A Generalized Matching Reconfiguration Problem

Authors Noam Solomon, Shay Solomon

Thumbnail PDF


  • Filesize: 483 kB
  • 20 pages

Document Identifiers

Author Details

Noam Solomon
  • Immunai, New York, NY, USA
Shay Solomon
  • School of Electrical Engineering, Tel Aviv University, Israel


The authors thank Thatchaphol Saranurak for fruitful discussions.

Cite AsGet BibTex

Noam Solomon and Shay Solomon. A Generalized Matching Reconfiguration Problem. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 57:1-57:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The goal in reconfiguration problems is to compute a gradual transformation between two feasible solutions of a problem such that all intermediate solutions are also feasible. In the Matching Reconfiguration Problem (MRP), proposed in a pioneering work by Ito et al. from 2008, we are given a graph G and two matchings M and M', and we are asked whether there is a sequence of matchings in G starting with M and ending at M', each resulting from the previous one by either adding or deleting a single edge in G, without ever going through a matching of size < min{|M|,|M'|}-1. Ito et al. gave a polynomial time algorithm for the problem, which uses the Edmonds-Gallai decomposition. In this paper we introduce a natural generalization of the MRP that depends on an integer parameter Δ ≥ 1: here we are allowed to make Δ changes to the current solution rather than 1 at each step of the {transformation procedure}. There is always a valid sequence of matchings transforming M to M' if Δ is sufficiently large, and naturally we would like to minimize Δ. We first devise an optimal transformation procedure for unweighted matching with Δ = 3, and then extend it to weighted matchings to achieve asymptotically optimal guarantees. The running time of these procedures is linear. We further demonstrate the applicability of this generalized problem to dynamic graph matchings. In this area, the number of changes to the maintained matching per update step (the recourse bound) is an important quality measure. Nevertheless, the worst-case recourse bounds of almost all known dynamic matching algorithms are prohibitively large, much larger than the corresponding update times. We fill in this gap via a surprisingly simple black-box reduction: Any dynamic algorithm for maintaining a β-approximate maximum cardinality matching with update time T, for any β ≥ 1, T and ε > 0, can be transformed into an algorithm for maintaining a (β(1 +ε))-approximate maximum cardinality matching with update time T + O(1/ε) and worst-case recourse bound O(1/ε). This result generalizes for approximate maximum weight matching, where the update time and worst-case recourse bound grow from T + O(1/ε) and O(1/ε) to T + O(ψ/ε) and O(ψ/ε), respectively; ψ is the graph aspect-ratio. We complement this positive result by showing that, for β = 1+ε, the worst-case recourse bound of any algorithm produced by our reduction is optimal. As a corollary, several key dynamic approximate matching algorithms - with poor worst-case recourse bounds - are strengthened to achieve near-optimal worst-case recourse bounds with no loss in update time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Dynamic algorithms
  • graph matching
  • reconfiguration problem
  • recourse bound


