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...
Uložené v:
| Vydané v: | Theoretical computer science Ročník 915; s. 26 - 44 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Elsevier B.V
14.05.2022
|
| Predmet: | |
| ISSN: | 0304-3975, 1879-2294 |
| On-line prístup: | Získať plný text |
| Tagy: |
Pridať tag
Žiadne tagy, Buďte prvý, kto otaguje tento záznam!
|
| 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 |
|---|---|
| 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 |
| BookMark | eNp9kM1qwzAQhEVJoUnaB-hNL2BXkh3LpqcS-geBXNKzkORVq2BLRlJK8_a14567DCwLM8vwrdDCeQcI3VOSU0Krh2OedMwZYSwnk-ortKQ1bzLGmnKBlqQgZVY0fHODVjEeyTgbXi3RYe9w-gKsfT908GPTGXuDpWtx9N0pWe8iTv5i6a2z_anHMflhsO7z4kpBzkeEhIfgVQd9vEXXRnYR7v72Gn28PB-2b9lu__q-fdplmpVNyigtgKuyLWqQRilNqOKsAbOpipKT1jDJtCy5VIwoo2lhuFIVNbomspaMN8Ua0fmvDj7GAEYMwfYynAUlYsIijmLEIiYsgkyqx8zjnIGx2LeFIKK24DS0NoBOovX2n_Qv8_VurQ |
| 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 |
| ContentType | Journal Article |
| Copyright | 2022 Elsevier B.V. |
| Copyright_xml | – notice: 2022 Elsevier B.V. |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.tcs.2022.02.028 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics Computer Science |
| EISSN | 1879-2294 |
| EndPage | 44 |
| ExternalDocumentID | 10_1016_j_tcs_2022_02_028 S030439752200127X |
| GroupedDBID | --K --M -~X .DC .~1 0R~ 123 1B1 1RT 1~. 1~5 4.4 457 4G. 5VS 7-5 71M 8P~ 9JN AABNK AACTN AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAXUO AAYFN ABAOU ABBOA ABJNI ABMAC ABYKQ ACAZW ACDAQ ACGFS ACRLP ACZNC ADBBV ADEZE AEBSH AEKER AENEX AFKWA AFTJW AGUBO AGYEJ AHHHB AHZHX AIALX AIEXJ AIKHN AITUG AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD ARUGR AXJTR BKOJK BLXMC CS3 DU5 EBS EFJIC EFLBG EO8 EO9 EP2 EP3 F5P FDB FEDTE FIRID FNPLU FYGXN G-Q GBLVA GBOLZ HVGLF IHE IXB J1W KOM LG9 M26 M41 MHUIS MO0 N9A O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. Q38 ROL RPZ SCC SDF SDG SES SPC SPCBC SSV SSW T5K TN5 WH7 YNT ZMT ~G- 29Q 9DU AAEDT AAQXK AATTM AAXKI AAYWO AAYXX ABDPE ABEFU ABFNM ABWVN ABXDB ACLOT ACNNM ACRPL ACVFH ADCNI ADMUD ADNMO ADVLN AEIPS AEUPX AEXQZ AFJKZ AFPUW AGHFR AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP ASPBG AVWKF AZFZN CITATION EFKBS EJD FGOYB G-2 HZ~ R2- SEW SSZ TAE WUQ ZY4 ~HD |
| ID | FETCH-LOGICAL-c249t-113e7b4d38eafbbc01b729ef563470df2a2ca47ab20bfc13f7bb61fc80a8a2793 |
| ISICitedReferencesCount | 0 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000795611200003&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0304-3975 |
| IngestDate | Sat Nov 29 07:18:37 EST 2025 Fri Feb 23 02:40:58 EST 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Trapping set Exact exponential algorithm Stopping set NP-complete Parameterized algorithm |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c249t-113e7b4d38eafbbc01b729ef563470df2a2ca47ab20bfc13f7bb61fc80a8a2793 |
| PageCount | 19 |
| ParticipantIDs | crossref_primary_10_1016_j_tcs_2022_02_028 elsevier_sciencedirect_doi_10_1016_j_tcs_2022_02_028 |
| PublicationCentury | 2000 |
| PublicationDate | 2022-05-14 |
| PublicationDateYYYYMMDD | 2022-05-14 |
| PublicationDate_xml | – month: 05 year: 2022 text: 2022-05-14 day: 14 |
| PublicationDecade | 2020 |
| PublicationTitle | Theoretical computer science |
| PublicationYear | 2022 |
| Publisher | Elsevier B.V |
| Publisher_xml | – name: Elsevier B.V |
