Towards Low-Complexity Linear-Programming Decoding

TL;DR

This paper explores structured linear programming decoding for LDPC codes, proposing methods that leverage problem structure to achieve low-complexity algorithms similar to min-sum and sum-product algorithms.

Contribution

It introduces coordinate-ascent and soft-minimum based methods for LP decoding that exploit problem structure, enabling low-complexity solutions comparable to existing message-passing algorithms.

Findings

Coordinate-ascent methods lead to simple update rules.

Soft-minimum based updates connect to sum-product algorithm.

Low-complexity LP decoding algorithms are feasible.

Abstract

We consider linear-programming (LP) decoding of low-density parity-check (LDPC) codes. While it is clear that one can use any general-purpose LP solver to solve the LP that appears in the decoding problem, we argue in this paper that the LP at hand is equipped with a lot of structure that one should take advantage of. Towards this goal, we study the dual LP and show how coordinate-ascent methods lead to very simple update rules that are tightly connected to the min-sum algorithm. Moreover, replacing minima in the formula of the dual LP with soft-minima one obtains update rules that are tightly connected to the sum-product algorithm. This shows that LP solvers with complexity similar to the min-sum algorithm and the sum-product algorithm are feasible. Finally, we also discuss some sub-gradient-based methods.