Decision Problems in Multilevel Linear Programming

TL;DR

This paper establishes the exact computational complexity of decision problems in multilevel linear programming, proving they are complete for certain levels of the polynomial hierarchy.

Contribution

It provides the first matching upper bounds for the complexity of these problems, closing open questions in multilevel programming.

Findings

Determined the complexity of optimal value threshold decision problems as ;complete.

Proved the unboundedness decision problem is ;complete.

Extended results to mixed-binary cases.

Abstract

We study the computational complexity of decision problems in $k$-level linear programming (LP). Seminal work by Jeroslow establishes that determining whether the optimal objective value of a $k$-level LP is at least as good as a given threshold is $\Sigma^{\mathrm{p}}_{k-1}$-hard. In this paper, we demonstrate the matching upper bound and thereby prove that this problem is $\Sigma^{\mathrm{p}}_{k-1}$-complete. To this end, we show that the feasible region of a $k$-level LP can be expressed as a union of sets defined by weak and strict linear inequalities. Moreover, we show that the decision of the unboundedness is $\Sigma^{\mathrm{p}}_{k-1}$-complete. Finally, we discuss the extension of our results to the mixed-binary cases. In short, this work closes lasting open questions in multilevel programming.