Mathematical Programming

, Volume 128, Issue 1–2, pp 355–372 | Cite as

Budgeted matching and budgeted matroid intersection via the gasoline puzzle

  • André Berger
  • Vincenzo Bonifaci
  • Fabrizio Grandoni
  • Guido Schäfer
Full Length Paper Series A


Many polynomial-time solvable combinatorial optimization problems become NP-hard if an additional complicating constraint is added to restrict the set of feasible solutions. In this paper, we consider two such problems, namely maximum-weight matching and maximum-weight matroid intersection with one additional budget constraint. We present the first polynomial-time approximation schemes for these problems. Similarly to other approaches for related problems, our schemes compute two solutions to the Lagrangian relaxation of the problem and patch them together to obtain a near-optimal solution. However, due to the richer combinatorial structure of the problems considered here, standard patching techniques do not apply. To circumvent this problem, we crucially exploit the adjacency relations on the solution polytope and, surprisingly, the solution to an old combinatorial puzzle.


Matching Matroid intersection Budgeted optimization Lagrangian relaxation 

Mathematics Subject Classification (2000)

05C70 05B35 90C27 68R99 


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  1. 1.
    Aggarwal V., Aneja Y.P., Nair K.P.K.: Minimal spanning tree subject to a side constraint. Comput. Oper. Res. 9, 287–296 (1982)CrossRefGoogle Scholar
  2. 2.
    Barahona F., Pulleyblank W.R.: Exact arborescences, matchings and cycles. Discrete. Appl. Math. 16, 91–99 (1987)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beasley J.E., Christofides N.: An algorithm for the resource constrained shortest path problem. Networks 19, 379–394 (1989)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Camerini P., Galbiati G., Maffioli F.: Random pseudo-polynomial algorithms for exact matroid problems. J. Algorithms 13, 258–273 (1992)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chaudhuri, K., Rao, S., Riesenfeld, S., Talwar, K.: What would Edmonds do? Augmenting paths and witnesses for degree-bounded MSTs. In: Proceedings of the 8th International Workshop on Approximation Algorithms for Combinatorial Optimization, pp. 26–39 (2005)Google Scholar
  6. 6.
    Chaudhuri, K., Rao, S., Riesenfeld, S.,Talwar, K.: A push-relabel algorithm for approximating degree bounded MSTs. In: Proceedings of the 33rd International Colloquium on Automata, Languages and Programming, pp. 191–201 (2006)Google Scholar
  7. 7.
    Chvátal V.: On certain polytopes associated with graphs. J. Comb. Theory B. 18, 138–154 (1975)MATHCrossRefGoogle Scholar
  8. 8.
    Frank A., Tardos É.: Generalized polymatroids and submodular flows. Math. Program. 42, 489–563 (1988)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fürer M., Raghavachari B.: Approximating the minimum-degree Steiner tree to within one of optimal. J. Algorithms 17, 409–423 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Garey M.R., Johnson D.S.: Computers and Intractability: a Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco, CA (1979)MATHGoogle Scholar
  11. 11.
    Goemans, M.X.: Minimum bounded-degree spanning trees. In: Proceedings of the 47th IEEE Symposium on Foundations of Computer Science, pp. 273–282 (2006)Google Scholar
  12. 12.
    Guignard M., Rosenwein M.B.: An application of Lagrangean decomposition to the resource-constrained minimum weighted arborescence problem. Networks 20, 345–359 (1990)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hong S.-P., Chung S.-J., Park B.-H.: A fully polynomial bicriteria approximation scheme for the constrained spanning tree problem. Oper. Res. Lett. 32, 233–239 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Iwata S.: On matroid intersection adjacency. Discrete Math. 242, 277–281 (2002)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Könemann J., Ravi R.: A matter of degree: Improved approximation algorithms for degree bounded minimum spanning trees. SIAM J. Comput. 