Subgradient optimization, matroid problems and heuristic evaluation

  • F. Maffioli
Mathematical Programming
Part of the Lecture Notes in Computer Science book series (LNCS, volume 41)


Many polynomial complete problems can be reduced efficiently to three matroids intersection problems. Subgradient methods are shown to yield very good algorithms for computing tight lower bounds to the solution of these problems. The bounds may be used either to construct heuristically guided (branch-and-bound) methods for solving the problems, or to obtain an upper bound to the difference between exact and approx imate solutions by heuristic methods. The existing experience tend to indicate that such bounds would be quite precise.


Greedy Algorithm Travel Salesman Problem Subgradient Method Heuristic Evaluation Subgradient Optimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    R.M. Karp, "Reducibility among combinatorial problems", Symp. on Complexity of Computations (Miller & Tatcher eds.) Plenum Press 1972Google Scholar
  2. [2]
    H. Whitney, "On the abstract properties of linear dependence", Amer. J. Math. 57 (1935) 509–533.Google Scholar
  3. [3]
    J. Edmonds, "Matroids and the greedy algorithm", Math. Programm. 1 (1971) 127–136.Google Scholar
  4. [4]
    E.L. Lawler, "Polynomial bounded and (apparently) non-polynomial bounded matroid computations", Combinatorial Algorithms (Rustin ed.) Algorithmics Press 1972.Google Scholar
  5. [5]
    M. Held & R.M. Karp, "The traveling salesman problem and minimum spanning trees: part II", Math. Programm. 1 (1971) 6–25.Google Scholar
  6. [6]
    P.M. Camerini, L. Fratta & F. Maffioli, "A heuristically guided algorithm for the traveling salesman problem", J. of the Institution of Computer Sciences 4 (1973) 31–35.Google Scholar
  7. [7]
    K.H. Hansen & J. Krarup, "Improvements of the Held-Karp algorithm for the symmetric traveling salesman problem" Math. Progr. 7 (1974) 87–96.Google Scholar
  8. [8]
    M. Held, P. Wolfe & H.P. Crowder, "Validation of subgradient optimization", Math. Programm. 6 (1974) 62–88.Google Scholar
  9. [9]
    P.M. Camerini, L. Fratta & F. Maffioli, "Relaxation methods improved by modified gradient techniques", Conference on Opns. Res., Eger (Hungary) August 1974.Google Scholar
  10. [10]
    J. Edmonds & D.R. Fulkerson, "Transversals and matroid partition", J. Res. NBS 69B (1965) 147–153.Google Scholar
  11. [11]
    H. Crowder, "Computational improvements for subgradient optimization", IBM Res. Rep. RC 4907 (21841) 1974.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • F. Maffioli
    • 1
  1. 1.Istituto di Elettrotecnica ed ElettronicaPolitecnico di MilanoMilanoItaly

Personalised recommendations