Skip to main content

Total Matching and Subdeterminants

  • Conference paper
  • First Online:
Combinatorial Optimization (ISCO 2024)

Abstract

In the total matching problem, one is given a graph G with weights on the vertices and edges. The goal is to find a maximum weight set of vertices and edges that is the non-incident union of a stable set and a matching. We consider the natural formulation of the problem as an integer program (IP), with variables corresponding to vertices and edges. Let \(M = M(G)\) denote the constraint matrix of this IP. We define \(\varDelta (G)\) as the maximum absolute value of the determinant of a square submatrix of M. We show that the total matching problem can be solved in strongly polynomial time provided \(\varDelta (G) \le \varDelta \) for some constant \(\varDelta \in \mathbb {Z}_{\ge 1}\). We also show that the problem of computing \(\varDelta (G)\) admits an FPT algorithm. We also establish further results on \(\varDelta (G)\) when G is a forest.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    We use \(\delta (v)\) to denote the set of edges incident to vertex v.

  2. 2.

    Notice that \(B'\) is not always a true incidence matrix since there might be bichromatic edges e such that only one endpoint of e is bichromatic.

References

  1. Alavi, Y., Behzad, M., Lesniak-Foster, L.M., Nordhaus, E.: Total matchings and total coverings of graphs. J. Graph Theory 1(2), 135–140 (1977)

    Article  MathSciNet  Google Scholar 

  2. Artmann, S., Eisenbrand, F., Glanzer, C., Oertel, T., Vempala, S., Weismantel, R.: A note on non-degenerate integer programs with small sub-determinants. Oper. Res. Lett. 44(5), 635–639 (2016)

    Article  MathSciNet  Google Scholar 

  3. Artmann, S., Weismantel, R., Zenklusen, R.: A strongly polynomial algorithm for bimodular integer linear programming. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pp. 1206–1219. Association for Computing Machinery (2017)

    Google Scholar 

  4. Ferrarini, L., Gualandi, S.: Total coloring and total matching: polyhedra and facets. Eur. J. Oper. Res. 303(1), 129–142 (2022)

    Article  MathSciNet  Google Scholar 

  5. Fiorini, S., Joret, G., Weltge, S., Yuditsky, Y.: Integer programs with bounded subdeterminants and two nonzeros per row. In: 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 13–24 (2022)

    Google Scholar 

  6. Glanzer, C., Stallknecht, I., Weismantel, R.: Notes on \(\{\)a, b, c\(\}\)-modular matrices. Vietnam J. Math. 50(2), 469–485 (2022)

    Article  MathSciNet  Google Scholar 

  7. Gribanov, D., Shumilov, I., Malyshev, D., Pardalos, P.: On \(\Delta \)-modular integer linear problems in the canonical form and equivalent problems. J. Glob. Optim. 1–61 (2022)

    Google Scholar 

  8. Gribanov, D.V., Veselov, S.I.: On integer programming with bounded determinants. Optim. Lett. 10, 1169–1177 (2016)

    Article  MathSciNet  Google Scholar 

  9. Kawarabayashi, K.I., Reed, B.: Odd cycle packing. In: Proceedings of the Forty-Second ACM Symposium on Theory of Computing, pp. 695–704 (2010)

    Google Scholar 

  10. Leidner, M.E.: A study of the total coloring of graphs. Ph.D. thesis, University of Louisville (2012)

    Google Scholar 

  11. Manlove, D.F.: On the algorithmic complexity of twelve covering and independence parameters of graphs. Disc. Appl. Math. 91(1–3), 155–175 (1999)

    Article  MathSciNet  Google Scholar 

  12. Nägele, M., Nöbel, C., Santiago, R., Zenklusen, R.: Advances on strictly \(\Delta \)-modular IPs. In: International Conference on Integer Programming and Combinatorial Optimization, pp. 393–407. Springer, Heidelberg (2023). https://doi.org/10.1007/978-3-031-32726-1_28

  13. Nägele, M., Santiago, R., Zenklusen, R.: Congruency-constrained TU problems beyond the bimodular case. In: Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2743–2790. SIAM (2022)

    Google Scholar 

  14. Nägele, M., Sudakov, B., Zenklusen, R.: Submodular minimization under congruency constraints. Combinatorica 39(6), 1351–1386 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stefan Kober .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ferrarini, L., Fiorini, S., Kober, S., Yuditsky, Y. (2024). Total Matching and Subdeterminants. In: Basu, A., Mahjoub, A.R., Salazar González, J.J. (eds) Combinatorial Optimization. ISCO 2024. Lecture Notes in Computer Science, vol 14594. Springer, Cham. https://doi.org/10.1007/978-3-031-60924-4_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-60924-4_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-60923-7

  • Online ISBN: 978-3-031-60924-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics