Abstract
Good bounds on the optimum value of hard optimization problems can often be efficiently obtained by “pricing out” certain “bad” constraints and incorporating them into the objective function. The resulting problem is known as the Lagrangian dual. Here we give improved algorithms for solving the Lagrangian duals of problems that have both of the following properties. First, in the absence of the bad constraints, the problems can be solved in strongly polynomial time by combinatorial algorithms. Second, the number of bad constraints is fixed. As part of our solution to these problems, we extend Cole's circuit simulation approach and develop a weighted version of Megiddo's multidimensional search technique.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
M. Ajtai, J. Komlós, and E. Szemerédi: A O(n log n) sorting network. Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pp 1–9, 1983.
Y. P. Aneja and S. N. Kabadi: Polynomial algorithms for lagrangean relaxations in combinatorial problems (Manuscript).
D. Bertsimas and J.B. Orlin: A technique for speeding up the solution of the Lagrangean dual. In proceedings of IPCO 92.
P.M. Camerini, F. Maffioli, and C. Vercellis: Multi-constrained matroidal knapsack problems. Mathematical Programming 45:211–231 (1989).
V. Chvátal: Linear Programming. W.H. Freeman, San Francisco 1983.
R. Cole: Slowing down sorting networks to obtain faster sorting algorithms. J. Assoc. Comput. Mach. 34(1):200–208, 1987.
M. E. Dyer: On a multidimensional search technique and its application to the Euclidean one-center problem. SLAM J. Comput. 15(3):725–738 (1986).
H. Edelsbrunner: Algorithms in Combinatorial Geometry. Springer-Verlag, Heidelberg 1987
D. Fernández-Baca and G. Slutzki: Parametric problems on graphs of bounded tree-width. To appear in Proceedings of the 3rd Scandinavian Workshop on Algorithm Theory, Springer-Verlag LNCS, 1992.
M. L. Fisher: The Lagrangian relaxation method for solving integer programming problems. Management Science 27(1):1–18, (1981).
M. Held and R.M. Karp: The traveling salesman problem and minimum spanning trees. Operations Research 18:1138–1162.
M. Held and R.M. Karp: The traveling salesman problem and minimum spanning trees: part II. Mathematical Programming 6:6–25 (1971).
E. Lawler: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and Winston, 1976.
N. Megiddo: Combinatorial optimization with rational objective functions. Math. Oper. Res. 4:414–424.
N. Megiddo: Applying parallel computation algorithms in the design of serial algorithms. J. Assoc. Comput. Mach. 30(4):852–865, (1983).
N. Megiddo: Linear programming in linear time when the dimension is fixed. J. Assoc. Comput. Mach., 31:114–127, 1984.
A. Schrijver: Theory of Linear and Integer Programming. Wiley, Chichester 1986.
R.E. Tarjan: Data Structures and Network Algorithms. SIAM Press, Philadelphia 1983.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Agarwala, R., Fernández-Baca, D. (1992). Solving the Lagrangian dual when the number of constraints is fixed. In: Shyamasundar, R. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1992. Lecture Notes in Computer Science, vol 652. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56287-7_103
Download citation
DOI: https://doi.org/10.1007/3-540-56287-7_103
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56287-0
Online ISBN: 978-3-540-47507-1
eBook Packages: Springer Book Archive