A (1.4+ϵ)-approximation algorithm for the 2-Max-Duo problem
The maximum duo-preservation string mapping ( Max-Duo ) problem is the complement of the well studied minimum common string partition problem, both of which have applications in many fields including text compression and bioinformatics. k - Max-Duo is the restricted version of Max-Duo , where every...
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| Vydáno v: | Journal of combinatorial optimization Ročník 40; číslo 3; s. 806 - 824 |
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01.10.2020
Springer Nature B.V |
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| Abstract | The
maximum duo-preservation string mapping
(
Max-Duo
) problem is the complement of the well studied
minimum common string partition
problem, both of which have applications in many fields including text compression and bioinformatics.
k
-
Max-Duo
is the restricted version of
Max-Duo
, where every letter of the alphabet occurs at most
k
times in each of the strings, which is readily reduced into the well known
maximum independent set
(
MIS
) problem on a graph of maximum degree
Δ
≤
6
(
k
-
1
)
. In particular, 2-
Max-Duo
can then be approximated arbitrarily close to 1.8 using the state-of-the-art approximation algorithm for the
MIS
problem on bounded-degree graphs. 2-
Max-Duo
was proved APX-hard and very recently a
(
1.6
+
ϵ
)
-approximation algorithm was claimed, for any
ϵ
>
0
. In this paper, we present a vertex-degree reduction technique, based on which, we show that 2-
Max-Duo
can be approximated arbitrarily close to 1.4. |
|---|---|
| AbstractList | The
maximum duo-preservation string mapping
(
Max-Duo
) problem is the complement of the well studied
minimum common string partition
problem, both of which have applications in many fields including text compression and bioinformatics.
k
-
Max-Duo
is the restricted version of
Max-Duo
, where every letter of the alphabet occurs at most
k
times in each of the strings, which is readily reduced into the well known
maximum independent set
(
MIS
) problem on a graph of maximum degree
Δ
≤
6
(
k
-
1
)
. In particular, 2-
Max-Duo
can then be approximated arbitrarily close to 1.8 using the state-of-the-art approximation algorithm for the
MIS
problem on bounded-degree graphs. 2-
Max-Duo
was proved APX-hard and very recently a
(
1.6
+
ϵ
)
-approximation algorithm was claimed, for any
ϵ
>
0
. In this paper, we present a vertex-degree reduction technique, based on which, we show that 2-
Max-Duo
can be approximated arbitrarily close to 1.4. The maximum duo-preservation string mapping (Max-Duo) problem is the complement of the well studied minimum common string partition problem, both of which have applications in many fields including text compression and bioinformatics. k-Max-Duo is the restricted version of Max-Duo, where every letter of the alphabet occurs at most k times in each of the strings, which is readily reduced into the well known maximum independent set (MIS) problem on a graph of maximum degree Δ≤6(k-1). In particular, 2-Max-Duo can then be approximated arbitrarily close to 1.8 using the state-of-the-art approximation algorithm for the MIS problem on bounded-degree graphs. 2-Max-Duo was proved APX-hard and very recently a (1.6+ϵ)-approximation algorithm was claimed, for any ϵ>0. In this paper, we present a vertex-degree reduction technique, based on which, we show that 2-Max-Duo can be approximated arbitrarily close to 1.4. |
| Author | Xu, Yao Zhang, Peng Lin, Guohui Su, Bing Liu, Tian Chen, Yong Luo, Taibo |
| Author_xml | – sequence: 1 givenname: Yong surname: Chen fullname: Chen, Yong organization: Department of Mathematics, Hangzhou Dianzi University – sequence: 2 givenname: Guohui orcidid: 0000-0003-4283-3396 surname: Lin fullname: Lin, Guohui email: guohui@ualberta.ca organization: Department of Computing Science, University of Alberta – sequence: 3 givenname: Tian surname: Liu fullname: Liu, Tian organization: Key Laboratory of High Confidence Software Technologies (MOE), Department of Computer Science and Technology, Peking University – sequence: 4 givenname: Taibo surname: Luo fullname: Luo, Taibo organization: School of Economics and Management, Xidian University – sequence: 5 givenname: Bing surname: Su fullname: Su, Bing organization: School of Economics and Management, Xi’an Technological University – sequence: 6 givenname: Yao surname: Xu fullname: Xu, Yao organization: Department of Computer Science, Kettering University – sequence: 7 givenname: Peng surname: Zhang fullname: Zhang, Peng organization: School of Computer Science and Technology, Shandong University |
