About strongly polynomial time algorithms for quadratic optimization over submodular constraints

Abstract

We present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints. The main results are: an O(NM log(N 2/M)) algorithm for the problemNetwork defined on a network onM arcs andN nodes; an O(n logn) algorithm for thetree problem onn variables; an O(n logn) algorithm for theNested problem, and a linear time algorithm for theGeneralized Upper Bound problem. These algorithms are the best known so far for these problems. The status of the general problem and open questions are presented as well.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    R. Baldick and F.F. Wu, “Efficient integer optimization algorithms for optimal coordination of capacitators and regulators,”IEEE Transactions on Power Systems 5 (1990) 805–812.

    Google Scholar 

  2. [2]

    M.J. Best and R.Y. Tan, “An O(n 2 logn) strongly polynomial algorithm for quadratic program with two equations and lower and upper bounds,” Research Report CORR 90-04, Department of Combinatorics and Optimization, University of Waterloo (1990).

  3. [3]

    M. Blum, R.W. Floyd, V.R. Pratt, R.L. Rivest and R.E. Tarjan, “Time bounds for selection,”Journal of Computer and Systems Sciences 7 (1972) 448–461.

    Google Scholar 

  4. [4]

    P. Brucker, “An O(n) algorithm for quadratic knapsack problems,”Operations Research Letters 3 (1984) 163–166.

    Google Scholar 

  5. [5]

    W. Cook, A.M.H. Gerards, A. Schrijver and E. Tardos, “Sensitivity results in integer linear programming,”Mathematical Programming 34 (1986) 251–264.

    Google Scholar 

  6. [6]

    S. Cosares and D.S. Hochbaum, “Strongly polynomial algorithms for the quadratic transportation problem with fixed number of sources,”Mathematics of Operations Research 19(1) (1994) 94–111.

    Google Scholar 

  7. [7]

    M.E. Dyer and A.M. Frieze, “On an optimization problem with nested constraints,”Discrete Applied Mathematics 26 (1990) 159–173.

    Google Scholar 

  8. [8]

    G.N. Frederickson and D.B. Johnson, “The complexity of selection and ranking inX+Y and matrices with sorted columns,”Journal of Computer and Systems Sciences 24 (1982) 197–208.

    Google Scholar 

  9. [9]

    S. Fujishige, “Lexicographically optimal base of a polymatroid with respect to a weight vector,”Mathematics of Operations Research 5 (1980) 186–196.

    Google Scholar 

  10. [10]

    S. Fujishige,Submodular Functions and Optimization, Annals of Discrete Mathematics 47 (North-Holland, Amsterdam, 1991).

    Google Scholar 

  11. [11]

    H.N. Gabow and R.E. Tarjan, “A linear-time algorithm for a special case of disjoint set union,”Journal of Computer and System Sciences 30 (1985) 209–221.

    Google Scholar 

  12. [12]

    G. Gallo, M.E. Grigoriadis and R.E. Tarjan, “A fast parametric maximum flow algorithm and applications,”SIAM Journal of Computing 18 (1989) 30–55.

    Google Scholar 

  13. [13]

    A.V. Goldberg and R.E. Tarjan, “A new approach to the maximum flow problem,” in:Proceedings of the 18th Annual ACM Symposium on Theory of Computing (1986) pp. 136–146.

  14. [14]

    F. Granot and J. Skorin-Kapov, “Some proximity and sensitivity results in quadratic integer programming,” Working Paper No. 1207, University of British Columbia (1986).

  15. [15]

    F. Granot and J. Skorin-Kapov, “Strongly polynomial solvability of a nonseparable quadratic integer program with applications to toxic waste disposal,” Manuscript (1990).

  16. [16]

    H. Groenevelt, “Two algorithms for maximizing a separable concave function over a polymatroidal feasible region,” Technical Report, The Graduate School of Management, University of Rochester (1985).

  17. [17]

    D.S. Hochbaum, “On the impossibility of strongly polynomial algorithms for the allocation problem and its extensions,” in:Proceedings of the 1st Integer Programming and Combinatorial Optimization Conference (1990) pp. 261–274; D.S. Hochbaum, “Lower and upper bounds for the allocation problem and other nonlinear optimization problems,”Mathematics of Operations Research 19 (1994) 390–409.

    Google Scholar 

  18. [18]

    D.S. Hochbaum and J.G. Shanthikumar, “Convex separable optimization is not much harder than linear optimization,”Journal of ACM 37 (1990) 843–862.

    Google Scholar 

  19. [19]

    D.S. Hochbaum, R. Shamir and J.G. Shanthikumar, “A polynomial algorithm for an integer quadratic nonseparable transportation problem,”Mathematical Programming 55(3) (1992) 359–372.

    Google Scholar 

  20. [20]

    T. Ibaraki and N. Katoh,Resource Allocation Problems: Algorithmic Approaches (MIT Press, Cambridge, MA, 1988).

    Google Scholar 

  21. [21]

    N. Megiddo and A. Tamir, “Linear time algorithms for some separable quadratic programming problems,”Operations Research Letters 13 (1993) 203–211.

    Google Scholar 

  22. [22]

    G.L. Nemhauser and L.A. Wolsey,Integer and Combinatorial Optimization (Wiley, New York, 1988).

    Google Scholar 

  23. [23]

    A. Tamir, “A strongly polynomial algorithm for minimum convex separable quadratic cost flow problems on series-parallel networks,”Mathematical Programming 59 (1993) 117–132.

    Google Scholar 

  24. [24]

    E. Tardos, “A strongly polynomial algorithm to solve combinatorial linear programs,”Operations Research 34 (1986) 250–256.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

This research has been supported in part by ONR grant N00014-91-J-1241.

Corresponding author.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hochbaum, D.S., Hong, S. About strongly polynomial time algorithms for quadratic optimization over submodular constraints. Mathematical Programming 69, 269–309 (1995). https://doi.org/10.1007/BF01585561

Download citation

Keywords

  • Quadratic programming
  • Submodular constraints
  • Kuhn-Tucker conditions
  • Lexicographically optimal flow
  • Parametric maximum flow