Diameter Polytopes of Feasible Binary Programs

Abstract

Feasible binary programs often have multiple optimal solutions, which is of interest in applications as they allow the user to choose between alternative optima without deteriorating the objective function. In this article, we present the optimal diameter of a feasible binary program as a metric for measuring the diversity among all optimal solutions. In addition, we present the diameter binary program whose optima contains two optimal solutions of the given feasible binary program that are as diverse as possible with respect to the optimal diameter. Our primary interest is in the study of the diameter polytope, i.e., the polytope underlying the diameter binary program. Under suitable conditions, we show that much of the structure of the diameter polytope is inherited from the polytope underlying the given binary program. Finally, we apply our results on the diameter binary program and diameter polytope to cases where the given binary program corresponds to the linear ordering problem and the symmetric traveling salesman problem.

On the Linear Ordering Problem and the Rankability of Data

“…In applications, the number of optimal solutions can be too large for enumeration to be practical. Therefore, in [5], the authors present the optimal diameter of a feasible binary program as a metric for measuring the diversity among all optimal solutions. In particular, a binary program is developed whose optima contain two optimal solutions of the given binary program that are as diverse as possible with respect to the optimal diameter.…”

“…Here, we develop a binary program whose optima contain two optimal solutions of a given LOP that are as far apart as possible with respect to the Kendall tau ranking distance. This program is similar to but more straightforward than the optimal diameter binary program in [5]. Also, this program is similar to those developed in [2,23], where the author in [23] is interested in two optimal solutions of separate LOPs that are as close together as possible, and the authors in [2] are interested in two optimal solutions of the same LOP that are as far apart as possible.…”

“…(10,7,8,5,1,6,9,2,4,3), σ 2 = (10, 7, 5, 8, 1, 6, 9, 2, 4, 3), σ 3 = (10, 7, 5, 1, 8, 6, 9, 2, 4, 3), σ 4 = (10, 7, 8, 5, 1, 6, 9, 2, 3, 4), σ 5 = (10, 7, 5, 8, 1, 6, 9, 2, 3, 4), σ 6 = (10, 7, 5, 1, 8, 6, 9, 2, 3, 4).…”