The Complexity of Recognizing Facets for the Knapsack Polytope

TL;DR

This paper proves that recognizing facets of the knapsack polytope is DP-complete and provides a polynomial-time algorithm for certain fixed-coefficient inequalities, advancing understanding of polyhedral recognition complexity.

Contribution

It confirms the DP-completeness of facet recognition for the knapsack polytope and introduces a polynomial-time algorithm for fixed-coefficient inequalities.

Findings

Facet recognition is DP-complete.

Polynomial algorithm for fixed-coefficient inequalities.

Advances understanding of polyhedral recognition complexity.

Abstract

The complexity class DP is the class of all languages that are the intersection of a language in NP and a language in coNP. It was conjectured that recognizing a facet for the knapsack polytope is DP-complete. We provide a positive answer to this conjecture. Moreover, despite the \DP-hardness of the recognition problem, we give a polynomial time algorithm for deciding if an inequality with a fixed number of distinct coefficients defines a facet of a knapsack polytope.