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

Abstract

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.

Keywords

Matching Matroid intersection Budgeted optimization Lagrangian relaxation 

Mathematics Subject Classification (2000)

05C70 05B35 90C27 68R99 

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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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