The author wishes to express deepest appreciation to Professor G. T. ... Nevertheless, there isa close connection between the optimal solutions to P1 and.
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Then X contains a disk D so that fj D is a homeomorphism of D onto Y. These results indicate a direction which may lead to a characterization of a certain class of light open mappings as generalized fiber spaces with totally disconnected fibers. The author wishes to express deepest appreciation to Professor G. T. Whyburn for stimulating conversations and encouragement. * Initial results were presented to the American Mathematical Society, January 1963, and research for these was accomplished at the University of Virginia while the author held an ONR Research Fellowship. 1 Hu, Sze-Tsen, Homotopy Theory (New York: Academic Press, 1959). 2 Browder, Felix E., "Covering spaces, fiber spaces, and local homeomorphisms," Duke Math. J., 21, 329-336 (1954). 3 Whyburn, G. T., Analytic Topology (Providence: American Mathematical Society, 1942). 4Floyd, E. E., "Some characterizations of interior maps," Ann. Math., 51, 571-575 (1950). 5 Whyburn, G. T., Topological Analysis (Princeton: Princeton University Press, 1958). 6 Whyburn, G. T., "Open mappings on 2-dimensional manifolds," J. Math. Mech., 10, 181198 (1961). Related references: Anderson, R. D., "A characterization of the universal curve and a proof of its homogeneity," Ann. Math., 67, 313-324 (1958); Cernavskii, A. V., "Finitely multiple open mappings of manifolds," Soviet Math. Doklady, 4, 946-949 (1963); Dyer, E., and M.-E. Hamstrom, "Completely regular mappings," Fund. Math., 45, 103-118 (1957); Whyburm, G. T., "On sequences and limiting sets," Fund. Math., 25, 408-426 (1935).
ON THE RELATION BETWEEN INTEGER AND NONINTEGER SOLUTIONS TO LINEAR PROGRAMS* BY R. E. GOMORY THOMAS J. WATSON RESEARCH CENTER, YORKTOWN HEIGHTS, NEW YORK
Communicated by R. Courant, December 22, 1964
We will refer to the ordinary linear programming problem
maximize zi Ax
(1)
= cx =
b,
x
> 0
as problem P1. In (1) b is an integer m-vector, c is an m + n vector, and A is an m X (m + n) integer matrix. x is an m + n vector, all of whose components are required to be nonnegative. We assume that A is of the form (A', I) with I an m X m identity matrix, so that in (1) Ax = b is equivalent to the m inequalities in n variables A 'x' < b. We will say that x is feasible if it satisfies the equality and nonnegativity conditions of (1) and optimal if it also maximizes. A problem closely related to P1 is the integer programming problem P2 which is P1 with the added condition that the components of x be integers. Because of the comparative ease with which P1 is solved' and the comparative difficulty of P2,2, it is natural to consider getting from the solution of P1 to the solution to P2 by some sort of a "rounding" process through which the noninteger components of the x solving P1 are rounded either up or down to produce a solution to P2. This
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procedure seems particularly plausible when the components xi of x are reasonably large numbers. However, it is easily shown by examples that a nearest-neighbor rounding process cannot generally produce the optimal solution to P2. These examples are neither pathological nor uncommon; it is simply not the case that the optimal solution can be obtained by simple rounding to some vector x' with |x' xil < 1, even if the rounding is followed by some sort of optimization on the residual problem and even if the b and x of (1) become arbitrarily large. Nevertheless, there is a close connection between the optimal solutions to P1 and P2 for a wide range of right-hand sides b. We first give some theorems on this connection and then an algorithm which for these b obtains the optimal solution of P2 from the optimal solution to P1. If B is a basis, i.e., an m X m nonsingular submatrix of A, we will assume that A has been rearranged and partitioned into matrices B and N with A = (B,N). We will also partition x = (XE, XN) and c = (CE, CN). The columns of A will be referred to as a,, B = (a,, ..., a,). We confine ourselves to right-hand side vectors b in that part of m-space for which (1) is solvable. If B is the optimal basis for P1 with right-hand side b, then it is also the optimal basis for all b' such that B-'b' > 0. These b form a cone in m-space, and in fact all solvable m-space is partitioned into such cones KE. On removing from KE all points within a distance d of its boundary, we have the reduced cone KB(d). With this notation we can now state Theorem 1. THEOREM 1. Let I = max ajjill, i = m + 1, .. .,m + n, D = Idet B, and zi(b) be the value of the solution to P1. Then if beKB(l(D - 1)), the value z2(b) of the solution to P2 is given by z2(b) = zi(b) + sp'(b), (2) and an optimal solution vector is given by x(b) = (xB(b), xN(b)) = (B'-(b - NyB(b)), yB(b)), (3) where both the scalar function (,B(b) and the n-vector function yB(b) are m-periodic, i.e.,