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| Keywords | String partition Duo-preservation string mapping G.4 Algorithm design and analysis Independent set F.2.2 Pattern matching Approximation algorithm G.2.1 Combinatorial algorithms |
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| References | BermanPFujitoTOn approximation properties of the independent set problem for low degree graphsTheory Comput Syst199932115132166390810.1007/s002240000113 BerettaSCastelliMDondiRParameterized tractability of the maximum-duo preservation string mapping problemTheoret Comput Sci20166461625354196510.1016/j.tcs.2016.07.011 Brubach B (2016) Further improvement in approximating the maximum duo-preservation string mapping problem. In: Proceedings of the 16th international workshop on algorithms in bioinformatics (WABI 2016), volume 9838 of LNBI, pp 52–64 Damaschke P (2008) Minimum common string partition parameterized. In: Proceedings of the 8th international workshop on algorithms in bioinformatics (WABI 2008), volume 5251 of LNBI, pp 87–98 Bulteau L, Komusiewicz C (2014) Minimum common string partition parameterized by partition size is fixed-parameter tractable. In: Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms (SODA’14), pp 102–121 SwensonKMMarronMEarnest-DeYoungJVMoretBMApproximating the true evolutionary distance between two genomesJ Exp Algorithmics20081235244370110.1145/1227161.1402297 Boria N, Cabodi G, Camurati P, Palena M, Pasini P, Quer S (2016) A 7/2-approximation algorithm for the maximum duo-preservation string mapping problem. In: Proceedings of the 27th annual symposium on combinatorial pattern matching (CPM 2016), volume 54 of LIPIcs, pp 11:1–11:8 ChenWChenZSamatovaNFPengLWangJTangMSolving the maximum duo-preservation string mapping problem with linear programmingTheoret Comput Sci2014530111318136610.1016/j.tcs.2014.02.017 Goldstein A, Kolman P, Zheng J (2004) Minimum common string partition problem: Hardness and approximations. In: Proceedings of the 15th international symposium on algorithms and computation (ISAAC 2004), volume 3341 of LNCS, pp 484–495 KolmanPWaleńTApproximating reversal distance for strings with bounded number of duplicatesDiscrete Appl Math2007155327336230315610.1016/j.dam.2006.05.011 Xu Y, Chen Y, Luo T, Lin G (2017) A local search 2.917-approximation algorithm for duo-preservation string mapping. arXiv:1702.01877 Kolman P, Waleń T (2006) Reversal distance for strings with duplicates: linear time approximation using hitting set. In: Proceedings of the 4th international workshop on approximation and online algorithms (WAOA 2006), volume 4368 of LNCS, pp 279–289 ChenXZhengJFuZNanPZhongYLonardiSJiangTAssignment of orthologous genes via genome rearrangementIEEE/ACM Trans Comput Biol Bioinf2005230231510.1109/TCBB.2005.48 CormodeGMuthukrishnanSThe string edit distance matching problem with movesACM Trans Algorithms200732:12:19230182810.1145/1186810.1186812 Bulteau L, Fertin G, Komusiewicz C, Rusu I (2013) A fixed-parameter algorithm for minimum common string partition with few duplications. In: Proceedings of the 13th international workshop on algorithms in bioinformatics (WABI 2013), volume 8126 of LNBI, pp 244–258 Dudek B, Gawrychowski P, Ostropolski-Nalewaja P (2017) A family of approximation algorithms for the maximum duo-preservation string mapping problem. In: Proceedings of the 28th annual symposium on combinatorial pattern matching (CPM 2017), volume 78 of LIPIcs, pp 10:1–10:14. arXiv:1702.02405 JiangHZhuBZhuDZhuHMinimum common string partition revisitedJ Comb Optim201223519527290396510.1007/s10878-010-9370-2 Berman P, Karpinski M (1999) On some tighter inapproximability results. In: Proceedings of the of 26th international colloquium on automata, languages and programming (ICALP’99), pp 200–209 