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We show super-linear bounds for the problem of computing the Inner-Product function on AC0 circuits with parity gates just above the input level.
We show that the problem of deciding if a lattice has an orthogonal basis can be solved efficiently in Construction-A lattices.
We show that it is NP-hard to decode RS codes if the amount of error is very close to the covering radius of the code.
We show a tight adaptive, two-sided lower bound for testing linear isomorphism to the inner-product function. This is the first lower bound for testing linear isomorphism to a specific function that matches the trivial upper bound. We also show an exponential query lower bound for any adaptive, two-sided tester for membership in a large subclass of bent functions.
We provide an unbiased estimator for the score of matches between a text string and a pattern string, of smaller variance than in previous algorithms.
We introduce a framework for proving lower bounds on computational problems over distributions, based on defining a restricted class of algorithms called statistical algorithms. For well-known problems over distributions, we give lower bounds on the complexity of any statistical algorithm. These include a nearly optimal lower bound for detecting planted bipartite clique distributions (or planted dense subgraph distributions) when the planted clique has small size.
Motivated by the similar discrete linear structure of linear codes and point lattices, and their many shared applications, we initiate the study of list decoding for lattices. We focus on combinatorial and algorithmic questions related to list decoding for the well-studied family of Barnes-Wall lattices. Our results imply a polynomial-time list-decoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for natural error-correcting codes posed by the Johnson radius.
We continue the study of boolean linear-invariant properties defined over the hypercube and draw more connections to the testability of dense graphs. We show that an important family in this class (called `odd-cycle free') is efficiently testable. We also show that a canonical tester for this property only blows up the query complexity polynomially. These results suggest new open questions that attempt to cast light into the larger program of characterizing testable linear-invariant properties.
Motivated by applications in property testing, we investigate explicit low-weight codewords and bases of low-weight codewords for the well-studied BCH codes. We exhibit such codewords in some restricted settings.
We propose an improved algorithm for learning sparse parities over arbitrary distributions.
Linear codes (properties) that are also affine-invariant represent well-studied objects in the areas of local-testing, local-correcting, and local-decoding. An important open question in algebraic property testing is: what are neccessary and sufficient conditions that unify the testability of all these properties? The notion of `single-orbit' has been identified in the literature as a promising structural feature that leads to testability. This notion allows us in this work to broaden the class of known testable affine invariant properties by considering a basic, yet nontrivial operation on properties that have a single orbit, namely summation. Our results here together with previous results from the literature suggest the first conjecture that attempts to essentially capture the structure of all testable affine invariant properties.
Motivated by applications to property reconstruction and access control hierarchies, we concentrate on Steiner TC-spanners for partially ordered sets. We present a nearly tight lower bound on the size of Steiner 2-TC-spanners of d-dimensional directed hypergrids. It implies better lower bounds on the complexity of local reconstructors of monotone functions and functions with low Lipschitz constant. We also show a lower bound on the size of Steiner k-TC-spanners of d-dimensional posets that almost matches the known upper bounds.
We focus again on boolean linear-invariant properties over the hypercube defined by forbidden patterns. We initiate a systematic study of these properties from the somewhat subtle point of view that different local characterizations by forbidden patterns can lead to properties that are essentially the same in a property testing sense. We identify a combinatorial tool (called `labelled matroid homomorphism') that captures the relationship between the constraints in a way relevant to the question of distinguishing the respective properties that they define.
We propose a framework for analyzing the testability of boolean linear-invariant properties defined over the hypercube by drawing ideas from mysterious syntactic connections to the testability of graph properties, which is an area well-understood. Our results show the testability of a large class of linear-invariant properties, defined by forbidding a possibly infinite collection of arbitrary patterns. Based on these results we formulate the first conjecture that attempts to unify the testability of all boolean properties that are invariant under linear transformations and are testable with one sided error.
We continue the study of Transitive Closure Spanners and reveal a connection to `local monotonicity reconstructors', which are algorithms that reconstruct monotone functions from corrupted versions, in a distributed manner. This connection allows us to derive lower bounds on the query complexity of local monotonicity reconstructors from lower bounds on the size of TC-spanners. We study such lower bounds on directed hypercubes and hypergrids.
We show a local test for deciding if a polynomial is sparse or not.
We construct efficient data structures that are resilient against a constant fraction of adversarial noise. Our model requires that the decoder answers most queries correctly with high probability. In this work, we study two data structure problems: membership and polynomial evaluation. We show that these two problems have constructions that are simultaneously efficient and error-correcting.
We show that the dual of every ``sparse'' binary code whose symmetry group includes the group of non-singular affine transformations, has the single local orbit property (i.e., it is specified by a single local constraint and its translations under the affine group. ) This class includes the dual-BCH codes for whose duals (i.e., for BCH codes) simple bases were not known.
We introduce the notion of Transitive-Closure spanners of a directed graph as a common abstraction to applications in data structures, monotonicity testing and access control. We present algorithms for approximating the size of the sparsest k-TC spanner, and prove strong hardness results for this problem. In addition, our structural bounds for path-separable (directed) graphs lead to improved monotonicity testers for these posets.
In this work we refute a conjecture from the literature, stating that the presence of a single low weight codeword in the dual of a code, and ``2-transitivity'' of the code (i.e., the code is invariant under a 2-transitive group of permutations on the coordinates of the code) suffice to imply local testability.
We show that the code whose codewords are the homomorphisms between any two finite abelian groups is locally list decodable from a fraction of errors arbitrarily close to its minimum distance. The heart of the argument is a combinatorial result which gives an upper bound on the number of codewords in within a certain distance from any given word.
We initiate a systematic study of local decoding of codes based on group homomorphisms. We present an efficient local list decoder for codes from any abelian group G to a fixed abelian group H. Our results give a new generalization of the classical work of Goldreich and Levin, and give a new abstraction of the list decoder of Sudan, Trevisan and Vadhan.
We consider certain generalizations of independent sets, called insulated sets, and completely characterize the possible orderings of an insulated sequence of a graph.
We improve some lower bounds on the number of edges to be added to a cycle in order to decrease its diameter to 2 or 3.
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