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. In Proc. of 57th FOCS, pages 335-344, 2016. Google Scholar
  2. Spyros Angelopoulos, Christoph Dürr, and Shendan Jin. Online maximum matching with recourse. In Proc. of 43rd MFCS, pages 8:1-8:15, 2018. Google Scholar
  3. Moab Arar, Shiri Chechik, Sarel Cohen, Cliff Stein, and David Wajc. Dynamic matching: Reducing integral algorithms to approximately-maximal fractional algorithms. In Proc. 45th ICALP, pages 7:1-7:16, 2018. Google Scholar
  4. Sepehr Assadi, MohammadHossein Bateni, Aaron Bernstein, Vahab S. Mirrokni, and Cliff Stein. Coresets meet EDCS: algorithms for matching and vertex cover on massive graphs. CoRR, abs/1711.03076, 2017. Google Scholar
  5. Sepehr Assadi and Sanjeev Khanna. Randomized composable coresets for matching and vertex cover. In Proc. of 29th SPAA, pages 3-12, 2017. Google Scholar
  6. Sepehr Assadi, Krzysztof Onak, Baruch Schieber, and Shay Solomon. Fully dynamic maximal independent set with sublinear update time. In Proc. 50th STOC, 2018. Google Scholar
  7. Giorgio Ausiello, Vincenzo Bonifaci, and Bruno Escoffier. Complexity and approximation in reoptimization. In Computability in Context: Computation and Logic in the Real World, pages 101-129. World Scientific, 2011. Google Scholar
  8. Giorgio Ausiello, Bruno Escoffier, Jérôme Monnot, and Vangelis Th. Paschos. Reoptimization of minimum and maximum traveling salesman’s tours. J. Discrete Algorithms, 7(4):453-463, 2009. Google Scholar
  9. Nikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Clifford Stein. A 2-competitive algorithm for online convex optimization with switching costs. In Proc. of APPROX-RANDOM, pages 96-109, 2015. Google Scholar
  10. S. Baswana, M. Gupta, and S. Sen. Fully dynamic maximal matching in O(log n) update time. In Proc. of 52nd FOCS, pages 383-392, 2011. Google Scholar
  11. Michael A Bender, Martin Farach-Colton, Sándor P Fekete, Jeremy T Fineman, and Seth Gilbert. Reallocation problems in scheduling. Algorithmica, 73(2):389-409, 2015. Google Scholar
  12. Michael A Bender, Martin Farach-Colton, Sándor P Fekete, Jeremy T Fineman, and Seth Gilbert. Cost-oblivious storage reallocation. TALG, 13(3):38, 2017. Google Scholar
  13. Aaron Bernstein, Sebastian Forster, and Monika Henzinger. A deamortization approach for dynamic spanner and dynamic maximal matching. In Proc. SODA, pages 1899-1918, 2019. Google Scholar
  14. Aaron Bernstein, Jacob Holm, and Eva Rotenberg. Online bipartite matching with amortized o(log^2 n) replacements. In Proc. of 28th SODA, pages 692-711, 2018. Google Scholar
  15. Aaron Bernstein, Tsvi Kopelowitz, Seth Pettie, Ely Porat, and Clifford Stein. Simultaneously load balancing for every p-norm, with reassignments. In Proc. ITCS, pages 51:1-51:14, 2017. Google Scholar
  16. Aaron Bernstein and Cliff Stein. Fully dynamic matching in bipartite graphs. In Proc. 42nd ICALP, pages 167-179, 2015. Google Scholar
  17. Aaron Bernstein and Cliff Stein. Faster fully dynamic matchings with small approximation ratios. In Proc. of 26th SODA, pages 692-711, 2016. Google Scholar
  18. S. Bhattacharya, M. Henzinger, and D. Nanongkai. Fully dynamic maximum matching and vertex cover in o(log³ n) worst case update time. In Proc. of 28th SODA, pages 470-489, 2017. Google Scholar
  19. Davide Bilò. New algorithms for steiner tree reoptimization. In Proc. of 45th ICALP, pages 19:1-19:14, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.19.
  20. Davide Bilò, Hans-Joachim Böckenhauer, Dennis Komm, Richard Královic, Tobias Mömke, Sebastian Seibert, and Anna Zych. Reoptimization of the shortest common superstring problem. Algorithmica, 61(2):227-251, 2011. Google Scholar
  21. Marthe Bonamy, Nicolas Bousquet, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Arnaud Mary, Moritz Mühlenthaler, and Kunihiro Wasa. The perfect matching reconfiguration problem. In Proc. of 44th MFCS, volume 138 of LIPIcs, pages 80:1-80:14, 2019. Google Scholar
  22. Paul Bonsma and Luis Cereceda. Finding paths between graph colourings: Pspace-completeness and superpolynomial distances. Theoretical Computer Science, 410(50):5215-5226, 2009. Google Scholar
  23. Nicolas Boria and Vangelis Th. Paschos. Fast reoptimization for the minimum spanning tree problem. J. Discrete Algorithms, 8(3):296-310, 2010. Google Scholar
  24. Bartlomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych. Online bipartite matching in offline time. In Proc. 55th FOCS, pages 384-393, 2014. Google Scholar
  25. Bartlomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych. Shortest augmenting paths for online matchings on trees. In Proc. of 13th WAOA, pages 59-71, 2015. Google Scholar
  26. Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych-Pawlewicz. A tight bound for shortest augmenting paths on trees. In Proc. 13th LATIN, pages 201-216, 2018. Google Scholar
  27. Nicolas Bousquet, Tatsuhiko Hatanaka, Takehiro Ito, and Moritz Mühlenthaler. Shortest reconfiguration of matchings. In Proc. of 45th WG, pages 162-174. Springer, 2019. Google Scholar
  28. Keren Censor-Hillel, Elad Haramaty, and Zohar S. Karnin. Optimal dynamic distributed MIS. In Proc. of PODC, pages 217-226, 2016. Google Scholar
  29. Moses Charikar and Shay Solomon. Fully dynamic almost-maximal matching: Breaking the polynomial worst-case time barrier. In Proc. 45th ICALP, pages 33:1-33:14, 2018. Google Scholar
