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Elements of information theory
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Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
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Approximating Maximum Clique by Removing Subgraphs
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Linear degree extractors and the inapproximability of max clique and chromatic number
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How to compress interactive communication
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An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem
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On the Complexity of Nonnegative Matrix Factorization
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A Randomized Rounding Approach to the Traveling Salesman Problem
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Computing a nonnegative matrix factorization -- provably
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Interactive information complexity
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Using separation algorithms to generate mixed integer model reformulations
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Approximation Limits of Linear Programs (Beyond Hierarchies)
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Lower Bounds on Information Complexity via Zero-Communication Protocols and Applications
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
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We prove an unconditional lower bound that any linear program that achieves an O(n1-ε) approximation for clique has size 2Ω(nε). There has been considerable recent interest in proving unconditional lower bounds against any linear program. Fiorini et al. proved that there is no polynomial sized linear program for traveling salesman. Braun et al. proved that there is no polynomial sized O(n1/2 - ε)-approximate linear program for clique. Here we prove an optimal and unconditional lower bound against linear programs for clique that matches Hastad's celebrated hardness result. Interestingly, the techniques used to prove such lower bounds have closely followed the progression of techniques used in communication complexity. Here we develop an information theoretic framework to approach these questions, and we use it to prove our main result. Also we resolve a related question: How many bits of communication are needed to get ε-advantage over random guessing for disjointness? Kalyanasundaram and Schnitger proved that a protocol that gets constant advantage requires Ω(n) bits of communication. This result in conjunction with amplification implies that any protocol that gets ε-advantage requires Ω(ε2 n) bits of communication. Here we improve this bound to Ω(ε n), which is optimal for any ε 0.