Comment on Two schemes for Secure Outsourcing of Linear Programming

TL;DR

This paper critiques two schemes for secure outsourcing of linear programming, highlighting their failure due to misunderstanding the essential nonnegativity constraints crucial for the simplex method.

Contribution

It clarifies the importance of the nonnegativity constraints in LP and demonstrates that the previously proposed schemes are invalid due to this fundamental oversight.

Findings

The schemes are unsolvable due to incorrect constraint assumptions.

The nonnegativity constraint x>0 is essential for the simplex method.

The proposed schemes failed because they confused inequality constraints with nonnegativity constraints.

Abstract

Recently, Wang et al. [IEEE INFOCOM 2011, 820-828], and Nie et al. [IEEE AINA 2014, 591-596] have proposed two schemes for secure outsourcing of large-scale linear programming (LP). They did not consider the standard form: minimize c^{T}x, subject to Ax=b, x>0. Instead, they studied a peculiar form: minimize c^{T}x, subject to Ax = b, Bx>0, where B is a non-singular matrix. In this note, we stress that the proposed peculiar form is unsolvable and meaningless. The two schemes have confused the functional inequality constraints Bx>0 with the nonnegativity constraints x>0 in the linear programming model. But the condition x>0 is indispensable to the simplex method. Therefore, both two schemes failed.