| References | Tanatmis, Ruzika, Punekar, Kienle (br0200) 2010 Punekar, Kienle, Wehn, Tanatmis, Ruzika, Hamacher (br0210) 2010 Koetter, Vontobel (br0270) 2003 Alimonti, Kann (br0320) 2000; 237 Klotz, Newman (br0350) 2013; 18 Crowston, Gutin, Jones (br0310) 2010; 110 Murali Krishnan, Shankar (br0050) 2006 Helmling, Ruzika, Tanatmis (br0220) 2012; 58 J. Rosenthal, P.O. Vontobel, Construction of ldpc codes based on ramanujan graphs and ideas from margulis, in: Proc. 38th Annual Allerton Conf. on Communication, Computing and Control, pp. 248–257. Karimi, Banihashemi (br0260) 2011 McGregor, Milenkovic (br0120) 2010; 56 Rosnes, Ytrehus, Ambroze, Tomlinson (br0180) 2012; 58 Wang (br0250) 2007 Karimi, Banihashemi (br0130) 2014; 60 Hashemi, Banihashemi (br0140) 2017 (br0340) 2014 Richardson (br0020) 1998 Murali Krishnan, Sunil Chandran (br0290) 2006 Hu, Eleftheriou (br0070) 2006 Rosnes, Ytrehus (br0090) 2009; 55 Helmling, Rosnes, Ruzika, Scholl (br0230) 2014 Di, Proietti, Emre Telatar, Richardson, Urbanke (br0040) 2002; 48 Wang, Kulkarni, Poor (br0160) 2006 Cygan, Fomin, Kowalik, Lokshtanov, Marx, Pilipczuk, Michal, Saurabh (br0330) 2015 Tian, Jones, Villasenor, Wesel (br0280) 2004; 52 Butler, Siegel (br0060) 2014; 60 Dehghan, Banihashemi (br0300) 2019; 65 MacKay (br0150) 2005 Hirotomo, Konishi, Morii (br0080) 2008 Vardy (br0240) 1997; 43 Velasquez, Subramani, Drager (br0010) 2018 Tanatmis, Ruzika, Hamacher, Punekar, Kienle, Wehn (br0190) 2009 Kyung, Wang (br0100) 2010 Xiao, Banihashemi (br0030) 2007; 55 Chih-Chun Wang, Sanjeev R. Kulkarni, H. Vincent Poor, Exhausting error-prone patterns in ldpc codes, arXiv preprint cs/0609046, 2006. Kyung (10.1016/j.tcs.2022.02.028_br0100) 2010 McGregor (10.1016/j.tcs.2022.02.028_br0120) 2010; 56 Butler (10.1016/j.tcs.2022.02.028_br0060) 2014; 60 Murali Krishnan (10.1016/j.tcs.2022.02.028_br0050) 2006 Wang (10.1016/j.tcs.2022.02.028_br0160) 2006 Wang (10.1016/j.tcs.2022.02.028_br0250) 2007 Rosnes (10.1016/j.tcs.2022.02.028_br0180) 2012; 58 Dehghan (10.1016/j.tcs.2022.02.028_br0300) 2019; 65 Hirotomo (10.1016/j.tcs.2022.02.028_br0080) 2008 Karimi (10.1016/j.tcs.2022.02.028_br0260) 2011 Tanatmis (10.1016/j.tcs.2022.02.028_br0200) 2010 Hu (10.1016/j.tcs.2022.02.028_br0070) 2006 Alimonti (10.1016/j.tcs.2022.02.028_br0320) 2000; 237 Punekar (10.1016/j.tcs.2022.02.028_br0210) 2010 Helmling (10.1016/j.tcs.2022.02.028_br0230) 2014 Murali Krishnan (10.1016/j.tcs.2022.02.028_br0290) 2006 Vardy (10.1016/j.tcs.2022.02.028_br0240) 1997; 43 Di (10.1016/j.tcs.2022.02.028_br0040) 2002; 48 Xiao (10.1016/j.tcs.2022.02.028_br0030) 2007; 55 Rosnes (10.1016/j.tcs.2022.02.028_br0090) 2009; 55 Velasquez (10.1016/j.tcs.2022.02.028_br0010) 2018 10.1016/j.tcs.2022.02.028_br0170 Hashemi (10.1016/j.tcs.2022.02.028_br0140) 2017 MacKay (10.1016/j.tcs.2022.02.028_br0150) 2005 10.1016/j.tcs.2022.02.028_br0110 Karimi (10.1016/j.tcs.2022.02.028_br0130) 2014; 60 Helmling (10.1016/j.tcs.2022.02.028_br0220) 2012; 58 Cygan (10.1016/j.tcs.2022.02.028_br0330) 2015 Klotz (10.1016/j.tcs.2022.02.028_br0350) 2013; 18 Richardson (10.1016/j.tcs.2022.02.028_br0020) 1998 Koetter (10.1016/j.tcs.2022.02.028_br0270) 2003 Crowston (10.1016/j.tcs.2022.02.028_br0310) 2010; 110 Tanatmis (10.1016/j.tcs.2022.02.028_br0190) 2009 Tian (10.1016/j.tcs.2022.02.028_br0280) 2004; 52 |
| References_xml | – volume: 60 start-page: 7416 year: 2014 end-page: 7441 ident: br0060 article-title: Error floor approximation for ldpc codes in the awgn channel publication-title: IEEE Trans. Inf. Theory – start-page: 411 year: 2006 end-page: 415 ident: br0160 article-title: Upper bounding the performance of arbitrary finite ldpc codes on binary erasure channels publication-title: Information Theory, 2006 IEEE International Symposium on – volume: 55 start-page: 2234 year: 2007 end-page: 2239 ident: br0030 article-title: Estimation of bit and frame error rates of finite-length low-density parity-check codes on binary symmetric channels publication-title: IEEE Trans. Commun. – start-page: 2589 year: 2014 