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Optimization Letters

, Volume 12, Issue 3, pp 649–659 | Cite as

Weighted matching with pair restrictions

  • Dorit S. Hochbaum
  • Asaf Levin
Original Paper
  • 79 Downloads

Abstract

The weighted matroid parity problems for the matching matroid and gammoids are among the very few cases for which the weighted matroid parity problem is polynomial time solvable. In this work we extend these problems to a general revenue function for each pair, and show that the resulting problem is still solvable in polynomial time via a standard weighted matching algorithm. We show that in many other directions, extending our results further is impossible (unless P = NP). One consequence of the new polynomial time algorithm is that it demonstrates, for the first time, that a prize-collecting assignment problem with “pair restriction” is solved in polynomial time. The prize collecting assignment problem is a relaxation of the prize-collecting traveling salesman problem which requires, for any prescribed pair of nodes, either both nodes of the pair are matched or none of them are. It is shown that the prize collecting assignment problem is equivalent to the prize collecting cycle cover problem which is hence solvable in polynomial time as well.

Keywords

Assignment Traveling salesman problem Prize collecting Weighted graph matching Matching matroid Gammoid Weighted matroid parity 

Notes

Acknowledgements

The topic of prize collecting 2-factor arose from a discussion the first author had with Alejandro Toriello concerning relaxations for the prize collecting traveling salesman problem.

References

  1. 1.
    Archetti, C., Speranza, M.G., Vigo, D.: Vehicle routing problems with profits. In: Vehicle Routing: Problems, Methods, and Applications, SIAM, pp. 273–298 (2014)Google Scholar
  2. 2.
    Balas, E.: The prize collecting traveling salesman problem. Networks 19, 621636 (1989)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Balas, E.: The prize collecting traveling salesman problem and its applications. In: The Traveling Salesman Problem and Its Variations, Springer, pp. 663–695 (2007)Google Scholar
  4. 4.
    Bienstock, D., Goemans, M.X., Simchi-Levi, D., Williamson, D.P.: A note on the prize collecting traveling salesman problem. Math. Program. 59, 413–420 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Camerini, P., Galbiati, G., Maffioli, F.: Random pseudo-polynomial algorithms for exact matroid problems. J. Algorithms 13, 258–273 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheung, H.Y., Lau, L.C., Leung, K.M.: Algebraic algorithms for linear matroid parity problems. ACM Trans. Algorithms (TALG), 10(3), article number 10 (2014)Google Scholar
  7. 7.
    Feillet, D., Dejax, P., Gendreau, M.: Traveling salesman problems with profits. Transp. Sci. 39(2), 188–205 (2005)CrossRefGoogle Scholar
  8. 8.
    Gabow, H.N., Stallmann, M.: An augmenting path algorithm for linear matroid parity. Combinatorica 6, 123–150 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  10. 10.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 296–317 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goemans, M.X.: Combining approximation algorithms for the prize-collecting TSP. arXiv:0910.0553v1 (2009)
  12. 12.
    Hoffman, K.L., Padberg, M., Rinaldi, G.: Traveling salesman problem. In: Encyclopedia of Operations Research and Management Science, Springer, pp. 1573–1578 (2013)Google Scholar
  13. 13.
    Lee, J., Sviridenko, M., Vondrak, J.: Matroid matching: the power of local search. SIAM J. Comput. 42, 357–379 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lovász, L.: The matroid matching problem. In: Algebraic Methods in Graph Theory, pp. 495–517, Szeged (1978)Google Scholar
  15. 15.
    Narayanan, H., Saran, H., Vazirani, V.: Randomized parallel algorithms for matroid union and intersection, with applications to arboresences and edge-disjoint spanning trees. SIAM J. Comput. 23, 387–397 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Orlin, J.B.: A fast, simpler algorithm for the matroid parity problem. Proc. IPCO 240–258, 2008 (2008)zbMATHGoogle Scholar
  17. 17.
    Orlin, J.B., Vate, J.H.V.: Solving the linear matroid parity problem as a sequence of matroid intersection problems. Math. Program. 47, 81–106 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Promel, H.J., Steger, A.: A new approximation algorithm for the Steiner tree problem with performance ratio 5/3. J. Algorithms 36, 89–101 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Riera-Ledesma, J., Salazar-Gonzalez, J.-J.: Solving school bus routing using the multiple vehicle traveling purchaser problem: a branch-and-cut approach. Comput. Oper. Res. 39(2), 391–404 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schrijver, A.: Combinatorial Optimization Polyhedra and Efficiency. Springer, Berlin (2003)zbMATHGoogle Scholar
  21. 21.
    Soto, A.J.: A simple PTAS for weighted matroid matching on strongly base orderable matroids. Discrete Appl. Math. 164, 406–412 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tong, P., Lawler, E.L., Vazirani, V.V.: Solving the Weighted Parity Problem for Gammoids by Reduction to Graphic Matching. University of California, Oakland (1982)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Operations ResearchUniversity of CaliforniaBerkeleyUSA
  2. 2.Faculty of Industrial Engineering and ManagementThe TechnionHaifaIsrael

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