On the generality of the greedy algorithm for solving matroid base problems (original) (raw)

Abstract

It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or k-sum objective functions. In this paper, we address matroid base problems with a more general-"universal"-objective function which contains the previous ones as special cases. This universal objective function is of the sum type and associates multiplicative weights with the ordered cost coefficients of the elements of matroid bases such that, by choosing appropriate weights, many different-classical and new-objectives can be modeled. We show that the greedy algorithm is applicable to a larger class of objective functions than commonly known and, as such, it solves universal matroid base problems with non-negative or non-positive weight coefficients. Based on problems with mixed weights and a single (−, +)-sign change in the universal weight vector, we give a characterization of uniform matroids. In case of multiple sign changes, we use partition matroids. For non-uniform matroids, single sign change problems can be reduced to problems in minors obtained by deletion and contraction. Finally, we discuss how special instances of universal bipartite matching and shortest

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (28)

  1. Bonin, J., de Mier, A.: Lattice path matroids: Structural properties. European Journal of Combinatorics 27(5), 701-738 (2006)
  2. Bonin, J., de Mier, A., Noy, M.: Lattice path matroids: Enumerative aspects and Tutte polynomials. Journal of Combinatorial Theory, Series A 104(1), 63-94 (2003)
  3. Brezovec, C., Cornuéjols, G., Glover, F.: Two algorithms for weighted matroid intersec- tion. Mathematical Programming 36(1), 39-53 (1986)
  4. Brezovec, C., Cornuéjols, G., Glover, F.: A matroid algorithm and its application to the efficient solution of two optimization problems on graphs. Mathematical Programming 42(1-3), 471-487 (1988)
  5. Burkard, R.E., Hamacher, H.W.: Minimal cost flows in regular matroids. In: H. König, B. Korte, K. Ritter (eds.) Mathematical Programming at Oberwolfach, Mathematical Programming Studies, vol. 14, pp. 32-47. Springer, Berlin (1981)
  6. Duin, C.W., Volgenant, A.: Minimum deviation and balanced optimization: A unified approach. Operations Research Letters 10(1), 43-48 (1991)
  7. Fernández, E., Puerto, J., Rodríguez-Chía, A.M.: On discrete optimization with ordering (2008). Unpublished manuscript
  8. Fernández, E., Puerto, J., Rodríguez-Chía, A.M.: On discrete optimization with order- ing. Annals of Operations Research (2012). Doi: 10.1007/s10479-011-1044-7
  9. Gabow, H.N., Tarjan, R.E.: Efficient algorithms for a family of matroid intersection problems. Journal of Algorithms 5(1), 80-131 (1984)
  10. Gorski, J., Ruzika, S.: On k-max-optimization. Operations Research Letters 37(1), 23-26 (2009)
  11. Gupta, S.K., Punnen, A.P.: Minimum deviation problems. Operations Research Letters 7(4), 201-204 (1988)
  12. Gupta, S.K., Punnen, A.P.: k-sum optimization problems. Operations Research Letters 9(2), 121-126 (1990)
  13. Hamacher, H.W.: Algebraic flows in regular matroids. Discrete Applied Mathematics 2(1), 27-38 (1980)
  14. Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Dover Publications, Mineola (2001)
  15. Martello, S., Pulleyblank, W.R., Toth, P., de Werra, D.: Balanced optimization prob- lems. Operations Research Letters 3(5), 275-278 (1984)
  16. Minoux, M.: Solving combinatorial problems with combined min-max-min-sum objec- tive and applications. Mathematical Programming 45(1-3), 361-372 (1989)
  17. Nickel, S., Puerto, J.: Location Theory: A Unified Approach. Springer, Berlin (2005)
  18. Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (1992)
  19. Punnen, A.P.: On combined minmax-minsum optimization. Computers & Operations Research 21(6), 707-716 (1994)
  20. Punnen, A.P., Aneja, Y.P.: On k-sum optimization. Operations Research Letters 18(5), 233-236 (1996)
  21. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)
  22. Turner, L.: Variants of the shortest path problem. Algorithmic Operations Research 6(2), 91-104 (2011)
  23. Turner, L.: Universal combinatorial optimization: Matroid bases and shortest paths. Ph.D. thesis, University of Kaiserslautern, Germany (2012)
  24. Turner, L., Hamacher, H.W.: Universal shortest paths. WIMA Report 128, University of Kaiserslatuern, Germany (2010)
  25. Turner, L., Hamacher, H.W.: On universal shortest paths. In: B. Hu, K. Morasch, S. Pickl, M. Siegle (eds.) Operations Research Proceedings 2010: Selected Papers of the Annual International Conference of the German Operations Research Society, Munich, September 1-3, 2010, Operations Research Proceedings, pp. 313-318. Springer, Berlin (2011)
  26. Welsh, D.J.A.: Matroid Theory, L.M.S. Monographs, vol. 8. Academic Press, London (1976)
  27. Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria de- cisionmaking. IEEE Transactions on Systems, Man and Cybernetics 18(1), 183-190 (1988)
  28. Zimmermann, U.: Linear and Combinatorial Optimization in Ordered Algebraic Struc- tures, Annals of Discrete Mathematics, vol. 10. North-Holland, Amsterdam (1981)