Abstract
We consider a system ofm linearly independent equality constraints inn nonnegative variables:Ax = b, x ≧ 0. The fundamental problem that we discuss is the following: suppose we are given a set ofr linearly independent column vectors ofA, known asthe special column vectors. The problem is to develop an efficient algorithm to determine whether there exists a feasible basis which contains all the special column vectors as basic column vectors and to find such a basis if one exists. Such an algorithm has several applications in the area of mathematical programming. As an illustration, we show that the famous travelling salesman problem can be solved efficiently using this algorithm. Recent published work indicates that this algorithm has applications in integer linear programming. An algorithm for this problem using a set covering approach is described.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Andrew, T. Hoffman and C. Krabek, “On the generalized set covering problem,” CDC Data Centers Division, Minneapolis, 1968.
E. Balas and M.W. Padberg, “On the set covering problem,” Management Science Research Report, No. 197, Carnegie Mellon University, 1970.
M.L. Balinski, “Integer programming: methods, uses, computation,”Management Science 12 (1965) 253–313.
A.V. Cabot and A.P. Hurter Jr., “An approach to zero–one integer programming,”Operations Research 16 (1968) 1206–1211.
G.B. Dantzig,Linear programming and extensions (Princeton University Press, 1963).
F. Glover, “A note on linear programming and integer feasibility,”Operations Research 16 (1968) 1212–1216.
R.E. Gomory, “The travelling salesman problem,” in:Proceedings of IBM scientific computing symposium on combinatorial problem, White Plains, New York, 1964, pp. 93–121.
I. Heller, “On the travelling salesman's problem,” in:Proceedings of the second symposium in linear programming (National Bureau of Standards, Washington, D.C., 1955) pp. 643–665.
K.G. Murty, “On the tours of a travelling salesman,”SIAM Journal on Control 7 (1969) 122–131.
K.G. Murty, “Adjacency on convex polyhedra,”SIAM Review 13 (1971) 377–386.
K.G. Murty, “On the set representation and set covering problems,” University of Michigan, 1972.
M. Raghavachari, “On connections between zero–one integer programming and concave programming under linear constraints,”Operations Research 17 (1969) 608–684.
R. Roth, “Computer solutions to minimum cover problems,”Operations Research 17 (1969) 455–465.
H.M. Salkin and R.D. Koncal, “A pseudo-dual all integer algorithm for the set covering problem,” Technical Report Memo, No. 204, Operations Research Department, Case Western Reserve University, 1970.
Author information
Authors and Affiliations
Additional information
This research has been partially supported by the ISDOS research project and the National Science Foundation under Grant GK-27872 with the University of Michigan.
Rights and permissions
About this article
Cite this article
Murty, K.G. A fundamental problem in linear inequalities with applications to the travelling salesman problem. Mathematical Programming 2, 296–308 (1972). https://doi.org/10.1007/BF01584550
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01584550