end-page: 2593 ident: br0230 article-title: Efficient maximum-likelihood decoding of linear block codes on binary memoryless channels publication-title: 2014 IEEE International Symposium on Information Theory – volume: 56 start-page: 1640 year: 2010 end-page: 1650 ident: br0120 article-title: On the hardness of approximating stopping and trapping sets publication-title: IEEE Trans. Inf. Theory – start-page: 295 year: 2008 end-page: 299 ident: br0080 article-title: On the probabilistic computation algorithm for the minimum-size stopping sets of ldpc codes publication-title: Information Theory, 2008. ISIT 2008. IEEE International Symposium on – start-page: 2216 year: 2009 end-page: 2220 ident: br0190 article-title: Valid inequalities for binary linear codes publication-title: 2009 IEEE International Symposium on Information Theory – volume: 110 start-page: 451 year: 2010 end-page: 454 ident: br0310 article-title: Note on max lin-2 above average publication-title: Inf. Process. Lett. – start-page: 1426 year: 1998 end-page: 1435 ident: br0020 article-title: Error Floors of ldpc Codes publication-title: Proceedings of the Annual Allerton Conference on Communication Control and Computing, Vol. 41 – reference: Chih-Chun Wang, Sanjeev R. Kulkarni, H. Vincent Poor, Exhausting error-prone patterns in ldpc codes, arXiv preprint cs/0609046, 2006. – start-page: 1 year: 2003 end-page: 5 ident: br0270 article-title: Graph-covers and iterative decoding of finite length codes publication-title: Proc. 3rd Intern. Symp. on Turbo Codes and Related Topics – volume: 58 start-page: 164 year: 2012 end-page: 171 ident: br0180 article-title: Addendum to ?an efficient algorithm to find all small-size stopping sets of low-density parity-check matrices? publication-title: IEEE Trans. Inf. Theory – year: 2018 ident: br0010 article-title: Finding minimum stopping and trapping sets: an integer linear programming approach publication-title: International Symposium on Combinatorial Optimization (ISCO), 2018, Marrakesh, Morocco, April 11-13, 2018, Proceedings – start-page: 2271 year: 2007 end-page: 2275 ident: br0250 article-title: On the exhaustion and elimination of trapping sets: algorithms & the suppressing effect publication-title: Information Theory, 2007. ISIT 2007. IEEE International Symposium on – start-page: 69 year: 2006 end-page: 80 ident: br0290 article-title: Hardness of approximation results for the problem of finding the stopping distance in tanner graphs publication-title: International Conference on Foundations of Software Technology and Theoretical Computer Science – start-page: 157 year: 2006 end-page: 158 ident: br0050 article-title: On the complexity of finding stopping set size in tanner graphs publication-title: 40th Annual Conference on Information Sciences and Systems – start-page: 739 year: 2010 end-page: 743 ident: br0100 article-title: Exhaustive search for small fully absorbing sets and the corresponding low error-floor decoder publication-title: Information Theory Proceedings (ISIT), 2010 IEEE International Symposium on – start-page: 1091 year: 2011 end-page: 1095 ident: br0260 article-title: An efficient algorithm for finding dominant trapping sets of irregular ldpc codes publication-title: Information Theory Proceedings (ISIT), 2011 IEEE International Symposium on – start-page: 1 year: 2006 end-page: 6 ident: br0070 article-title: A probabilistic subspace approach to the minimal stopping set problem publication-title: Turbo Codes&Related Topics; 6th International ITG-Conference on Source and Channel Coding (TURBOCODING), 2006 4th International Symposium on – volume: 55 start-page: 4167 year: 2009 end-page: 4178 ident: br0090 