On the complexity of and solutions to the minimum stopping and trapping set problems

In this paper, we discuss the problems of finding minimum stopping sets and trapping sets in Tanner graphs, using integer linear programming. These problems are important for establishing reliable communication across noisy channels. Indeed, stopping sets and trapping sets correspond to combinatoria...

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Vydáno v:Theoretical computer science Ročník 915; s. 26 - 44
Hlavní autoři: Velasquez, Alvaro, Subramani, K., Wojciechowski, Piotr
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 14.05.2022
Témata:
Exact exponential algorithm
NP-complete
Parameterized algorithm
Stopping set
Trapping set
Trapping set
Exact exponential algorithm
Stopping set
NP-complete
Parameterized algorithm
ISSN:0304-3975, 1879-2294
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Shrnutí:In this paper, we discuss the problems of finding minimum stopping sets and trapping sets in Tanner graphs, using integer linear programming. These problems are important for establishing reliable communication across noisy channels. Indeed, stopping sets and trapping sets correspond to combinatorial structures in information-theoretic codes, which lead to errors in decoding once a message is received. In particular, these sets correspond to variables that are not eventually corrected by the decoder, thus causing failures in decoding when using iterative algorithms. We present integer linear programs (ILPs) for finding stopping sets and several trapping set variants. Furthermore, we prove that two of these trapping set problem variants are NP-hard for the first time. Additionally, we analyze these variants from the parameterized perspective. Finally, we model stopping set and trapping set problems as Integer Linear Programs (ILPs). The effectiveness of our approach is demonstrated by finding stopping sets of size up to 48 in the (4896,2474) Margulis code. This compares favorably to the current state-of-the-art, which finds stopping sets of size up to 26. For the trapping set problems, we show for which cases an ILP produces solutions efficiently and for which cases it performs poorly. The proposed approach is applicable to codes represented by regular and irregular graphs alike.1
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2022.02.028