Convex hulls of superincreasing knapsacks and lexicographic orderings

TL;DR

This paper characterizes the convex hull of superincreasing knapsacks, showing it can be described with a linear number of facets and establishing a distributive property for combining different types of these knapsacks.

Contribution

It provides a complete convex hull description for superincreasing knapsacks and generalizes previous binary case results, including a distributive property for convex hulls.

Findings

Convex hull described with O(n) facets.

Distributive property for convex hulls of superincreasing knapsacks.

Generalization of binary case results.

Abstract

We consider bounded integer knapsacks where the weights and variable upper bounds together form a superincreasing sequence. The elements of this superincreasing knapsack are exactly those vectors that are lexicographically smaller than the greedy solution to optimizing over this knapsack. We describe the convex hull of this $n$-dimensional set with $\mathcal{O}(n)$ facets. We also establish a distributive property by proving that the convex hull of $\le$- and $\ge$-type superincreasing knapsacks can be obtained by intersecting the convex hulls of $\le$- and $\ge$-sets taken individually. Our proofs generalize existing results for the binary case.