Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomial cases

TL;DR

This paper proves that testing weak optimality in interval linear programming is NP-hard in general, corrects a previous misconception, and introduces an algorithm with polynomial complexity for certain cases.

Contribution

It establishes the NP-hardness of the problem, corrects prior claims of polynomial solvability, and presents a new algorithm with polynomial time for interval LPs with only inequalities.

Findings

NP-hardness of weak optimality testing in general interval LPs

Polynomial-time algorithm for interval LPs with only inequalities

Correction of previous claims about problem complexity

Abstract

We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running time of the algorithm is exponential in the number of equality constraints. Interval linear programs with inequality constraints only can be processed in polynomial time.