Inverse Fractional Knapsack Problem with Profits and Costs Modification

TL;DR

This paper introduces the inverse fractional knapsack problem, focusing on modifying profits and costs to achieve a desired optimal solution, and provides algorithms for different norm constraints.

Contribution

It formulates the inverse fractional knapsack problem and develops quadratic-time algorithms for profit-only modifications and under $l__$-norm, addressing computational complexity.

Findings

The problem is NP-hard under the $l_1$-norm.

Quadratic-time algorithm for profit-only modification.

Quadratic-time algorithm under $l__$-norm.

Abstract

We address in this paper the problem of modifying both profits and costs of a fractional knapsack problem optimally such that a prespecified solution becomes an optimal solution with prespect to new parameters. This problem is called the inverse fractional knapsack problem. Concerning the $l_1$-norm, we first prove that the problem is NP-hard. The problem can be however solved in quadratic time if we only modify profit parameters. Additionally, we develop a quadratic-time algorithm that solves the inverse fractional knapsack problem under $l_\infty$-norm.