Thinner is not Always Better: Cascade Knapsack Problems

TL;DR

This paper challenges the assumption that thin directions in integer programming are optimal for branching, showing that minimal width directions are not always the most effective for efficient problem solving.

Contribution

It reveals that thin directions and integer width are unreliable indicators for selecting effective branching directions in branch-and-bound algorithms.

Findings

Thin directions are not always optimal for branching.

Integer width may not correlate with branching efficiency.

Effective branching strategies require more than just minimal width considerations.

Abstract

In the context of branch-and-bound (B&B) for integer programming (IP) problems, a direction along which the polyhedron of the IP has minimum width is termed a thin direction. We demonstrate that a thin direction need not always be a good direction to branch on for solving the problem efficiently. Further, the integer width, which is the number of B&B nodes created when branching on the direction, may also not be an accurate indicator of good branching directions.