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Applying the max-product (and sum-product) algorithms to loopy graphs is now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence or optimality of converged solutions or both. This paper presents an analysis of both attenuated max-product decoding and weighted min-sum decoding for low-density parity-check (LDPC) codes, which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution. For (dv, dc)-regular LDPC codes, the weight factor must satisfy β(dv-1) <; 1 to guarantee convergence to a fixed point, whereas the results proposed by Frey and Koetter require instead that β(dv-1)(dc-1) ≤ 1. In addition, the range of the weight factor for a provable ML decoding solution is extended to 0 <; β(dv-1) 1. In addition, counterexamples that show a fixed point might not be the ML decoding solution if β(dv-1) > 1 are given. Finally, connections are explored with recent work on the threshold of LP decoding.