Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs
Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for exam...
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| Vydáno v: | Algorithmica Ročník 85; číslo 8; s. 2454 - 2481 |
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| Jazyk: | angličtina |
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01.08.2023
Springer Nature B.V |
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case. |
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| AbstractList | Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case. |
| Author | Köhler, Ekkehard Beisegel, Jesse Strehler, Martin Scheffler, Robert |
| Author_xml | – sequence: 1 givenname: Jesse orcidid: 0000-0002-8760-0169 surname: Beisegel fullname: Beisegel, Jesse organization: Institute of Mathematics, Brandenburg University of Technology – sequence: 2 givenname: Ekkehard surname: Köhler fullname: Köhler, Ekkehard organization: Institute of Mathematics, Brandenburg University of Technology – sequence: 3 givenname: Robert orcidid: 0000-0001-6007-4202 surname: Scheffler fullname: Scheffler, Robert email: robert.scheffler@b-tu.de organization: Institute of Mathematics, Brandenburg University of Technology – sequence: 4 givenname: Martin orcidid: 0000-0003-4241-6584 surname: Strehler fullname: Strehler, Martin organization: Department of Mathematics, Westsächsische Hochschule Zwickau |
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| Cites_doi | 10.1016/0022-2496(87)90014-9 10.1137/S0097539700372216 10.1007/978-3-030-60440-0_11 10.1016/j.tcs.2008.07.020 10.1137/0205021 10.1007/s00453-013-9835-7 10.1016/0166-218X(87)90050-3 10.1137/0206008 10.1137/0218010 10.1016/0167-6377(86)90026-X 10.1007/s00453-008-9273-0 10.1007/978-3-540-73545-8_40 10.1137/0603036 10.1145/1383369.1383371 10.1016/S0167-5060(08)70731-3 10.1137/1.9780898719796 10.1016/0166-218X(90)90092-Q 10.1137/0605055 10.1016/S0166-218X(03)00448-7 10.1145/2897518.2897568 10.1007/PL00009223 10.1007/BF01189067 10.1007/PL00009228 10.1145/335305.335345 10.1007/978-3-319-30139-6_25 10.1016/S0196-6774(03)00082-8 |
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| Keywords | 05C45 Hamiltonian cycles 05C85 Chain graphs Difference graphs Hamiltonian paths Fully dynamic algorithms Threshold graphs 68R10 |
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| References | HenzingerMRFredmanMLLower bounds for fully dynamic connectivity problems in graphsAlgorithmica1998223351362164611310.1007/PL000092280915.68132 IbarraLFully dynamic algorithms for chordal graphs and split graphsACM Trans. Algorithms200844140244695910.1145/1383369.13833711445.05103 YannakakisMThe complexity of the partial order dimension problemSIAM J. Algebr. Discrete Methods19823335135866686010.1137/06030360516.06001 KleinPNSubramanianSA fully dynamic approximation scheme for shortest paths in planar graphsAlgorithmica1998223235249164610810.1007/PL000092230915.68130 Calamoneri, T., Monti, A., Petreschi, R.: Fully dynamically maintaining minimal integral separator for threshold and difference graphs. In: WALCOM: Algorithms and Computation, LNCS, vol. 9627, pp. 313–324. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30139-6_25 HellPShamirRSharanRA fully dynamic algorithm for recognizing and representing proper interval graphsSIAM J. Comput.2002311289305185740110.1137/S00975397003722160992.68065 Beisegel, J., Chiarelli, N., Köhler, E., Krnc, M., Milanič, M., Pivač, N., Scheffler, R., Strehler, M.: Edge elimination and weighted graph classes. In: Adler, I., Müller, H. (eds.) Graph-Theoretic Concepts in Computer Science, LNCS, vol. 12301, pp. 134–147. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-60440-0_11 Bhattacharya, S., Henzinger, M., Nanongkai, D.: New deterministic approximation algorithms for fully dynamic matching. In: Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, pp. 398–411 (2016). https://doi.org/10.1145/2897518.2897568 Heggernes, P., Papadopoulos, C.: Single-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletions. In: International Computing and Combinatorics Conference, pp. 406–416. Springer (2007). https://doi.org/10.1007/978-3-540-73545-8_40 FrigioniDMarchetti-SpaccamelaANanniUFully dynamic shortest paths in digraphs with arbitrary arc weightsJ. Algorithms200349186113202706010.1016/S0196-6774(03)00082-81064.68068 Chvátal, V., Hammer, P.L.: Set-packing and threshold graphs. Tech. Rep. CORR 73-21, University of Waterloo (1973) CozzensMBLeibowitzRMultidimensional scaling and threshold graphsJ. Math. Psychol.19873117919190006710.1016/0022-2496(87)90014-90649.92021 BrandstädtALeVBSpinradJPGraph Classes: A Survey1999SIAM10.1137/1.97808987197960919.05001 HammerPLPeledUNSunXDifference graphsDiscrete Appl. Math.19902813544106482910.1016/0166-218X(90)90092-Q0716.05032 CozzensMBLeibowitzRThreshold dimension of graphsSIAM J. Algebr. Discrete Methods1984557959576398610.1137/06050550717.05069 ShamirRSharanRA fully dynamic algorithm for modular decomposition and recognition of cographsDiscrete Appl. Math.20041362–3329340204521910.1016/S0166-218X(03)00448-71062.68092 MahadevNVRPeledUNThreshold Graphs and Related Topics1995AmsterdamNorth-Holland Publishing Co.0852.05001 HeggernesPKratschDLinear-time certifying recognition algorithms and forbidden induced subgraphsNordic J. Comput.2007141–28710824605581169.68653 OrdmanETMinimal threshold separators and memory requirements for synchronizationSIAM J. Comput.198918115216597817210.1137/02180100677.68064 Ordman, E.T.: Threshold coverings and resource allocation. In: Proceedings of the 16th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pp. 99–113. Utilitas Mathematica Pub., Winnipeg (1985) SoulignacFJFully dynamic recognition of proper circular-arc graphsAlgorithmica2015714904968331880810.1007/s00453-013-9835-71323.05125 HararyFPeledUHamiltonian threshold graphsDiscrete Appl. Math.198716111586973410.1016/0166-218X(87)90050-30601.05033 HeggernesPPapadopoulosCSingle-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletionsTheor. Comput. Sci.20094101115248830710.1016/j.tcs.2008.07.0201161.68040 RoseDJTarjanRELuekerGSAlgorithmic aspects of vertex elimination on graphsSIAM J. Comput.19765226628340831210.1137/02050210353.65019 HenzingerMRFully dynamic biconnectivity in graphsAlgorithmica1995136503538132550310.1007/BF011890670826.68097 KoopGJCyclic scheduling of offweekendsOper. Res. Lett.1986425926383626110.1016/0167-6377(86)90026-X0598.90052 Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 343–350 (2000). https://doi.org/10.1145/335305.335345 CrespelleCPaulCFully dynamic algorithm for recognition and modular decomposition of permutation graphsAlgorithmica2010582405432267002310.1007/s00453-008-9273-01205.68258 HendersonPBZalcsteinYA graph-theoretic characterization of the PVchunk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{PV}_{\text{ chunk }}$$\end{document} class of synchronizing primitivesSIAM J. Comput.1977618810848894810.1137/02060080349.68022 ChvátalVHammerPLAggregations of inequalities in integer programmingStud. Integ. Program. Ann. Discrete Math.1977114516247938410.1016/S0167-5060(08)70731-3 M Yannakakis (1107_CR30) 1982; 3 MB Cozzens (1107_CR8) 1987; 31 P Hell (1107_CR16) 2002; 31 NVR Mahadev (1107_CR23) 1995 A Brandstädt (1107_CR3) 1999 1107_CR5 DJ Rose (1107_CR26) 1976; 5 1107_CR4 PB Henderson (1107_CR17) 1977; 6 GJ Koop (1107_CR22) 1986; 4 1107_CR1 1107_CR2 MR Henzinger (1107_CR18) 1995; 13 FJ Soulignac (1107_CR28) 2015; 71 PN Klein (1107_CR21) 1998; 22 1107_CR24 1107_CR29 C Crespelle (1107_CR9) 2010; 58 F Harary (1107_CR12) 1987; 16 PL Hammer (1107_CR11) 1990; 28 MR Henzinger (1107_CR19) 1998; 22 L Ibarra (1107_CR20) 2008; 4 ET Ordman (1107_CR25) 1989; 18 V Chvátal (1107_CR6) 1977; 1 MB Cozzens (1107_CR7) 1984; 5 P Heggernes (1107_CR13) 2007; 14 R Shamir (1107_CR27) 2004; 136 D Frigioni (1107_CR10) 2003; 49 1107_CR14 P Heggernes (1107_CR15) 2009; 410 |
| References_xml | – reference: ChvátalVHammerPLAggregations of inequalities in integer programmingStud. Integ. Program. Ann. Discrete Math.1977114516247938410.1016/S0167-5060(08)70731-3 – reference: SoulignacFJFully dynamic recognition of proper circular-arc graphsAlgorithmica2015714904968331880810.1007/s00453-013-9835-71323.05125 – reference: BrandstädtALeVBSpinradJPGraph Classes: A Survey1999SIAM10.1137/1.97808987197960919.05001 – reference: HammerPLPeledUNSunXDifference graphsDiscrete Appl. Math.19902813544106482910.1016/0166-218X(90)90092-Q0716.05032 – reference: CozzensMBLeibowitzRMultidimensional scaling and threshold graphsJ. Math. Psychol.19873117919190006710.1016/0022-2496(87)90014-90649.92021 – reference: HeggernesPPapadopoulosCSingle-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletionsTheor. Comput. Sci.20094101115248830710.1016/j.tcs.2008.07.0201161.68040 – reference: FrigioniDMarchetti-SpaccamelaANanniUFully dynamic shortest paths in digraphs with arbitrary arc weightsJ. Algorithms200349186113202706010.1016/S0196-6774(03)00082-81064.68068 – reference: OrdmanETMinimal threshold separators and memory requirements for synchronizationSIAM J. Comput.198918115216597817210.1137/02180100677.68064 – reference: HenzingerMRFully dynamic biconnectivity in graphsAlgorithmica1995136503538132550310.1007/BF011890670826.68097 – reference: IbarraLFully dynamic algorithms for chordal graphs and split graphsACM Trans. Algorithms200844140244695910.1145/1383369.13833711445.05103 – reference: ShamirRSharanRA fully dynamic algorithm for modular decomposition and recognition of cographsDiscrete Appl. Math.20041362–3329340204521910.1016/S0166-218X(03)00448-71062.68092 – reference: Chvátal, V., Hammer, P.L.: Set-packing and threshold graphs. Tech. Rep. CORR 73-21, University of Waterloo (1973) – reference: HenzingerMRFredmanMLLower bounds for fully dynamic connectivity problems in graphsAlgorithmica1998223351362164611310.1007/PL000092280915.68132 – reference: KleinPNSubramanianSA fully dynamic approximation scheme for shortest paths in planar graphsAlgorithmica1998223235249164610810.1007/PL000092230915.68130 – reference: Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 343–350 (2000). https://doi.org/10.1145/335305.335345 – reference: Beisegel, J., Chiarelli, N., Köhler, E., Krnc, M., Milanič, M., Pivač, N., Scheffler, R., Strehler, M.: Edge elimination and weighted graph classes. In: Adler, I., Müller, H. (eds.) Graph-Theoretic Concepts in Computer Science, LNCS, vol. 12301, pp. 134–147. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-60440-0_11 – reference: HararyFPeledUHamiltonian threshold graphsDiscrete Appl. Math.198716111586973410.1016/0166-218X(87)90050-30601.05033 – reference: MahadevNVRPeledUNThreshold Graphs and Related Topics1995AmsterdamNorth-Holland Publishing Co.0852.05001 – reference: CozzensMBLeibowitzRThreshold dimension of graphsSIAM J. Algebr. Discrete Methods1984557959576398610.1137/06050550717.05069 – reference: Ordman, E.T.: Threshold coverings and resource allocation. In: Proceedings of the 16th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pp. 99–113. Utilitas Mathematica Pub., Winnipeg (1985) – reference: HellPShamirRSharanRA fully dynamic algorithm for recognizing and representing proper interval graphsSIAM J. Comput.2002311289305185740110.1137/S00975397003722160992.68065 – reference: Calamoneri, T., Monti, A., Petreschi, R.: Fully dynamically maintaining minimal integral separator for threshold and difference graphs. In: WALCOM: Algorithms and Computation, LNCS, vol. 9627, pp. 313–324. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30139-6_25 – reference: HendersonPBZalcsteinYA graph-theoretic characterization of the PVchunk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{PV}_{\text{ chunk }}$$\end{document} class of synchronizing primitivesSIAM J. Comput.1977618810848894810.1137/02060080349.68022 – reference: KoopGJCyclic scheduling of offweekendsOper. Res. Lett.1986425926383626110.1016/0167-6377(86)90026-X0598.90052 – reference: HeggernesPKratschDLinear-time certifying recognition algorithms and forbidden induced subgraphsNordic J. Comput.2007141–28710824605581169.68653 – reference: RoseDJTarjanRELuekerGSAlgorithmic aspects of vertex elimination on graphsSIAM J. Comput.19765226628340831210.1137/02050210353.65019 – reference: Bhattacharya, S., Henzinger, M., Nanongkai, D.: New deterministic approximation algorithms for fully dynamic matching. In: Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, pp. 398–411 (2016). https://doi.org/10.1145/2897518.2897568 – reference: CrespelleCPaulCFully dynamic algorithm for recognition and modular decomposition of permutation graphsAlgorithmica2010582405432267002310.1007/s00453-008-9273-01205.68258 – reference: YannakakisMThe complexity of the partial order dimension problemSIAM J. Algebr. Discrete Methods19823335135866686010.1137/06030360516.06001 – reference: Heggernes, P., Papadopoulos, C.: Single-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletions. In: International Computing and Combinatorics Conference, pp. 406–416. 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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Apexes Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphs Mathematics of Computing Problem solving Recognition Shortest-path problems Theory of Computation |
| Title | Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs |
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