article-title: An efficient algorithm to find all small-size stopping sets of low-density parity-check matrices publication-title: IEEE Trans. Inf. Theory – year: 2017 ident: br0140 article-title: Lower bounds on the size of smallest elementary and non-elementary trapping sets in variable-regular ldpc codes publication-title: IEEE Commun. Lett. – year: 2014 ident: br0340 – volume: 60 start-page: 5188 year: 2014 end-page: 5203 ident: br0130 article-title: On characterization of elementary trapping sets of variable-regular ldpc codes publication-title: IEEE Trans. Inf. Theory – volume: 65 start-page: 4307 year: 2019 end-page: 4315 ident: br0300 article-title: Hardness results on finding leafless elementary trapping sets and elementary absorbing sets of ldpc codes publication-title: IEEE Trans. Inf. Theory – reference: J. Rosenthal, P.O. Vontobel, Construction of ldpc codes based on ramanujan graphs and ideas from margulis, in: Proc. 38th Annual Allerton Conf. on Communication, Computing and Control, pp. 248–257. – year: 2005 ident: br0150 article-title: Encyclopedia of Sparse Graph Codes – volume: 48 start-page: 1570 year: 2002 end-page: 1579 ident: br0040 article-title: Finite-length analysis of low-density parity-check codes on the binary erasure channel publication-title: IEEE Trans. Inf. Theory – start-page: 329 year: 2010 end-page: 333 ident: br0210 article-title: Calculating the minimum distance of linear block codes via integer programming publication-title: 2010 6th International Symposium on Turbo Codes & Iterative Information Processing – volume: 43 start-page: 1757 year: 1997 end-page: 1766 ident: br0240 article-title: The intractability of computing the minimum distance of a code publication-title: IEEE Trans. Inf. Theory – volume: 18 start-page: 18 year: 2013 end-page: 32 ident: br0350 article-title: Practical guidelines for solving difficult mixed integer linear programs publication-title: Surv. Oper. Res. Manag. Sci. – start-page: 1 year: 2010 end-page: 5 ident: br0200 article-title: Numerical comparison of ip formulations as ml decoders publication-title: 2010 IEEE International Conference on Communications – year: 2015 ident: br0330 article-title: Parameterized Algorithms – volume: 237 start-page: 123 year: 2000 end-page: 134 ident: br0320 article-title: Some apx-completeness results for cubic graphs publication-title: Theor. Comput. Sci. – volume: 52 start-page: 1242 year: 2004 end-page: 1247 ident: br0280 article-title: Selective avoidance of cycles in irregular ldpc code construction publication-title: IEEE Trans. Commun. – volume: 58 start-page: 4753 year: 2012 end-page: 4769 ident: br0220 article-title: Mathematical programming decoding of binary linear codes: theory and algorithms publication-title: IEEE Trans. Inf. Theory – year: 2017 ident: 10.1016/j.tcs.2022.02.028_br0140 article-title: Lower bounds on the size of smallest elementary and non-elementary trapping sets in variable-regular ldpc codes publication-title: IEEE Commun. Lett. doi: 10.1109/LCOMM.2017.2707553 – start-page: 69 year: 2006 ident: 10.1016/j.tcs.2022.02.028_br0290 article-title: Hardness of approximation results for the problem of finding the stopping distance in tanner graphs – start-page: 2589 year: 2014 ident: 10.1016/j.tcs.2022.02.028_br0230 article-title: Efficient maximum-likelihood decoding of linear block codes on binary memoryless channels – volume: 237 start-page: 123 issue: 1–2 year: 2000 ident: 10.1016/j.tcs.2022.02.028_br0320 article-title: Some apx-completeness results for cubic graphs publication-title: Theor. Comput. Sci. doi: 10.1016/S0304-3975(98)00158-3 – volume: 52 start-page: 1242 issue: 8 year: 2004 ident: 10.1016/j.tcs.2022.02.028_br0280 article-title: Selective avoidance of cycles in irregular ldpc code construction publication-title: IEEE Trans. Commun. doi: 10.1109/TCOMM.2004.833048 – start-page: 157 year: 2006 ident: 10.1016/j.tcs.2022.02.028_br0050 article-title: On the complexity of finding stopping set size in tanner graphs – volume: 65 start-page: 4307 issue: 7 year: 2019 ident: 10.1016/j.tcs.2022.02.028_br0300 article-title: Hardness results on finding leafless elementary trapping sets and elementary absorbing sets of ldpc codes publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2019.2904988 – start-page: 1 year: 2010 ident: 10.1016/j.tcs.2022.02.028_br0200 article-title: Numerical comparison of ip formulations as ml decoders – start-page: 2271 year: 2007 ident: 10.1016/j.tcs.2022.02.028_br0250 article-title: On the exhaustion and elimination of trapping sets: algorithms & the suppressing effect – start-page: 411 year: 2006 ident: 10.1016/j.tcs.2022.02.028_br0160 article-title: Upper bounding the performance of arbitrary finite ldpc codes on binary erasure channels – volume: 48 start-page: 1570 issue: 6 year: 2002 ident: 10.1016/j.tcs.2022.02.028_br0040 article-title: Finite-length analysis of low-density parity-check codes on the binary erasure channel publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2002.1003839 – ident: 10.1016/j.tcs.2022.02.028_br0170 – start-page: 295 year: 2008 ident: 10.1016/j.tcs.2022.02.028_br0080 article-title: On the probabilistic computation algorithm for the minimum-size stopping sets of ldpc codes – volume: 60 start-page: 7416 issue: 12 year: 2014 ident: 10.1016/j.tcs.2022.02.028_br0060 article-title: Error floor approximation for ldpc codes in the awgn channel publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2014.2363832 – volume: 58 start-page: 4753 issue: 7 year: 2012 ident: 10.1016/j.tcs.2022.02.028_br0220 article-title: Mathematical programming decoding of binary linear codes: theory and algorithms publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2012.2191697 – volume: 58 start-page: 164 issue: 1 year: 2012 ident: 10.1016/j.tcs.2022.02.028_br0180 article-title: Addendum to ?an efficient algorithm to find all small-size stopping sets of low-density parity-check matrices? publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2011.2171531 – year: 2018 ident: 10.1016/j.tcs.2022.02.028_br0010 article-title: Finding minimum stopping and trapping sets: an integer linear programming approach – volume: 110 start-page: 451 issue: 11 year: 2010 ident: 10.1016/j.tcs.2022.02.028_br0310 article-title: Note on max lin-2 above average publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2010.04.010 – start-page: 1 year: 2006 ident: 10.1016/j.tcs.2022.02.028_br0070 article-title: A probabilistic subspace approach to the minimal stopping set problem – year: 2015 ident: 10.1016/j.tcs.2022.02.028_br0330 – year: 2005 ident: 10.1016/j.tcs.2022.02.028_br0150 – volume: 55 start-page: 4167 issue: 9 year: 2009 ident: 10.1016/j.tcs.2022.02.028_br0090 article-title: An efficient algorithm to find all small-size stopping sets of low-density parity-check matrices publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2009.2025573 – start-page: 329 year: 2010 ident: 10.1016/j.tcs.2022.02.028_br0210 article-title: Calculating the minimum distance of linear block codes via integer programming – volume: 60 start-page: 5188 issue: 9 year: 2014 ident: 10.1016/j.tcs.2022.02.028_br0130 article-title: On characterization of elementary trapping sets of variable-regular ldpc codes publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2014.2334657 – start-page: 2216 year: 2009 ident: 10.1016/j.tcs.2022.02.028_br0190 