  30. Kamalika Chaudhuri, Constantinos Daskalakis, Robert D. Kleinberg, and Henry Lin. Online bipartite perfect matching with augmentations. In Proc. INFOCOM, pages 1044-1052, 2009. Google Scholar
  31. P. Gopalan, P. G Kolaitis, E. Maneva, and C. H Papadimitriou. The connectivity of boolean satisfiability: computational and structural dichotomies. SICOMP, 38(6):2330-2355, 2009. Google Scholar
  32. Fabrizio Grandoni, Stefano Leonardi, Piotr Sankowski, Chris Schwiegelshohn, and Shay Solomon. (1 + ε)-approximate incremental matching in constant deterministic amortized time. In Proc. of 30th SODA, pages 1886-1898, 2019. Google Scholar
  33. Edward F. Grove, Ming-Yang Kao, P. Krishnan, and Jeffrey Scott Vitter. Online perfect matching and mobile computing. In Proc. of 45th Wads, pages 194-205, 1995. Google Scholar
  34. Albert Gu, Anupam Gupta, and Amit Kumar. The power of deferral: maintaining a constant-competitive steiner tree online. In Proc. of STOC, pages 525-534, 2013. Google Scholar
  35. Anupam Gupta, Ravishankar Krishnaswamy, Amit Kumar, and Debmalya Panigrahi. Online and dynamic algorithms for set cover. In Proc. 49th STOC, pages 537-550, 2017. Google Scholar
  36. Anupam Gupta and Amit Kumar. Online steiner tree with deletions. In Proc. of 25th SODA, pages 455-467, 2014. Google Scholar
  37. Anupam Gupta, Amit Kumar, and Cliff Stein. Maintaining assignments online: Matching, scheduling, and flows. In Proc. 25th SODA, pages 468-479, 2014. Google Scholar
  38. M. Gupta and R. Peng. Fully dynamic (1+ε)-approximate matchings. In 54th FOCS, pages 548-557, 2013. Google Scholar
  39. Manoj Gupta, Hitesh Kumar, and Neeldhara Misra. On the complexity of optimal matching reconfiguration. In International Conference on Current Trends in Theory and Practice of Informatics, pages 221-233. Springer, 2019. Google Scholar
  40. Robert A. Hearn and Erik D. Demaine. The nondeterministic constraint logic model of computation: Reductions and applications. In Proc. ICALP, pages 401-413. Springer, 2002. Google Scholar
  41. T. Ito, E. D. Demaine, N. J. A. Harvey, C. H. Papadimitriou, M. Sideri, R. Uehara, and Y. Uno. On the complexity of reconfiguration problems. TCS, 412(12-14):1054-1065, 2011. Google Scholar
  42. T. Ito, N. Kakimura, N. Kamiyama, Y. Kobayashi, and Y. Okamoto. Reconfiguration of maximum-weight b-matchings in a graph. J. Comb. Optim., 37(2):454-464, 2019. Google Scholar
  43. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Shortest reconfiguration of perfect matchings via alternating cycles. In Proc. of 27th ESA, volume 144 of LIPIcs, pages 61:1-61:15, 2019. Google Scholar
  44. Marcin Kaminski, Paul Medvedev, and Martin Milanic. Complexity of independent set reconfigurability problems. Theor. Comput. Sci., 439:9-15, 2012. Google Scholar
  45. Bruce M. Kapron, Valerie King, and Ben Mountjoy. Dynamic graph connectivity in polylogarithmic worst case time. In Proc. of 24th SODA, pages 1131-1142, 2013. Google Scholar
  46. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci., 74(3):335-349, 2008. Google Scholar
  47. Jannik Matuschke, Ulrike Schmidt-Kraepelin, and José Verschae. Maintaining perfect matchings at low cost. CoRR, abs/1811.10580, 2018. Google Scholar
  48. Nicole Megow, Martin Skutella, José Verschae, and Andreas Wiese. The power of recourse for online MST and TSP. SIAM J. Comput., 45(3):859-880, 2016. Google Scholar
  49. Vahab S. Mirrokni and Morteza Zadimoghaddam. Randomized composable core-sets for distributed submodular maximization. In Proc. 47th STOC, pages 153-162, 2015. Google Scholar
  50. Ofer Neiman and Shay Solomon. Simple deterministic algorithms for fully dynamic maximal matching. In Proc. 45th STOC, pages 745-754, 2013. Google Scholar
  51. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. Google Scholar
  52. K. Onak and R. Rubinfeld. Maintaining a large matching and a small vertex cover. In Proc. of 42nd STOC, pages 457-464, 2010. Google Scholar
  53. D. Peleg and S. Solomon. Dynamic (1 + ε)-approximate matchings: A density-sensitive approach. In Proc. of 26th SODA, 2016. Google Scholar
  54. Baruch Schieber, Hadas Shachnai, Gal Tamir, and Tami Tamir. A theory and algorithms for combinatorial reoptimization. Algorithmica, 80(2):576-607, 2018. Google Scholar
  55. Noam Solomon and Shay Solomon. A generalized matching reconfiguration problem. CoRR, abs/1803.05825, 2020. URL: http://arxiv.org/abs/1803.05825.
  56. S. Solomon. Fully dynamic maximal matching in constant update time. In Proc. 57th FOCS, pages 325-334, 2016. Google Scholar
  57. Babacar Thiongane, Anass Nagih, and Gérard Plateau. Lagrangean heuristics combined with reoptimization for the 0-1 bidimensional knapsack problem. Discrete applied mathematics, 154(15):2200-2211, 2006. Google Scholar
  58. Jan van den Heuvel. The complexity of change. In Simon R. Blackburn, Stefanie Gerke, and Mark Wildon, editors, Surveys in Combinatorics 2013, volume 409 of London Mathematical Society Lecture Note Series, pages 127-160. Cambridge University Press, 2013. Google Scholar
  59. David Wajc. Rounding dynamic matchings against an adaptive adversary. In Proc. of 52nd STOC, pages 194-207. ACM, 2020. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail