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A new algebraic technique for the construction of interactive proof systems is presented. Our technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP = PSPACE [28] and that MIP = NEXP [4].

There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach utilizes geometric properties of the measurement matrix Phi. A notable example is the Restricted Isometry Property, which states that the mapping Phi preserves the Euclidean norm of sparse signals; it is known that random dense matrices… (More)

Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due to Dahlhaus, Johnson, Papadimitriou, Seymour, and… (More)

In recent years the MapReduce framework has emerged as one of the most widely used parallel computing platforms for processing data on terabyte and petabyte scales. Used daily at companies such as Yahoo!, Google, Amazon, and Facebook, and adopted more recently by several universities, it allows for easy parallelization of data intensive computations over… (More)

We prove that if a linear error-correcting code C : f0;1g n ! f0;1g 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 (n). We also present several extensions of this result. We show a reduction from the complexity of one-round, information-theoretic Private Information… (More)

We consider the on-line version of the original <italic>m</italic>-machine scheduling problem: given <italic>m</italic> machines and <italic>n</italic> positive real jobs, schedule the <italic>n</italic> jobs on the <italic>m</italic> machines so as to minimize the makespan, the completion time of the last job. In the on-line version, as soon as job… (More)

In the 0-<i>extension problem</i>, we are given a weighted graph with some nodes marked as <i>terminals</i> and a semi-metric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are… (More)

The celebrated semidetinite programming algorithm for MAX CUT introduced by Goemans and Williamson was known to hav; a performance ratio of at least a = — (0,87856 < cr < 0.87857); the exact per: minO<eSr ~_co~e formance ratio was unknown. We prove that the performance ratio of their algorithm is exactly a. Furthermore, we show that it is impossible to add… (More)

We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instance—a collection of clauses each of length at most three—is satisfi-able, then the expected weight of the assignment found is at least 7=8 of optimal. We provide strong evidence (but not a proof) that the algorithm performs equally well on arbitrary… (More)

We study the problem of nding a most prootable subset of n given tasks, each with a given start and nish time as well as proot and resource requirement, that at no time exceeds the quantity B of available resource. We show that this NP-hard Resource Allocation problem can be (1=2?")-approximated in polynomial time, which improves upon earlier approximation… (More)