31, 1783–1793 (2002)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Könemann J., Ravi R.: Primal-Dual meets local search: approximating MSTs with nonuniform degree bounds. SIAM J. Comput. 34, 763–773 (2005)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Krogdahl S.: The dependence graph for bases in matroids. Discrete Math. 19, 47–59 (1977)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Lin S., Kernighan B.W.: An effective heuristic algorithm for the traveling salesman problem. Oper. Res. 21, 498–516 (1973)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Lovász L.: Combinatorial Problems and Exercises. North-Holland, Amsterdam (1979)MATHGoogle Scholar
  20. 20.
    Magazine M.J., Chern M.S.: A note on approximation schemes for multidimensional knapsack problems. Math. Oper. Res. 9, 244–247 (1984)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Marathe, M.V., Ravi, R., Sundaram, R., Ravi, S.S., Rosenkrantz, D.J., Hunt, H.B. III: Bicriteria network design problems. In: Proceedings of the 22nd International Colloquium on Automata, Languages and Programming, pp. 487–498 (1995)Google Scholar
  22. 22.
    Megiddo N.: Combinatorial optimization with rational objective functions. Math. Oper. Res. 4, 414–424 (1979)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Mulmuley K., Vazirani U., Vazirani V.: Matching is as easy as matrix inversion. Combinatorica 7, 105–113 (1987)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Naor, J., Shachnai, H., Tamir, T.: Personal communication (2007)Google Scholar
  25. 25.
    Naor J., Shachnai H., Tamir T.: Real-time scheduling with a budget. Algorithmica 47, 343–364 (2007)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley (1988)Google Scholar
  27. 27.
    Papadimitriou C.H., Yannakakis M.: The complexity of restricted spanning tree problems. J. ACM 29, 285–309 (1982)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proceedings of the 41st IEEE Symposium on Foundations of Computer Science, pp. 86–92 (2000)Google Scholar
  29. 29.
    Ravi, R. Goemans, M.X.: The constrained minimum spanning tree problem (extended abstract). In: Proceedings of the 5th Scandinavian Workshop on Algorithms and Theory, pp. 66–75 (1996)Google Scholar
  30. 30.
    Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrants, D.J.,Hunt, H.B.: Many birds with one stone: Multi-objective approximation algorithms. In: Proceedings of the 25th ACM Symposium on the Theory of Computing, pp. 438–447 (1993)Google Scholar
  31. 31.
    Ravi, R., Singh, M.: Delegate and Conquer: An LP-based approximation algorithm for minimum degree MSTs. In: Proceedings of the 33rd International Colloquium on Automata, Languages and Programming, pp. 169–180 (2006)Google Scholar
  32. 32.
    Schrijver A.: Theory of Linear and Integer Programming. Wiley, New York (1986)MATHGoogle Scholar
  33. 33.
    Schrijver A.: Combinatorial. Optimization Polyhedra and Efficiency. Springer, New York (2003)MATHGoogle Scholar
  34. 34.
    Shmoys, D.B., Tardos, É.: Scheduling unrelated machines with costs. In: Proceedings of the 4th ACM-SIAM Symposium on Discrete Algorithms, pp. 448–454 (1993)Google Scholar
  35. 35.
    Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: Proceedings of the 39th ACM Symposium on the Theory of Computing, pp. 661–670 (2007)Google Scholar

Copyright information

© Springer and Mathematical Programming Society 2009

Authors and Affiliations

  • André Berger
    • 1
  • Vincenzo Bonifaci
    • 2
    • 3
  • Fabrizio Grandoni
    • 4
  • Guido Schäfer
    • 5
    • 6
  1. 1.Department of Quantitative EconomicsMaastricht UniversityMaastrichtThe Netherlands
  2. 2.Department of Electrical EngineeringUniversity of L’AquilaL’AquilaItaly
  3. 3.Department of Computer and Systems ScienceSapienza University of RomeRomeItaly
  4. 4.Department of Computer Science, Systems and ProductionUniversity of Rome Tor VergataRomeItaly
  5. 5.Center for Mathematics and Computer Science (CWI)AmsterdamThe Netherlands
  6. 6.Department of Econometrics and Operations ResearchVU University AmsterdamAmsterdamThe Netherlands

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