Compact Representation of Near-Optimal Integer Programming Solutions

It is often useful in practice to explore near-optimal solutions of an integer programming problem. We show how all solutions within a given tolerance of the optimal value can be efficiently and compactly represented in a weighted decision diagram, once the optimal value is known. The structure of a decision diagram facilitates rapid processing of a wide range of queries about the near-optimal solution space. To obtain a more compact diagram, we exploit the property that such diagrams may become paradoxically smaller when they contain more solutions. We use sound decision diagrams, which innocuously admit some solutions that are worse than near-optimal. We describe a simple “sound reduction” operation that, when applied repeatedly in any order, yields a smallest possible sound diagram for a given problem instance. We find that sound reduction yields a structure that is typically far smaller than a tree that represents the same set of near-optimal solutions.