GareyMRJohnsonDSComputers and intractability: a guide to the theory of NP-completeness1979San FranciscoW. H. Freeman and Company0411.68039 Chrobak M, Kolman P, Sgall J (2004) The greedy algorithm for the minimum common string partition problem. In: Proceedings of the 7th international workshop on approximation algorithms for combinatorial optimization problems (APPROX 2004) and the 8th international workshop on randomization and computation (RANDOM 2004), volume 3122 of LNCS, pp 84–95 Boria N, Kurpisz A, Leppänen S, Mastrolilli M (2014) Improved approximation for the maximum duo-preservation string mapping problem. In: Proceedings of the 14th international workshop on algorithms in bioinformatics (WABI 2014), volume 8701 of LNBI, pp 14–25 Beretta S, Castelli M, Dondi R (2016a) Corrigendum to “Parameterized tractability of the maximum-duo preservation string mapping problem” [646(2016), 16–25]. Theoret Comput Sci 653:108–110 621_CR9 G Cormode (621_CR13) 2007; 3 P Berman (621_CR3) 1999; 32 P Kolman (621_CR19) 2007; 155 W Chen (621_CR11) 2014; 530 621_CR17 621_CR14 MR Garey (621_CR16) 1979 621_CR15 621_CR12 X Chen (621_CR10) 2005; 2 621_CR22 621_CR1 621_CR20 621_CR4 H Jiang (621_CR18) 2012; 23 621_CR6 621_CR5 S Beretta (621_CR2) 2016; 646 621_CR8 621_CR7 KM Swenson (621_CR21) 2008; 12 |
| References_xml | – reference: Beretta S, Castelli M, Dondi R (2016a) Corrigendum to “Parameterized tractability of the maximum-duo preservation string mapping problem” [646(2016), 16–25]. Theoret Comput Sci 653:108–110 – reference: KolmanPWaleńTApproximating reversal distance for strings with bounded number of duplicatesDiscrete Appl Math2007155327336230315610.1016/j.dam.2006.05.011 – reference: Goldstein A, Kolman P, Zheng J (2004) Minimum common string partition problem: Hardness and approximations. In: Proceedings of the 15th international symposium on algorithms and computation (ISAAC 2004), volume 3341 of LNCS, pp 484–495 – reference: Xu Y, Chen Y, Luo T, Lin G (2017) A local search 2.917-approximation algorithm for duo-preservation string mapping. arXiv:1702.01877 – reference: CormodeGMuthukrishnanSThe string edit distance matching problem with movesACM Trans Algorithms200732:12:19230182810.1145/1186810.1186812 – reference: ChenXZhengJFuZNanPZhongYLonardiSJiangTAssignment of orthologous genes via genome rearrangementIEEE/ACM Trans Comput Biol Bioinf2005230231510.1109/TCBB.2005.48 – reference: Brubach B (2016) Further improvement in approximating the maximum duo-preservation string mapping problem. In: Proceedings of the 16th international workshop on algorithms in bioinformatics (WABI 2016), volume 9838 of LNBI, pp 52–64 – reference: Damaschke P (2008) Minimum common string partition parameterized. In: Proceedings of the 8th international workshop on algorithms in bioinformatics (WABI 2008), volume 5251 of LNBI, pp 87–98 – reference: BerettaSCastelliMDondiRParameterized tractability of the maximum-duo preservation string mapping problemTheoret Comput Sci20166461625354196510.1016/j.tcs.2016.07.011 – reference: Bulteau L, Fertin G, Komusiewicz C, Rusu I (2013) A fixed-parameter algorithm for minimum common string partition with few duplications. In: Proceedings of the 13th international workshop on algorithms in bioinformatics (WABI 2013), volume 8126 of LNBI, pp 244–258 – reference: GareyMRJohnsonDSComputers and intractability: a guide to the theory of NP-completeness1979San FranciscoW. H. Freeman and Company0411.68039 – reference: BermanPFujitoTOn approximation properties of the independent set problem for low degree graphsTheory Comput Syst199932115132166390810.1007/s002240000113 – reference: Chrobak M, Kolman P, Sgall J (2004) The greedy algorithm for the minimum common string partition problem. In: Proceedings of the 7th international workshop on approximation algorithms for combinatorial optimization problems (APPROX 2004) and the 8th international workshop on randomization and computation (RANDOM 2004), volume 3122 of LNCS, pp 84–95 – reference: Dudek B, Gawrychowski P, Ostropolski-Nalewaja P (2017) A family of approximation algorithms for the maximum duo-preservation string mapping problem. In: Proceedings of the 28th annual symposium on combinatorial pattern matching (CPM 2017), volume 