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Iterative Rounding for Multi-Objective Optimization Problems

  • Fabrizio Grandoni
  • R. Ravi
  • Mohit Singh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5757)

Abstract

In this paper we show that iterative rounding is a powerful and flexible tool in the design of approximation algorithms for multi-objective optimization problems. We illustrate that by considering the multi-objective versions of three basic optimization problems: spanning tree, matroid basis and matching in bipartite graphs. Here, besides the standard weight function, we are given k length functions with corresponding budgets. The goal is finding a feasible solution of maximum weight and such that, for all i, the ith length of the solution does not exceed the ith budget. For these problems we present polynomial-time approximation schemes that, for any constant ε> 0 and k ≥ 1, compute a solution violating each budget constraint at most by a factor (1 + ε). The weight of the solution is optimal for the first two problems, and (1 − ε)-approximate for the last one.

Keywords

Span Tree Budget Constraint Network Design Problem Linear Programming Relaxation Span Tree Problem 
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.

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References

  1. 1.
    Aggarwal, V., Aneja, Y.P., Nair, K.P.K.: Minimal spanning tree subject to a side constraint. Computers & Operations Research 9, 287–296 (1982)CrossRefGoogle Scholar
  2. 2.
    Bansal, N., Khandekar, R., Nagarajan, V.: Additive Guarantees for Degree Bounded Directed Network Design. In: STOC, pp. 769–778 (2008)Google Scholar
  3. 3.
    Barahona, F., Pulleyblank, W.R.: Exact arborescences, matchings and cycles. Discrete Applied Mathematics 16(2), 91–99 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barichard, V., Ehrgott, M., Gandibleux, X., T’Kindt, V. (eds.): Multiobjective Programming and Goal Programming: Theoretical Results and Practical Applications. Lecture Notes in Economics and Mathematical Systems, vol. 618. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  5. 5.
    Beasley, J.E., Christofides, N.: An algorithm for the resource constrained shortest path problem. Networks 19, 379–394 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berger, A., Bonifaci, V., Grandoni, F., Schäfer, G.: Budgeted matching and budgeted matroid intersection via the gasoline puzzle. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 273–287. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Bilò, V., Goyal, V., Ravi, R., Singh, M.: On the Crossing Spanning Tree Problem. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 51–64. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Cheriyan, J., Vempala, S., Vetta, A.: Network design via iterative rounding of setpair relaxations. Combinatorica 26(3), 255–275 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds.): Pareto Optimality, Game Theory and Equilibria. Optimization and Its Applications, vol. 17 (2008)Google Scholar
  10. 10.
    Climacao, J.: Multicriteria Analysis. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  11. 11.
    Cunningham, W.H.: Testing Membership in Matroid Polyhedra. Journal of Combinatorial Theory B 36(2), 161–188 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fleischer, L., Jain, K., Williamson, D.P.: Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems. Journal of Computer and System Sciences 72(5), 838–867 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Guignard, M., Rosenwein, M.B.: An application of Lagrangean decomposition to the resource-constrained minimum weighted arborescence problem. Networks 20, 345–359 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hartley, R.: Survey of Algorithms for Vector Optimization Problems. In: French, S., Hartley, R., Thomas, L.C., White, D.J. (eds.) Multiobjective Decision Making, pp. 1–34. Academic Press, London (1983)Google Scholar
  15. 15.
    Jain, K.: A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21, 39–60 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lau, L.C., Naor, S., Salavatipour, M., Singh, M.: Survivable network design with degree or order constraints. In: STOC, pp. 651–660 (2007)Google Scholar
  17. 17.
    Lau, L.C., Singh, M.: Additive approximation for bounded degree survivable network design. In: STOC, pp. 759–768 (2008)Google Scholar
  18. 18.
    Levin, A., Woeginger, G.J.: The constrained minimum weight sum of job completion times. Mathematical Programming 108, 115–126 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Marathe, M.V., Ravi, R., Sundaram, R., Ravi, S.S., Rosenkrantz, D.J., Hunt III, H.B.: Bicriteria network design problems. In: Fülöp, Z., Gecseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 487–498. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  20. 20.
    Melkonian, V., Tardos, E.: Algorithms for a Network Design Problem with Crossing Supermodular Demands. Networks 43, 4 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mulmuley, K., Vazirani, U., Vazirani, V.: Matching is as Easy as Matrix Inversion. Combinatorica 7(1), 101–104 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of Web sources. In: FOCS, pp. 86–92 (2000)Google Scholar
  23. 23.
    Ravi, R.: Rapid rumor ramification: Approximating the minimum broadcast time. In: FOCS, pp. 202–213 (1994)Google Scholar
  24. 24.
    Ravi, R.: Matching Based Augmentations for Approximating Connectivity Problems. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 13–24. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  25. 25.
    Ravi, R., Goemans, M.X.: The constrained minimum spanning tree problem (extended abstract). In: Karlsson, R., Lingas, A. (eds.) SWAT 1996. LNCS, vol. 1097, pp. 66–75. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  26. 26.
    Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Hunt, H.B.: Many Birds with One Stone: Multi-objective Approximation Algorithms. In: STOC, pp. 438–447 (1993)Google Scholar
  27. 27.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)zbMATHGoogle Scholar
  28. 28.
    Shmoys, D.B., Tardos, É.: Scheduling unrelated machines with costs. In: SODA, pp. 448–454 (1993)Google Scholar
  29. 29.
    Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: STOC, pp. 661–670 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Fabrizio Grandoni
    • 1
  • R. Ravi
    • 2
  • Mohit Singh
    • 3
  1. 1.Department of Computer Science, Systems and ProductionUniversity of Rome Tor VergataItaly
  2. 2.Tepper School of Business, Carnegie Mellon University, Pittsburgh, USA, Supported in part by NSF grant CCF-0728841USA
  3. 3.Microsoft Research, New EnglandCambridgeUSA

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