A Fixed-Parameter Perspective on #BIS
The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class # RH Π 1 . It is believed that #BIS does not have an efficient approximation algorithm but also that it is not...
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| Vydané v: | Algorithmica Ročník 81; číslo 10; s. 3844 - 3864 |
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01.10.2019
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| Abstract | The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class
#
RH
Π
1
. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in
#
RH
Π
1
, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms. |
|---|---|
| AbstractList | The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class
#
RH
Π
1
. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in
#
RH
Π
1
, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms. The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class #RHΠ1. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in #RHΠ1, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms. |
| Author | Fomin, Fedor Lapinskas, John Curticapean, Radu Goldberg, Leslie Ann Dell, Holger |
| Author_xml | – sequence: 1 givenname: Radu surname: Curticapean fullname: Curticapean, Radu organization: Basic Algorithms Research Copenhagen, IT University of Copenhagen – sequence: 2 givenname: Holger surname: Dell fullname: Dell, Holger organization: Department of Computer Science, IT University of Copenhagen – sequence: 3 givenname: Fedor surname: Fomin fullname: Fomin, Fedor organization: Department of Informatics, University of Bergen – sequence: 4 givenname: Leslie Ann surname: Goldberg fullname: Goldberg, Leslie Ann organization: Department of Computer Science, University of Oxford – sequence: 5 givenname: John orcidid: 0000-0003-3197-0854 surname: Lapinskas fullname: Lapinskas, John email: john.lapinskas@cs.ox.ac.uk organization: Department of Computer Science, University of Oxford |
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| Cites_doi | 10.1016/j.jcss.2015.11.009 10.1007/978-1-4471-5559-1 10.1090/coll/060 10.1007/978-3-319-21275-3 10.1137/S0097539703427203 10.1145/2746539.2746598 10.1007/s00453-003-1073-y 10.1145/3055399.3055502 10.1016/0001-8708(85)90121-5 10.1137/140997580 10.1002/9781118032718 10.1006/jcss.2000.1727 10.1109/FOCS.2014.22 10.1007/s00224-003-1111-9 10.1007/11847250_5 10.1073/pnas.1505664112 10.1016/j.tcs.2007.05.023 10.1137/S0097539797321602 |
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| References | Curticapean, R., Dell, H., Fomin, F., Goldberg, L.A., Lapinskas, J.: A fixed-parameter perspective on #BIS. In: 12th International Symposium on Parameterized and Exact Computation (IPEC), pp. 13:1–13:13 (2017) FlumJGroheMThe parameterized complexity of counting problemsSIAM J. Comput.2004334892922206533810.1137/S00975397034272031105.68042 XiaMZhangPZhaoWComputational complexity of counting problems on 3-regular planar graphsTheor. Comput. Sci.20073841111125235422710.1016/j.tcs.2007.05.0231124.68083 FrickMGeneralized model-checking over locally tree-decomposable classesTheory Comput. Syst.2004371157191203840710.1007/s00224-003-1111-91101.68727 GoldbergLAJerrumMA complexity classification of spin systems with an external fieldProc. Natl. Acad. Sci.2015112431316113166342176210.1073/pnas.15056641121355.68120 Liu, J., Lu, P.: FPTAS for #BIS with degree bounds on one side. In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, 14–17 June 2015, pp. 549–556 (2015) VadhanSPThe complexity of counting in sparse, regular, and planar graphsSIAM J. Comput.2001312398427186128210.1137/S00975397973216020994.68070 LovászLLarge Networks and Graph Limits2012ProvidenceAmerican Mathematical Society1292.05001 Müller, M.: Randomized approximations of parameterized counting problems. In: Proceedings of the Second International Conference on Parameterized and Exact Computation, IWPEC’06, pp. 50–59. Springer, Berlin (2006) SipserMIntroduction to the Theory of Computation19961StamfordInternational Thomson Publishing1169.68300 Patel, V., Regts, G.: Computing the number of induced copies of a fixed graph in a bounded degree graph. CoRR, abs/1707.05186 (2017) CaiJ-YGalanisAGoldbergLAGuoHJerrumMStefankovicDVigodaE#BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness regionJ. Comput. Syst. Sci.2016825690711348036010.1016/j.jcss.2015.11.0091338.68086 Curticapean, R., Dell, H., Marx, D.: Homomorphisms are a good basis for counting small subgraphs. In: Proceedings of the 49th Annual ACM Symposium on Theory of Computing, pp. 210–213 (2017) JansonSŁuczakTRucinskiARandom Graphs2000New YorkWiley10.1002/97811180327180968.05003 FlumJGroheMParameterized Complexity Theory2006BerlinSpringer1143.68016 Curticapean, R., Marx, D.: Complexity of counting subgraphs: only the boundedness of the vertex-cover number counts. In: 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, 18–21 Oct 2014, pp. 130–139 (2014) CyganMFominFKowalikŁLokshtanovDMarxDPilipczukMPilipczukMSaurabhSParameterized Algorithms2015BerlinSpringer10.1007/978-3-319-21275-31334.90001 DyerMEGoldbergLAGreenhillCSJerrumMThe relative complexity of approximate counting problemsAlgorithmica2003383471500204488610.1007/s00453-003-1073-y1138.68424 DowneyRGFellowsMRFundamentals of Parameterized Complexity2013BerlinSpringer10.1007/978-1-4471-5559-11358.68006 GalanisAStefankovicDVigodaEYangLFerromagnetic Potts model: refined #BIS-hardness and related resultsSIAM J. Comput.201645620042065357237510.1137/1409975801355.68198 ImpagliazzoRPaturiROn the complexity of k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-SATJ. Comput. Syst. Sci.2001622367375182059710.1006/jcss.2000.17270990.68079 GesselIViennotGBinomial determinants, paths, and hook length formulaeAdv. Math.198558330032181536010.1016/0001-8708(85)90121-50579.05004 ArvindVRamanVApproximation Algorithms for Some Parameterized Counting Problems2002BerlinSpringer4534641019.68135 Patel, V., Regts, G.: Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. CoRR, abs/1607.01167 (2016) M Cygan (606_CR6) 2015 I Gessel (606_CR13) 1985; 58 S Janson (606_CR16) 2000 L Lovász (606_CR18) 2012 M Sipser (606_CR22) 1996 RG Downey (606_CR7) 2013 ME Dyer (606_CR8) 2003; 38 J Flum (606_CR9) 2004; 33 R Impagliazzo (606_CR15) 2001; 62 M Xia (606_CR24) 2007; 384 LA Goldberg (606_CR14) 2015; 112 606_CR19 606_CR17 M Frick (606_CR11) 2004; 37 J-Y Cai (606_CR2) 2016; 82 606_CR20 V Arvind (606_CR1) 2002 606_CR21 606_CR4 SP Vadhan (606_CR23) 2001; 31 606_CR5 A Galanis (606_CR12) 2016; 45 606_CR3 J Flum (606_CR10) 2006 |
| References_xml | – reference: FlumJGroheMParameterized Complexity Theory2006BerlinSpringer1143.68016 – reference: Curticapean, R., Dell, H., Marx, D.: Homomorphisms are a good basis for counting small subgraphs. In: Proceedings of the 49th Annual ACM Symposium on Theory of Computing, pp. 210–213 (2017) – reference: JansonSŁuczakTRucinskiARandom Graphs2000New YorkWiley10.1002/97811180327180968.05003 – reference: LovászLLarge Networks and Graph Limits2012ProvidenceAmerican Mathematical Society1292.05001 – reference: XiaMZhangPZhaoWComputational complexity of counting problems on 3-regular planar graphsTheor. Comput. Sci.20073841111125235422710.1016/j.tcs.2007.05.0231124.68083 – reference: GoldbergLAJerrumMA complexity classification of spin systems with an external fieldProc. Natl. Acad. Sci.2015112431316113166342176210.1073/pnas.15056641121355.68120 – reference: CaiJ-YGalanisAGoldbergLAGuoHJerrumMStefankovicDVigodaE#BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness regionJ. Comput. Syst. Sci.2016825690711348036010.1016/j.jcss.2015.11.0091338.68086 – reference: DowneyRGFellowsMRFundamentals of Parameterized Complexity2013BerlinSpringer10.1007/978-1-4471-5559-11358.68006 – reference: FrickMGeneralized model-checking over locally tree-decomposable classesTheory Comput. Syst.2004371157191203840710.1007/s00224-003-1111-91101.68727 – reference: Patel, V., Regts, G.: Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. CoRR, abs/1607.01167 (2016) – reference: GalanisAStefankovicDVigodaEYangLFerromagnetic Potts model: refined #BIS-hardness and related resultsSIAM J. Comput.201645620042065357237510.1137/1409975801355.68198 – reference: Müller, M.: Randomized approximations of parameterized counting problems. In: Proceedings of the Second International Conference on Parameterized and Exact Computation, IWPEC’06, pp. 50–59. Springer, Berlin (2006) – reference: GesselIViennotGBinomial determinants, paths, and hook length formulaeAdv. Math.198558330032181536010.1016/0001-8708(85)90121-50579.05004 – reference: Liu, J., Lu, P.: FPTAS for #BIS with degree bounds on one side. In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, 14–17 June 2015, pp. 549–556 (2015) – reference: Patel, V., Regts, G.: Computing the number of induced copies of a fixed graph in a bounded degree graph. CoRR, abs/1707.05186 (2017) – reference: CyganMFominFKowalikŁLokshtanovDMarxDPilipczukMPilipczukMSaurabhSParameterized Algorithms2015BerlinSpringer10.1007/978-3-319-21275-31334.90001 – reference: ImpagliazzoRPaturiROn the complexity of k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-SATJ. Comput. Syst. Sci.2001622367375182059710.1006/jcss.2000.17270990.68079 – reference: VadhanSPThe complexity of counting in sparse, regular, and planar graphsSIAM J. Comput.2001312398427186128210.1137/S00975397973216020994.68070 – reference: DyerMEGoldbergLAGreenhillCSJerrumMThe relative complexity of approximate counting problemsAlgorithmica2003383471500204488610.1007/s00453-003-1073-y1138.68424 – reference: Curticapean, R., Dell, H., Fomin, F., Goldberg, L.A., Lapinskas, J.: A fixed-parameter perspective on #BIS. In: 12th International Symposium on Parameterized and Exact Computation (IPEC), pp. 13:1–13:13 (2017) – reference: FlumJGroheMThe parameterized complexity of counting problemsSIAM J. Comput.2004334892922206533810.1137/S00975397034272031105.68042 – reference: Curticapean, R., Marx, D.: Complexity of counting subgraphs: only the boundedness of the vertex-cover number counts. In: 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, 18–21 Oct 2014, pp. 130–139 (2014) – reference: ArvindVRamanVApproximation Algorithms for Some Parameterized Counting Problems2002BerlinSpringer4534641019.68135 – reference: SipserMIntroduction to the Theory of Computation19961StamfordInternational Thomson Publishing1169.68300 – volume: 82 start-page: 690 issue: 5 year: 2016 ident: 606_CR2 publication-title: J. Comput. Syst. Sci. doi: 10.1016/j.jcss.2015.11.009 – volume-title: Parameterized Complexity Theory year: 2006 ident: 606_CR10 – volume-title: Fundamentals of Parameterized Complexity year: 2013 ident: 606_CR7 doi: 10.1007/978-1-4471-5559-1 – volume-title: Large Networks and Graph Limits year: 2012 ident: 606_CR18 doi: 10.1090/coll/060 – ident: 606_CR3 – volume-title: Parameterized Algorithms year: 2015 ident: 606_CR6 doi: 10.1007/978-3-319-21275-3 – volume: 33 start-page: 892 issue: 4 year: 2004 ident: 606_CR9 publication-title: SIAM J. Comput. doi: 10.1137/S0097539703427203 – ident: 606_CR17 doi: 10.1145/2746539.2746598 – volume: 38 start-page: 471 issue: 3 year: 2003 ident: 606_CR8 publication-title: Algorithmica doi: 10.1007/s00453-003-1073-y – ident: 606_CR4 doi: 10.1145/3055399.3055502 – volume: 58 start-page: 300 issue: 3 year: 1985 ident: 606_CR13 publication-title: Adv. Math. doi: 10.1016/0001-8708(85)90121-5 – ident: 606_CR21 – ident: 606_CR20 – volume: 45 start-page: 2004 issue: 6 year: 2016 ident: 606_CR12 publication-title: SIAM J. Comput. doi: 10.1137/140997580 – volume-title: Random Graphs year: 2000 ident: 606_CR16 doi: 10.1002/9781118032718 – volume: 62 start-page: 367 issue: 2 year: 2001 ident: 606_CR15 publication-title: J. Comput. Syst. Sci. doi: 10.1006/jcss.2000.1727 – start-page: 453 volume-title: Approximation Algorithms for Some Parameterized Counting Problems year: 2002 ident: 606_CR1 – ident: 606_CR5 doi: 10.1109/FOCS.2014.22 – volume: 37 start-page: 157 issue: 1 year: 2004 ident: 606_CR11 publication-title: Theory Comput. Syst. doi: 10.1007/s00224-003-1111-9 – ident: 606_CR19 doi: 10.1007/11847250_5 – volume: 112 start-page: 13161 issue: 43 year: 2015 ident: 606_CR14 publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1505664112 – volume: 384 start-page: 111 issue: 1 year: 2007 ident: 606_CR24 publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2007.05.023 – volume-title: Introduction to the Theory of Computation year: 1996 ident: 606_CR22 – volume: 31 start-page: 398 issue: 2 year: 2001 ident: 606_CR23 publication-title: SIAM J. Comput. doi: 10.1137/S0097539797321602 |
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