Hardness Results on Finding Leafless Elementary Trapping Sets and Elementary Absorbing Sets of LDPC Codes

TL;DR

This paper proves that finding leafless elementary trapping sets and elementary absorbing sets of LDPC codes with specific parameters is NP-hard to approximate, highlighting the computational difficulty of analyzing these problematic structures in error correction.

Contribution

The paper establishes NP-hardness of approximating minimal leafless elementary trapping and absorbing sets in LDPC codes, filling a gap in complexity results for these structures.

Findings

NP-hardness of approximating LETS with given size and unsatisfied check nodes

NP-hardness of approximating minimal size LETS for given unsatisfied check nodes

NP-hardness results also apply to elementary absorbing sets

Abstract

Leafless elementary trapping sets (LETSs) are known to be the problematic structures in the error floor region of low-density parity-check (LDPC) codes over the additive white Gaussian (AWGN) channel under iterative decoding algorithms. While problems involving the general category of trapping sets, and the subcategory of elementary trapping sets (ETSs), have been shown to be NP-hard, similar results for LETSs, which are a subset of ETSs are not available. In this paper, we prove that, for a general LDPC code, finding a LETS of a given size a with minimum number of unsatisfied check nodes b is NP-hard to approximate with any guaranteed precision. We also prove that finding the minimum size a of a LETS with a given b is NP-hard to approximate. Similar results are proved for elementary absorbing sets, a popular subcategory of LETSs.