepted to IEEE Transactions on Information Theory (June 2018

Derandomized Construction of Combinatorial Batch Codes

Combinatorial Batch Codes (CBCs), replication-based variant of Batch Codes introduced by Ishai et al. in STOC 2004, abstracts the following data distribution problem: $n$ data items are to be replicated among $m$ servers in such a way that any $k$ of the $n$ data items can be retrieved by reading at most one item from each server with the total amount of storage over $m$ servers restricted to $N$. Given parameters $m, c,$ and $k$, where $c$ and $k$ are constants, one of the challenging problems is to construct $c$-uniform CBCs (CBCs where each data item is replicated among exactly $c$ servers) which maximizes the value of $n$. In this work, we present explicit construction of $c$-uniform CBCs with $\Omega(m^{c-1+{1 \over k}})$ data items. The construction has the property that the servers are almost regular, i.e., number of data items stored in each server is in the range $[{nc \over m}-\sqrt{{n\over 2}\ln (4m)}, {nc \over m}+\sqrt{{n \over 2}\ln (4m)}]$. The construction is obtained through better analysis and derandomization of the randomized construction presented by Ishai et al. Analysis reveals almost regularity of the servers, an aspect that so far has not been addressed in the literature. The derandomization leads to explicit construction for a wide range of values of $c$ (for given $m$ and $k$) where no other explicit construction with similar parameters, i.e., with $n = \Omega(m^{c-1+{1 \over k}})$, is known. Finally, we discuss possibility of parallel derandomization of the construction