Forbidden vertices

TL;DR

This paper introduces the forbidden-vertices problem, analyzing its complexity and tractability depending on polytope encoding, with applications to binary and integral polytopes.

Contribution

It defines the forbidden-vertices problem, explores its complexity, and provides new tractability results and formulations for specific polytope classes.

Findings

Complexity varies with polytope and subset encoding.

Extended formulations improve tractability for binary polytopes.

Applications to integral polytopes are discussed.

Abstract

In this work, we introduce and study the forbidden-vertices problem. Given a polytope P and a subset X of its vertices, we study the complexity of linear optimization over the subset of vertices of P that are not contained in X. This problem is closely related to finding the k-best basic solutions to a linear problem. We show that the complexity of the problem changes significantly depending on the encoding of both P and X. We provide additional tractability results and extended formulations when P has binary vertices only. Some applications and extensions to integral polytopes are discussed.