On taking advantage of multiple requests in error correcting codes

In most notions of locality in error correcting codes -- notably locally recoverable codes (LRCs) and locally decodable codes (LDCs) -- a decoder seeks to learn a single symbol of a message while looking at only a few symbols of the corresponding codeword. However, suppose that one wants to recover r > 1 symbols of the message. The two extremes are repeating the single-query algorithm r times (this is the intuition behind LRCs with availability, primitive multiset batch codes, and PIR codes) or simply running a global decoding algorithm to recover the whole thing. In this paper, we investigate what can happen in between these two extremes: at what value of r does repetition stop being a good idea? In order to begin to study this question we introduce robust batch codes, which seek to find r symbols of the message using m queries to the codeword, in the presence of erasures. We focus on the case where r = m, which can be seen as a generalization of the MDS property. Surprisingly, we show that for this notion of locality, repetition is optimal even up to very large values of $r = \Omega(k)$

Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows $x_{n/2+1},\dots,x_n$, and both parties know $\varphi$. The goal is to have Alice and Bob jointly write a PCP that $x$ satisfies $\varphi$, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of $x$. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of $2^{(\log n)^{1-o(1)}}$; only $(1+o(1))$-factor lower bounds (under SETH) were known before