Venkatesan Guruswami

Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma et al. [2007] required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy [2005]. Our expanders can be interpreted as near-optimal “randomness condensers,” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. [2003] and improving upon it when the error parameter is small (e.g., 1/poly(n)).

Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems

We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths problem (EDP), we are given a network G with source-sink pairs (si; ti), 1 i k, and the goal is to nd a largest subset of source-sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any > 0, EDP is NP-hard to approximate within m 1=2 . We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any > 0, bounded length EDP is hard to approximate within m 1=2 even in undirected networks, and give an O( p m)-approximation algorithm for it. For directed networks, we show that even the single source-sink pair case (i.e. nd the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1=2 , for any > 0.

Multiclass learning, boosting, and error-correcting codes

We focus on methods to solve multiclass learning problems by using only simple and efficient binary learners. We investigate the approach of Dietterich and Bakiri [2] based on error-correcting codes (which we call ECC). We distill error correlation as one of the key parameters influencing the performance of the ECC approach, and prove upper and lower bounds on the training error of the final hypothesis in terms of the error-correlation between the various binary hypotheses. Boosting is a powerful and well-studied learning technique that appears to annul error correlation disadvantages by cleverly weighting training examples and hypotheses. An interesting algorithm called ADABOOST.OC [12] combines boosting with the ECC approach and gives an algorithm that has the performance advantages of boosting and at the same time relies only on simple binary weak learners. We propose a variant of this algorithm, which we call ADABOOST.ECC, that, by using a different weighting of the votes of the weak hypotheses, is able to improve on the performance of ADABOOST.OC, both theoretically and experimentally, and in addition is arguably a more direct reduction of multiclass learning to binary learning problems than previous multiclass boosting algorithms.

A new multilayered PCP and the hardness of hypergraph vertex cover

Given a k-uniform hyper-graph, the Ek-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 Ek-Vertex-Cover is NP-hard to approximate within factor (k-1-ε) for any k ≥ 3 and any ε>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.

Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy

In this paper, we present error-correcting codes that achieve the information-theoretically best possible tradeoff between the rate and error-correction radius. Specifically, for every 0 < R < 1 and epsiv < 0, we present an explicit construction of error-correcting codes of rate that can be list decoded in polynomial time up to a fraction (1- R - epsiv) of worst-case errors. At least theoretically, this meets one of the central challenges in algorithmic coding theory. Our codes are simple to describe: they are folded Reed-Solomon codes, which are in fact exactly Reed-Solomon (RS) codes, but viewed as a code over a larger alphabet by careful bundling of codeword symbols. Given the ubiquity of RS codes, this is an appealing feature of our result, and in fact our methods directly yield better decoding algorithms for RS codes when errors occur in phased bursts. The alphabet size of these folded RS codes is polynomial in the block length. We are able to reduce this to a constant (depending on epsiv) using existing ideas concerning ldquolist recoveryrdquo and expander-based codes. Concatenating the folded RS codes with suitable inner codes, we get binary codes that can be efficiently decoded up to twice the radius achieved by the standard GMD decoding.

A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

Given a k-uniform hypergraph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyperedge. We present a new multilayered probabilistically checkable proof (PCP) construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within a factor of

Correlation Clustering with a Fixed Number of Clusters.

(k-1-\epsilon)

Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph

for arbitrary constants

Hardness of Learning Halfspaces with Noise

\epsilon>0

Optimal column-based low-rank matrix reconstruction

and