Advertisement

On the Connected Spanning Cubic Subgraph Problem

  • Damien Massé
  • Reinhardt Euler
  • Laurent Lemarchand
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 148)

Abstract

Given an \(n \times n\)-distance matrix D, the connected spanning cubic subgraph problem (\(\mathcal {CSC}\)) is to determine a connected cubic graph \(G=(V,E)\) of order n with total distance \(\sum _{ij \in E} d_{ij}\) as small as possible. Restricting matrix D to have 0–1 entries only leads to the problem of deciding whether a given graph contains a connected spanning cubic subgraph. We present some first results on the facial structure of the associated polytope including several classes of valid inequalities some of which are shown to be facet-defining. To solve problem \(\mathcal {CSC}\), two procedures are formulated: the first is based on a binary linear program, that iteratively constructs an optimal solution, the second on a linear program, that iteratively exploits additional cutting planes from different families to accelerate the solution process. All formulations have been implemented and tested on series of randomly generated problem instances.

Keywords

Connected spanning cubic subgraph 0–1 polytope 0–1 programming Branch and Cut 

AMS Subject Classification:

90C27 90C35 90C57 05C99 

References

  1. 1.
    R. Bagnara, P.M. Hill, E. Zaffanella, The Parma Polyhedra Library: toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Sci. Comput. Program. 72(1–2), 3–21 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    F. Barahona, On the k-cut problem. Oper. Res. Lett. 26, 99–105 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    F. Cheah, D.G. Corneil, The complexity of regular subgraph recognition. Discrete Appl. Math. 27, 59–68 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    K. Cornelissen, R. Hoeksma, B. Manthey, N.S. Narayanaswamy and C.S. Rahul, Approximability of connected factors, to appear, in Proceedings of the 11th Workshop on Approximation and Online Algorithms (WAOA 2013), Sophia Antipolis, France, 5–6 Sep 2013Google Scholar
  5. 5.
    G. Csardi and T. Nepusz, The igraph software package for complex network research. Int. J. Complex Syst. 1695 (2006). http://igraph.sf.net
  6. 6.
    J. Edmonds, Maximum matching and a polyhedron with 0–1 vertices. J. Res. Nat. Bur. Stan. 69B, 125130 (1965)MathSciNetGoogle Scholar
  7. 7.
    M. Grötschel, Polyedrische Charakterisierungen kombinatorischer Optimierungsprobleme. Math. Syst. Econ. 36, (A. Hain, Meisenheim, 1977)Google Scholar
  8. 8.
    D. Gusfield, Very simple methods for all pairs network flow analysis. SIAM J. Comput. 19(1), 143–155 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    A.N. Letchford, G. Reinelt, D.O. Theis, Odd minimum cut sets and b-matchings revisited. SIAM J. Discrete Math. 22(4), 14801487 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    L. Lemarchand, R. Euler, C. Lin, I. Sparkes, Modeling the geometry of the Endoplasmic Reticulum Network, in Proceedings of the 1st International Conference on Algorithms for Computational Biology—AlCob’14, LNBI 8542, (Springer, 2014), pp. 132–146Google Scholar
  11. 11.
    C. Lin, L. Lemarchand, R. Euler, I. Sparkes, Modeling the geometry and dynamics of the Endoplasmic Reticulum network (2015). doi: 10.1109/TCBB.2015.2389226, to appear in IEEE/ACM Trans. Comput. Biol. Bio
  12. 12.
    D. Naddef, Y. Pochet, The symmetric traveling salesman polytope revisited. Math. Oper. Res. 26(4), 700–722 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    M. Stoer, F. Wagner, A simple min-cut algorithm. J. ACM 44, 585–591 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    F. Viger and M. Latapy, Efficient and simple generation of random simple connected graphs with prescribed degree sequence, in Proceedings of the 11th Conference of Computing & Combinatorics (COCOON 2005), LNCS 3595, (Springer, 2005), pp 440–449Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Damien Massé
    • 1
  • Reinhardt Euler
    • 1
  • Laurent Lemarchand
    • 1
  1. 1.Lab-STICC UMR 6285UBO-Université Européenne de BretagneBrestFrance

Personalised recommendations