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 |
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Elsevier B.V
14.05.2022
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| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | 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 |
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| AbstractList | 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 |
| Author | Wojciechowski, Piotr Subramani, K. Velasquez, Alvaro |
| Author_xml | – sequence: 1 givenname: Alvaro surname: Velasquez fullname: Velasquez, Alvaro email: alvaro.velasquez.1@us.af.mil organization: Information Directorate, Air-Force Research Laboratory, Rome, NY, United States of America – sequence: 2 givenname: K. surname: Subramani fullname: Subramani, K. email: k.subramani@mail.wvu.edu organization: LCSEE, West Virginia University, Morgantown, WV, United States of America – sequence: 3 givenname: Piotr surname: Wojciechowski fullname: Wojciechowski, Piotr email: pwojciec@mail.wvu.edu organization: LCSEE, West Virginia University, Morgantown, WV, United States of America |
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| Cites_doi | 10.1109/LCOMM.2017.2707553 10.1016/S0304-3975(98)00158-3 10.1109/TCOMM.2004.833048 10.1109/TIT.2019.2904988 10.1109/TIT.2002.1003839 10.1109/TIT.2014.2363832 10.1109/TIT.2012.2191697 10.1109/TIT.2011.2171531 10.1016/j.ipl.2010.04.010 10.1109/TIT.2009.2025573 10.1109/TIT.2014.2334657 10.1109/TCOMM.2007.910589 10.1109/18.641542 10.1109/TIT.2010.2040941 |
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| Keywords | Trapping set Exact exponential algorithm Stopping set NP-complete Parameterized algorithm |
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| Snippet | In this paper, we discuss the problems of finding minimum stopping sets and trapping sets in Tanner graphs, using integer linear programming. These problems... |
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| SubjectTerms | Exact exponential algorithm NP-complete Parameterized algorithm Stopping set Trapping set |
| Title | On the complexity of and solutions to the minimum stopping and trapping set problems |
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