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Greedy in Approximation Algorithms

  • Julián Mestre
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)

Abstract

The objective of this paper is to characterize classes of problems for which a greedy algorithm finds solutions provably close to optimum. To that end, we introduce the notion of k-extendible systems, a natural generalization of matroids, and show that a greedy algorithm is a \(\frac{1}{k}\)-factor approximation for these systems. Many seemly unrelated problems fit in our framework, e.g.: b-matching, maximum profit scheduling and maximum asymmetric TSP.

In the second half of the paper we focus on the maximum weight b-matching problem. The problem forms a 2-extendible system, so greedy gives us a \(\frac{1}{2}\)-factor solution which runs in O(m logn) time. We improve this by providing two linear time approximation algorithms for the problem: a \(\frac{1}{2}\)-factor algorithm that runs in O(bm) time, and a \(\left(\frac{2}{3} -- \epsilon\right)\)-factor algorithm which runs in expected \(O\left(b m \log \frac{1}{\epsilon}\right)\) time.

Keywords

Approximation Algorithm Greedy Algorithm Maximum Weight Factor Algorithm Matched Edge 
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.
    Angelopoulos, S., Borodin, A.: The power of priority algorithms for facility location and set cover. Algorithmica 40(4), 271–291 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arkin, E.M., Hassin, R.: On local search for weighted k-set packing. Mathematics of Operations Research 23(3), 640–648 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arora, V., Vempala, S., Saran, H., Vazirani, V.V.: A limited-backtrack greedy schema for approximation algorithms. In: Thiagarajan, P.S. (ed.) FSTTCS 1994. LNCS, vol. 880, pp. 318–329. Springer, Heidelberg (1994)Google Scholar
  4. 4.
    Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J.S., Schieber, B.: A unified approach to approximating resource allocation and scheduling. Journal of the ACM 48(5), 1069–1090 (2001)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Borodin, A., Nielsen, M.N., Rockoff, C.: (Incremental) Priority algorithms. Algorithmica 37(4), 295–326 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Borovik, A.V., Gelfand, I., White, N.: Symplectic matroid. Journal of Algebraic Combinatorics 8, 235–252 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bouchet, A.: Greedy algorithm and symmetric matroids. Mathematical Programming 38, 147–159 (1987)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Drake, D.E., Hougardy, S.: Improved linear time approximation algorithms for weighted matchings. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 14–23. Springer, Heidelberg (2003)Google Scholar
  9. 9.
    Drake, D.E., Hougardy, S.: A simple approximation algorithm for the weighted matching problem. Information Processing Letters 85, 211–213 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Edmonds, J.: Minimum partition of a matroid into independent subsets. J. of Research National Bureau of Standards 69B, 67–77 (1965)MathSciNetGoogle Scholar
  11. 11.
    Edmonds, J.: Matroids and the greedy algorithm. Mathematical Programming 1, 36–127 (1971)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Gabow, H.N.: An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In: STOC, pp. 448–456 (1983)Google Scholar
  13. 13.
    Jenkyns, T.A.: The greedy travelling salesman’s problem. Networks 9, 363–373 (1979)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Korte, B., Hausmann, D.: An analysis of the greedy algorithm for independence systems. Ann. Disc. Math. 2, 65–74 (1978)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Korte, B., Lovász, L.: Greedoids—a structural framework for the greedy algorithm. In: Progress in Combinatorial Optimization, pp. 221–243 (1984)Google Scholar
  16. 16.
    Lewenstein, M., Sviridenko, M.: Approximating asymmetric maximum TSP. In: SODA, pp. 646–654 (2003)Google Scholar
  17. 17.
    Lovász, L.: The matroid matching problem. In: Algebraic Methods in Graph Theory, Colloquia Mathematica Societatis Janos Bolyai (1978)Google Scholar
  18. 18.
    Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (1992)MATHGoogle Scholar
  19. 19.
    Pettie, S., Sanders, P.: A simpler linear time 2/3 − ε approximation to maximum weight matching. Information Processing Letters 91(6), 271–276 (2004)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Preis, R.: Linear time 1/2-approximation algorithm for maximum weighted matching in general graphs. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 259–269. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  21. 21.
    Rado, R.: A theorem on independence relations. Quart. J. Math. 13, 83–89 (1942)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Schrijver, A.: Combinatorial Optimization. Springer, Heidelberg (2003)MATHGoogle Scholar
  23. 23.
    Vince, A.: A framework for the greedy algorithm. Discrete Applied Mathematics 121(1–3), 247–260 (2002)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Whitney, H.: On the abstract properties of linear dependence. American Journal of Mathematic 57, 509–533 (1935)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Julián Mestre
    • 1
  1. 1.Department of Computer ScienceUniversity of MarylandCollege ParkUSA

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