article-title: Valid inequalities for binary linear codes – start-page: 1091 year: 2011 ident: 10.1016/j.tcs.2022.02.028_br0260 article-title: An efficient algorithm for finding dominant trapping sets of irregular ldpc codes – start-page: 1 year: 2003 ident: 10.1016/j.tcs.2022.02.028_br0270 article-title: Graph-covers and iterative decoding of finite length codes – start-page: 739 year: 2010 ident: 10.1016/j.tcs.2022.02.028_br0100 article-title: Exhaustive search for small fully absorbing sets and the corresponding low error-floor decoder – volume: 55 start-page: 2234 issue: 12 year: 2007 ident: 10.1016/j.tcs.2022.02.028_br0030 article-title: Estimation of bit and frame error rates of finite-length low-density parity-check codes on binary symmetric channels publication-title: IEEE Trans. Commun. doi: 10.1109/TCOMM.2007.910589 – volume: 43 start-page: 1757 issue: 6 year: 1997 ident: 10.1016/j.tcs.2022.02.028_br0240 article-title: The intractability of computing the minimum distance of a code publication-title: IEEE Trans. Inf. Theory doi: 10.1109/18.641542 – start-page: 1426 year: 1998 ident: 10.1016/j.tcs.2022.02.028_br0020 article-title: Error Floors of ldpc Codes – ident: 10.1016/j.tcs.2022.02.028_br0110 – volume: 56 start-page: 1640 issue: 4 year: 2010 ident: 10.1016/j.tcs.2022.02.028_br0120 article-title: On the hardness of approximating stopping and trapping sets publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2010.2040941 – volume: 18 start-page: 18 issue: 1 year: 2013 ident: 10.1016/j.tcs.2022.02.028_br0350 article-title: Practical guidelines for solving difficult mixed integer linear programs publication-title: Surv. Oper. Res. Manag. Sci. |
| SSID | ssj0000576 |
| Score | 2.361329 |
| 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... |
| SourceID | crossref elsevier |
| SourceType | Index Database Publisher |
| StartPage | 26 |
| 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 |
| URI | https://dx.doi.org/10.1016/j.tcs.2022.02.028 |
| Volume | 915 |
| WOSCitedRecordID | wos000795611200003&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1879-2294 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000576 issn: 0304-3975 databaseCode: AIEXJ dateStart: 20211207 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3NT9swFLf42AEOjBUQZRvyYadVqRInjZMjQkzb0IBDmXqLbMeWqNSkStJS8dfzEttpYCCNA1IVNVHitHm_vA_7vfdD6BuEsWAFReS4lPpOIMPY4Yoqhwap4CJ1Vaw02QS9uoomk_jGpA6VDZ0AzbJotYrn7ypqOAbCrktn3yDudlA4AN9B6LAFscP2vwR_rRMXm1xxuTIZF3p63NzW-pt1W5HZYjaomwvMbbFiVTC9U8pqYOhmyq4LO-6UPgrDCTEwhtSK768EpxwszoOuoVmyIl-vPkF8PtNMUoPLYWsU8ikMAbr43hBp39zlVdGdk4Bwtm5nGnRUl1-vt8SaE8Xq2dgbdTVl2LG5ugXkP9pcTyxMh5WoG6sT0nRXNbXkTzpnP7NobZ6hTWGbJjBEUg-RuPUn2kTbhI5iUIPbZ78uJr_XxntE9fK2-QN2IbxJCXz2O152ZTruyXgf7Zm4Ap9pPHxCGzLroY-WswMbFd5Du3_aPr3lARpfZxh28RosOFcYYIBbsOAqb04xYMEWLM1ZFiwYwIItWA7R7Y-L8flPx_BsOAKC78rxPF9SHqR-JJniXLgeh5BLqlHoB9RNFWFEsIAyTlyuhOcrynnoKRG5LGIEFPwR2sryTB4jDA4poyFl8KYLcE0ZDwj3iYSwWKUUbFkffbePLJnrdirJq0Lqo8A-1MTAWPt5CQDk9ctO3nKPz2hnDeAvaKsqFvIr-iCW1V1ZnBp0PAJkBITn |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+the+complexity+of+and+solutions+to+the+minimum+stopping+and+trapping+set+problems&rft.jtitle=Theoretical+computer+science&rft.au=Velasquez%2C+Alvaro&rft.au=Subramani%2C+K.&rft.au=Wojciechowski%2C+Piotr&rft.date=2022-05-14&rft.issn=0304-3975&rft.volume=915&rft.spage=26&rft.epage=44&rft_id=info:doi/10.1016%2Fj.tcs.2022.02.028&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_tcs_2022_02_028 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0304-3975&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0304-3975&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0304-3975&client=summon |