78 of LIPIcs, pp 10:1–10:14. arXiv:1702.02405 – reference: JiangHZhuBZhuDZhuHMinimum common string partition revisitedJ Comb Optim201223519527290396510.1007/s10878-010-9370-2 – reference: Berman P, Karpinski M (1999) On some tighter inapproximability results. In: Proceedings of the of 26th international colloquium on automata, languages and programming (ICALP’99), pp 200–209 – reference: Boria N, Kurpisz A, Leppänen S, Mastrolilli M (2014) Improved approximation for the maximum duo-preservation string mapping problem. In: Proceedings of the 14th international workshop on algorithms in bioinformatics (WABI 2014), volume 8701 of LNBI, pp 14–25 – reference: Boria N, Cabodi G, Camurati P, Palena M, Pasini P, Quer S (2016) A 7/2-approximation algorithm for the maximum duo-preservation string mapping problem. In: Proceedings of the 27th annual symposium on combinatorial pattern matching (CPM 2016), volume 54 of LIPIcs, pp 11:1–11:8 – reference: SwensonKMMarronMEarnest-DeYoungJVMoretBMApproximating the true evolutionary distance between two genomesJ Exp Algorithmics20081235244370110.1145/1227161.1402297 – reference: ChenWChenZSamatovaNFPengLWangJTangMSolving the maximum duo-preservation string mapping problem with linear programmingTheoret Comput Sci2014530111318136610.1016/j.tcs.2014.02.017 – reference: Bulteau L, Komusiewicz C (2014) Minimum common string partition parameterized by partition size is fixed-parameter tractable. In: Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms (SODA’14), pp 102–121 – reference: Kolman P, Waleń T (2006) Reversal distance for strings with duplicates: linear time approximation using hitting set. In: Proceedings of the 4th international workshop on approximation and online algorithms (WAOA 2006), volume 4368 of LNCS, pp 279–289 – ident: 621_CR20 doi: 10.1007/11970125_22 – ident: 621_CR6 – ident: 621_CR7 doi: 10.1007/978-3-319-43681-4_5 – volume: 23 start-page: 519 year: 2012 ident: 621_CR18 publication-title: J Comb Optim doi: 10.1007/s10878-010-9370-2 – volume: 530 start-page: 1 year: 2014 ident: 621_CR11 publication-title: Theoret Comput Sci doi: 10.1016/j.tcs.2014.02.017 – ident: 621_CR14 doi: 10.1007/978-3-540-87361-7_8 – ident: 621_CR8 doi: 10.1007/978-3-642-40453-5_19 – volume: 2 start-page: 302 year: 2005 ident: 621_CR10 publication-title: IEEE/ACM Trans Comput Biol Bioinf doi: 10.1109/TCBB.2005.48 – volume-title: Computers and intractability: a guide to the theory of NP-completeness year: 1979 ident: 621_CR16 – volume: 3 start-page: 2:1 year: 2007 ident: 621_CR13 publication-title: ACM Trans Algorithms doi: 10.1145/1186810.1186812 – ident: 621_CR5 doi: 10.1007/978-3-662-44753-6_2 – volume: 155 start-page: 327 year: 2007 ident: 621_CR19 publication-title: Discrete Appl Math doi: 10.1016/j.dam.2006.05.011 – ident: 621_CR1 doi: 10.1016/j.tcs.2016.09.015 – volume: 12 start-page: 3 year: 2008 ident: 621_CR21 publication-title: J Exp Algorithmics doi: 10.1145/1227161.1402297 – volume: 32 start-page: 115 year: 1999 ident: 621_CR3 publication-title: Theory Comput Syst doi: 10.1007/s002240000113 – ident: 621_CR15 – ident: 621_CR9 doi: 10.1137/1.9781611973402.8 – volume: 646 start-page: 16 year: 2016 ident: 621_CR2 publication-title: Theoret Comput Sci doi: 10.1016/j.tcs.2016.07.011 – ident: 621_CR12 doi: 10.1007/978-3-540-27821-4_8 – ident: 621_CR22 – ident: 621_CR4 doi: 10.1007/3-540-48523-6_17 – ident: 621_CR17 doi: 10.1007/978-3-540-30551-4_43 |
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| Snippet | The
maximum duo-preservation string mapping
(
Max-Duo
) problem is the complement of the well studied
minimum common string partition
problem, both of which... The maximum duo-preservation string mapping (Max-Duo) problem is the complement of the well studied minimum common string partition problem, both of which have... |
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| SubjectTerms | Algorithms Approximation Bioinformatics Combinatorics Convex and Discrete Geometry Degree reduction Mapping Mathematical analysis Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Strings Theory of Computation |
| Title | A (1.4+ϵ)-approximation algorithm for the 2-Max-Duo problem |
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