Chernoff-Hoeffding bounds are fundamental tools used in bounding the tail probabilities of the sums of bounded and independent random variables. We present a simple technique which gives slightly better bounds than these and which, more importantly, requires only limited independence among the random variables, thereby importing a variety of standard results to the case of limited independence for free. Additional methods are also presented, and the aggregate results are very sharp and provide a better understanding of the proof techniques behind these bounds. They also yield improved bounds for various tail probability distributions and enable improved approximation algorithms for jobshop scheduling. The "limited independence" result implies that weaker sources of randomness are sufficient for randomized algorithms whose analyses use the Chernoff-Hoeffding bounds; further, it leads to algorithms that require a reduced amount of randomness for any analysis which uses the Chernoff-Hoeffding bounds, e.g., the analysis of randomized algorithms for random sampling and oblivious packet routing.
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