Improved Approximations for k-Exchange Systems

(Extended Abstract)
  • Moran Feldman
  • Joseph (Seffi) Naor
  • Roy Schwartz
  • Justin Ward
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6942)


Submodular maximization and set systems play a major role in combinatorial optimization. It is long known that the greedy algorithm provides a 1/(k + 1)-approximation for maximizing a monotone submodular function over a k-system. For the special case of k-matroid intersection, a local search approach was recently shown to provide an improved approximation of 1 / (k + δ) for arbitrary δ > 0. Unfortunately, many fundamental optimization problems are represented by a k-system which is not a k-intersection. An interesting question is whether the local search approach can be extended to include such problems.

We answer this question affirmatively. Motivated by the b-matching and k-set packing problems, as well as the more general matroid k-parity problem, we introduce a new class of set systems called k-exchange systems, that includes k-set packing, b-matching, matroid k-parity in strongly base orderable matroids, and additional combinatorial optimization problems such as: independent set in (k + 1)-claw free graphs, asymmetric TSP, job interval selection with identical lengths and frequency allocation on lines. We give a natural local search algorithm which improves upon the current greedy approximation, for this new class of independence systems. Unlike known local search algorithms for similar problems, we use counting arguments to bound the performance of our algorithm.

Moreover, we consider additional objective functions and provide improved approximations for them as well. In the case of linear objective functions, we give a non-oblivious local search algorithm, that improves upon existing local search approaches for matroid k-parity.


Local Search Local Search Algorithm Free Graph Submodular Function Linear Objective Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anstee, R.P.: A polynomial algorithm for b-matchings: An alternative approach. Information Processing Letters 24(3), 153–157 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berman, P.: A d/2 approximation for maximum weight independent set in d-claw free graphs. Nordic J. of Computing 7, 178–184 (2000)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brualdi, R.A.: Comments on bases in dependence structures. Bull. of the Australian Math. Soc. 1(02), 161–167 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brualdi, R.A.: Common transversals and strong exchange systems. J. of Combinatorial Theory 8(3), 307–329 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brualdi, R.A.: Induced matroids. Proc. of the American Math. Soc. 29, 213–221 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brualdi, R.A., Scrimger, E.B.: Exchange systems, matchings, and transversals. J. of Combinatorial Theory 5(3), 244–257 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Calinescu, G., Chekuri, C., Pal, M., Vondrák, J.: Maximizing a submodular set function subject to a matroid constraint. To appear in SIAM Journal on Computing, Special Issue for STOC 2008 (2008)Google Scholar
  8. 8.
    Chandra, B., Halldórsson, M.: Greedy local improvement and weighted set packing approximation. In: SODA, pp. 169–176 (1999)Google Scholar
  9. 9.
    Chekuri, C., Vondrák, J., Zenklusen, R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes. In: STOC, pp. 783–792 (2011)Google Scholar
  10. 10.
    Conforti, M., Cornuèjols, G.: Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the rado-edmonds theorem. Disc. Appl. Math. 7(3), 251–274 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Edmonds, J.: Matroid intersection. In: Hammer, P., Johnson, E., Korte, B. (eds.) Discrete Optimization I, Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium. Annals of Discrete Mathematics, vol. 4, pp. 39–49. Elsevier, Amsterdam (1979)CrossRefGoogle Scholar
  12. 12.
    Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. In: FOCS, pp. 461–471 (2007)Google Scholar
  13. 13.
    Feldman, M., Naor, J.S., Schwartz, R.: Nonmonotone submodular maximization via a structural continuous greedy algorithm. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 342–353. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Feldman, M., Naor, J.S., Schwartz, R.: A unified continuous greedy algorithm for submodular maximization. To appear in FOCS 2011 (2011)Google Scholar
  15. 15.
    Fisher, M., Nemhauser, G., Wolsey, L.: An analysis of approximations for maximizing submodular set functions - ii. Math. Prog. Study 8, 73–87 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gharan, S.O., Vondrák, J.: Submodular maximization by simulated annealing. In: SODA, pp. 1098–1116 (2011)Google Scholar
  17. 17.
    Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained non-monotone submodular maximization: Offline and secretary algorithms. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 246–257. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Hausmann, D., Korte, B.: K-greedy algorithms for independence systems. Oper. Res. Ser. A-B 22(1), 219–228 (1978)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hausmann, D., Korte, B., Jenkyns, T.: Worst case analysis of greedy type algorithms for independence systems. Math. Prog. Study 12, 120–131 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Comput. Complex. 15, 20–39 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an sdr, with an application to the worst case ratio of heuristics for packing problems. SIAM J. Disc. Math. 2(1), 68–72 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jenkyns, T.: The efficacy of the greedy algorithm. Cong. Num. 17, 341–350 (1976)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jensen, P.M., Korte, B.: Complexity of matroid property algorithms. SIAM J. Computing 11(1), 184 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kalyanasundaram, B., Pruhs, K.: An optimal deterministic algorithm for online b-matching. In: Chandru, V., Vinay, V. (eds.) FSTTCS 1996. LNCS, vol. 1180, pp. 193–199. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  25. 25.
    Korte, B., Hausmann, D.: An analysis of the greedy heuristic for independence systems. Annals of Discrete Math. 2, 65–74 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lee, J., Sviridenko, M., Vondrák, J.: Submodular maximization over multiple matroids via generalized exchange properties. To appear in Mathematics of Operations Research (2009)Google Scholar
  27. 27.
    Lee, J., Sviridenko, M., Vondrak, J.: Matroid matching: the power of local search. In: STOC, pp. 369–378 (2010)Google Scholar
  28. 28.
    Lovász, L.: The matroid matching problem. In: Lovász, L., Sós, V.T. (eds.) Algebraic Methods in Graph Theory, Amsterdam (1981)Google Scholar
  29. 29.
    Marsh III., A.B.: Matching algorithms. PhD thesis, The Johns Hopkins University (1979)Google Scholar
  30. 30.
    Mestre, J.: Greedy in approximation algorithms. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 528–539. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  31. 31.
    Nemhauser, G., Wolsey, L.: Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res. 3(3), 177–188 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nemhauser, G., Wolsey, L., Fisher, M.: An analysis of approximations for maximizing submodular set functions - i. Math. Prog. 14(1), 265–294 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Pulleyblank, W.: Faces of matching polyhedra. PhD thesis, Deptartment of Combinatorics and Optimization, University of Waterloo (1973)Google Scholar
  34. 34.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  35. 35.
    Simon, H.: Approximation algorithms for channel assignment in cellular radio networks. In: Csirik, J., Demetrovics, J., Gècseg, F. (eds.) FCT 1989. LNCS, vol. 380, pp. 405–415. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  36. 36.
    Soto, J.A.: A simple PTAS for weighted matroid matching on strongly base orderable matroids. To appear in LAGOS (2011)Google Scholar
  37. 37.
    Tong, P., Lawler, E.L., Vazirani, V.V.: Solving the weighted parity problem for gammoids by reduction to graphic matching. Technical Report UCB/CSD-82-103, EECS Department, University of California, Berkeley (April 1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Moran Feldman
    • 1
  • Joseph (Seffi) Naor
    • 1
  • Roy Schwartz
    • 1
  • Justin Ward
    • 2
  1. 1.Computer Science Dept.TechnionHaifaIsrael
  2. 2.Computer Science Dept.University of TorontoTorontoCanada

Personalised recommendations