Low-density parity-check (LDPC) codes are one of the most promising error-correcting codes approaching Shannon capacity and have been adopted in many applications. However, the efficient implementation of high-throughput LDPC decoders adaptable for various channel conditions still remains challenging. In this paper, a low-complexity reconfigurable VLSI architecture for high-speed LDPC decoders is presented. Shift-LDPC codes are incorporated within the design and have shown not only comparable decoding performance to computer-generated random codes but also high hardware efficiency in high-speed applications. The single-minimum Min-Sum decoding scheme and the nonuniform quantization scheme are explored to reduce the complexity of computing core and the memory requirement. The well-known Benes network is employed to construct the configurable permutation network to support multiple shift-LDPC codes with various code parameters. The ASIC implementation results of an (8192, 7168) (4, 32)-regular shift-LDPC decoder demonstrate a maximum decoding throughput of 3.6 Gbits/s at 16 iterations, which outperforms the state-of-the-art design for high-speed flexible LDPC decoders by many times with even less hardware.