Efficient storage of Pareto points in biobjective mixed integer programming

TL;DR

This paper introduces a new binary tree-based data structure for efficiently storing and updating the set of nondominated solutions in biobjective mixed integer linear programming, significantly improving performance over traditional methods.

Contribution

A novel data structure for storing Pareto points in BOMILPs that enhances efficiency in solution management within branch-and-bound algorithms.

Findings

Handles up to 10^7 points or segments efficiently

Outperforms dynamic list in storing solutions

Improves branch-and-bound solution process

Abstract

In biobjective mixed integer linear programs (BOMILPs), two linear objectives are minimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for finding or approximating the Pareto set of a BOMILP use and update a subset of nondominated solutions, it is highly desirable to efficiently store this subset. We present a new data structure, a variant of a binary tree that takes as input points and line segments in $\R^2$ and stores the nondominated subset of this input. When used within an exact solution procedure, such as branch-and-bound (BB), at termination this structure contains the set of Pareto optimal solutions. We compare the efficiency of our structure in storing solutions to that of a dynamic list which updates via pairwise comparison. Then we use our data structure in two biobjective BB techniques available in the literature and solve three classes of instances of BOMILP, one of which is generated by us. The first experiment shows that our data structure handles up to $10^7$ points or segments much more efficiently than a dynamic list. The second experiment shows that our data structure handles points and segments much more efficiently than a list when used in a BB.