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The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes

Authors Guillaume Ducoffe, Alexandru Popa

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Design and analysis of algorithms
Keywords
  • maximum matching
  • FPT in P
  • modular decomposition
  • pruned graphs
  • one-vertex extensions
  • P_4-structure

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Abstract

We address the following general question: given a graph class C on which we can solve Maximum Matching in (quasi) linear time, does the same hold true for the class of graphs that can be modularly decomposed into C? As a way to answer this question for distance-hereditary graphs and some other superclasses of cographs, we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. We remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a "pruned modular decomposition", that can be computed in quasi linear time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. The latter result strictly extends a recent framework in (Coudert et al., SODA'18). Our work is the first to explain why the existence of some nice ordering over the modules of a graph, instead of just over its vertices, can help to speed up the computation of maximum matchings on some graph classes.

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Guillaume Ducoffe and Alexandru Popa. The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ISAAC.2018.6

Author Details

Guillaume Ducoffe
  • ICI – National Institute for Research and Development in Informatics, Bucharest, Romania , The Research Institute of the University of Bucharest ICUB, Bucharest, Romania
Alexandru Popa
  • University of Bucharest, Bucharest, Romania , ICI – National Institute for Research and Development in Informatics, Bucharest, Romania

Funding

This work was supported by the Institutional research programme PN 1819 "Advanced IT resources to support digital transformation processes in the economy and society - RESINFO-TD" (2018), project PN 1819-01-01 "Modeling, simulation, optimization of complex systems and decision support in new areas of IT&C research", funded by the Ministry of Research and Innovation, Romania.

References

  1. H.-J. Bandelt and H. Mulder. Distance-hereditary graphs. J. of Combinatorial Theory, Series B, 41(2):182-208, 1986. Google Scholar
  2. C. Berge. Two theorems in graph theory. Proceedings of the National Academy of Sciences, 43(9):842-844, 1957. Google Scholar
  3. J. A. Bondy and U. S. R. Murty. Graph theory. Grad. Texts in Math., 2008. Google Scholar
  4. A. Brandstädt and V. Le. Tree-and forest-perfect graphs. Discrete applied mathematics, 95(1-3):141-162, 1999. Google Scholar
  5. M. Chang. Algorithms for maximum matching and minimum fill-in on chordal bipartite graphs. In ISAAC, pages 146-155. Springer, 1996. Google Scholar
  6. V. Chvátal. A semi-strong perfect graph conjecture. In North-Holland mathematics studies, volume 88, pages 279-280. Elsevier, 1984. Google Scholar
  7. O. Cogis and E. Thierry. Computing maximum stable sets for distance-hereditary graphs. Discrete Optimization, 2(2):185-188, 2005. Google Scholar
  8. D. Corneil, Y. Perl, and L. Stewart. A linear recognition algorithm for cographs. SIAM Journal on Computing, 14(4):926-934, 1985. Google Scholar
  9. D. Coudert, G. Ducoffe, and A. Popa. Fully polynomial FPT algorithms for some classes of bounded clique-width graphs. In SODA'18, pages 2765-2784. SIAM, 2018. Google Scholar
  10. E. Dahlhaus and M. Karpinski. Matching and multidimensional matching in chordal and strongly chordal graphs. Discrete Applied Mathematics, 84(1-3):79-91, 1998. Google Scholar
  11. G. Damiand, M. Habib, and C. Paul. A simple paradigm for graph recognition: application to cographs and distance hereditary graphs. Theoretical Computer Science, 263(1-2):99-111, 2001. Google Scholar
  12. F. Dragan. On greedy matching ordering and greedy matchable graphs. In WG'97, volume 1335 of LNCS, pages 184-198. Springer, 1997. Google Scholar
  13. S. Dubois, V. Giakoumakis, and C. El Mounir. On co-distance hereditary graphs. In CTW, pages 94-97, 2008. Google Scholar
  14. G. Ducoffe and A. Popa. The use of a pruned modular decomposition for Maximum Matching algorithms on some graph classes. Technical Report arXiv:1804.09407, arXiv, 2018. Google Scholar
  15. J. Edmonds. Paths, trees, and flowers. Canadian J. of mathematics, 17(3):449-467, 1965. Google Scholar
  16. H. Gabow and R. Tarjan. A linear-time algorithm for a special case of disjoint set union. In STOC'83, pages 246-251. ACM, 1983. Google Scholar
  17. Tibor Gallai. Transitiv orientierbare graphen. Acta Math. Hungarica, 18(1):25-66, 1967. Google Scholar
  18. M. Habib and C. Paul. A survey of the algorithmic aspects of modular decomposition. Computer Science Review, 4(1):41-59, 2010. Google Scholar
  19. J. Hopcroft and R. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on computing, 2(4):225-231, 1973. Google Scholar
  20. J. Lanlignel, O. Raynaud, and E. Thierry. Pruning graphs with digital search trees. application to distance hereditary graphs. In STACS, pages 529-541. Springer, 2000. Google Scholar
  21. G. Mertzios, A. Nichterlein, and R. Niedermeier. Linear-time algorithm for maximum-cardinality matching on cocomparability graphs. arXiv, 2017. URL: http://arxiv.org/abs/1703.05598.
  22. S. Micali and V. Vazirani. An O(√V E) Algorithm for Finding Maximum Matching in General Graphs. In FOCS'80, pages 17-27. IEEE, 1980. Google Scholar
  23. D. Paulusma, F. Slivovsky, and S. Szeider. Model counting for cnf formulas of bounded modular treewidth. Algorithmica, 76(1):168-194, 2016. Google Scholar
  24. M. Rao. Clique-width of graphs defined by one-vertex extensions. Discrete Mathematics, 308(24):6157-6165, 2008. Google Scholar
  25. M. Tedder, D. Corneil, M. Habib, and C. Paul. Simpler Linear-Time Modular Decomposition Via Recursive Factorizing Permutations. In ICALP, pages 634-645. Springer, 2008. Google Scholar
  26. M.-S. Yu and C.-H. Yang. An O(n)-time algorithm for maximum matching on cographs. Information processing letters, 47(2):89-93, 1993. Google Scholar
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