We give a linear programming (LP) decoder that achieves the capacity (optimal rate) of a wide range of probabilistic binary communication channels. This is the first such result for LP decoding. More generally, as far as the authors are aware this is the first known polynomial-time capacity-achieving decoder with the <i>maximum-likelihood (ML) certificate</i> property---where output codewords come with a proof of optimality. Additionally, this result extends the capacity-achieving property of expander codes beyond the binary symmetric channel to a larger family of communication channels.Perhaps most importantly, since LP decoding performs well in practice on <i>turbo</i> codes and <i>low-density parity-check (LDPC)</i> codes (comparable to the popular "belief propagation" algorithm), this result exhibits the power of a new, widely applicable "dual witness" technique (Feldman, Malkin, Servedio, Stein and Wainwright, ISIT '04) for bounding decoder performance.For expander codes over an adversarial channel, we prove that LP decoding corrects a constant fraction of errors. To show this, we provide a new combinatorial characterization of error events that is of independent interest, and which we expect will lead to further improvements.
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