The Complexity of Recognizing Facets for the Knapsack Polytope
Rui Chen, Haoran Zhu
TL;DR
The paper addresses the complexity of recognizing facet-defining inequalities for the knapsack polytope, proving KNAPSACK FACETS is $D^\text{p}$-complete and that the related knapsack hyperplane problem and CSS inherit this hardness. It constructs a chain of reductions, notably from Exact Vertex Cover to CSS and from CSS to KNAPSACK FACETS, with a key reduction that uses a shifted Fibonacci sequence to encode the CSS instance. Additionally, it provides a polynomial-time algorithm in $n^{K+O(1)}$ for inequalities with a fixed number $K$ of distinct positive coefficients, using minimal basic knapsack solutions to verify validity and facet-defining properties. These results advance our understanding of polyhedral facet recognition, offering practical XP-time methods for special cases while highlighting inherent DP-completeness in the general problem and leaving fixed-parameter tractability as an open question.
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.