On bounded pitch inequalities for the min-knapsack polytope

TL;DR

This paper explores bounded pitch inequalities for the min-knapsack problem, demonstrating polynomial-time approximate separation for pitch-1 and pitch-2 inequalities and analyzing the impact on integrality gaps of relaxations.

Contribution

It introduces the concept of bounded pitch inequalities for min-knapsack, providing polynomial-time approximate separation algorithms and studying their effect on linear relaxation gaps.

Findings

Approximate separation over pitch-1 and pitch-2 inequalities is polynomial-time solvable.

The t-th CG closure of the natural relaxation has an unbounded integrality gap for any fixed t.

Bounded pitch inequalities generalize unweighted cover inequalities, enriching the polyhedral understanding.

Abstract

In the min-knapsack problem one aims at choosing a set of objects with minimum total cost and total profit above a given threshold. In this paper, we study a class of valid inequalities for min-knapsack known as bounded pitch inequalities, which generalize the well-known unweighted cover inequalities. While separating over pitch-1 inequalities is NP-hard, we show that approximate separation over the set of pitch-1 and pitch-2 inequalities can be done in polynomial time. We also investigate integrality gaps of linear relaxations for min-knapsack when these inequalities are added. Among other results, we show that, for any fixed $t$, the $t$-th CG closure of the natural linear relaxation has the unbounded integrality gap.