Impact of Redundant Checks on the LP Decoding Thresholds of LDPC Codes

Feldman et al. [11] asked whether the performance of the linear programming (LP) decoder can be improved by adding redundant parity checks to tighten the LP relaxation. We prove in this paper that for low-density parity-check codes, even if we include all redundant parity checks, asymptotically there is no gain in the LP decoder threshold on the binary symmetric channel under certain conditions on the base Tanner graph. First, we show that if the Tanner graph has bounded check-degree and satisfies a condition which we call asymptotic strength, then including high degree redundant parity checks in the LP does not significantly improve the threshold of the LP decoder in the following sense. For each constant delta > 0, there is a constant k > 0 such that the threshold of the LP decoder containing all redundant checks of degree at most k improves by at most delta upon adding to the LP all redundant checks of degree larger than k. We conclude that if the graph satisfies an additional condition which we call rigidity, then including all redundant checks does not improve the threshold of the base LP. We call the graph asymptotically strong if the LP decoder corrects a constant fraction of errors even if the log-likelihood-ratios of the correct variables are arbitrarily small. By building on a construction due Feldman et al. [9] and its recent improvement by Viderman [24], we show that asymptotic strength follows from sufficiently large variable-to-check expansion. We also give a geometric interpretation of asymptotic strength in terms pseudocodewords. We call the graph rigid if the minimum weight of a sum of check nodes involving a cycle tends to infinity as the block length tends to infinity. Under the assumptions that the graph girth is logarithmic and the minimum check degree is at least 3, rigidity is equivalent to the nondegeneracy property that adding at least logarithmically many checks does not give a constant weight check. We argue that nondegeneracy is a typical property of random check-regular Tanner graphs.