Diameter Polytopes of Feasible Binary Programs

TL;DR

This paper introduces the diameter polytope as a new metric for measuring diversity among optimal solutions in feasible binary programs, providing structural insights and applications to classic combinatorial problems.

Contribution

It defines the diameter polytope, analyzes its structure, and applies it to the linear ordering and traveling salesman problems.

Findings

The diameter of feasible binary programs can be characterized by a new polytope.

Structural properties of the diameter polytope mirror those of the original program under certain conditions.

Applications demonstrate the utility of the diameter polytope in combinatorial optimization problems.

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.