Lower Bounds for Linear Locally Decodable Codes and Private Information Retrieval

We prove that if a linear error-correcting code C: {0, 1}^n → {0, 1}^m is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then m = 2^(Ω(n)). We also present several extensions of this result. We show a reduction from the complexity, of one-round, information-theoretic private information retrieval systems (with two servers) to locally decodable codes, and conclude that if all the servers' answers are linear combinations of the database content, then t = Ω(n/2^a), where t is the length of the user's query and a is the length of the servers' answers. Actually, 2^a can be replaced by O(a^k), where k is the number of bit locations in the answer that are actually inspected in the reconstruction

Weighted Matchings via Unweighted Augmentations

We design a generic method for reducing the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the streaming model and in the massively parallel computation (MPC) model. In the context of streaming with random edge arrivals, our techniques yield a $(1/2+c)$-approximation algorithm thus breaking the natural barrier of $1/2$. For multi-pass streaming and the MPC model, we show that any algorithm computing a $(1-\delta)$-approximate unweighted matching in bipartite graphs can be translated into an algorithm that computes a $(1-\varepsilon(\delta))$-approximate maximum weighted matching. Furthermore, this translation incurs only a constant factor (that depends on $\varepsilon> 0$) overhead in the complexity. Instantiating this with the current best multi-pass streaming and MPC algorithms for unweighted matchings yields the following results for maximum weighted matchings: * A $(1-\varepsilon)$-approximation streaming algorithm that uses $O_\varepsilon(1)$ passes and $O_\varepsilon(n\, \text{poly} (\log n))$ memory. This is the first $(1-\varepsilon)$-approximation streaming algorithm for weighted matchings that uses a constant number of passes (only depending on $\varepsilon$). * A $(1 - \varepsilon)$-approximation algorithm in the MPC model that uses $O_\varepsilon(\log \log n)$ rounds, $O(m/n)$ machines per round, and $O_\varepsilon(n\, \text{poly}(\log n))$ memory per machine. This improves upon the previous best approximation guarantee of $(1/2-\varepsilon)$ for weighted graphs

Lower bounds for linear locally decodable codes and private information retrieval

We prove that if a linear error-correcting code C:{0, 1}^n→{0, 1}^m is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then m = 2^(Ω (n)). We also present several extensions of this result. We show a reduction from the complexity of one-round, information-theoretic Private Information Retrieval Systems (with two servers) to Locally Decodable Codes, and conclude that if all the servers’ answers are linear combinations of the database content, then t = Ω (n/2^a), where t is the length of the user’s query and a is the length of the servers’ answers. Actually, 2^a can be replaced by O(a^k), where k is the number of bit locations in the answer that are actually inspected in the reconstruction

Weighted Matchings via Unweighted Augmentations

We design a generic method to reduce the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the massively parallel computation (MPC) model and in the streaming model.For the MPC and the multi-pass streaming model, we show that any algorithm computing a (1- delta)-approximate unweighted matching in bipartite graphs can be translated into an algorithm that computes a (1 - epsilon(delta))-approximate maximum weighted matching. Furthermore, this translation incurs only a constant factor (that depends on epsilon > 0) overhead in the complexity. Instantiating this with the current best MPC algorithm for unweighted matching yields a (1 - epsilon)-approximation algorithm for maximum weighted matching that uses O-epsilon (log logn) rounds, O(m/n) machines per round, and Oe (n poly(logn)) memory per machine. This improves upon the previous best approximation guarantee of (1/2 - epsilon) for weighted graphs. In the context of single-pass streaming with random edge arrivals, our techniques yield a (1/2 + c)-approximation algorithm thus breaking the natural barrier of 1/2