We introduce three new innovations for compression using LDPCs for the Slepian-Wolf problem. The first is a general iterative Slepian-Wolf decoding algorithm that incorporates the graphical structure of all the encoders and operates in a 'turbo-like' fashion. The second innovation introduces source-splitting to enable low-complexity pipelined implementations of Slepian-Wolf decoding at rates besides corner points of the Slepian-Wolf region. This innovation can also be applied to single-source block coding for reduced decoder complexity. The third approach is a linear programming relaxation to maximum-likelihood sequence decoding that exhibits the ML-certificate property. This can be used for decoding a single binary block-compressed source as well as decoding at vertex points for the binary Slepian-Wolf problem. All three of these innovations were motivated by recent analogous results in the channel coding domain.