Decision Problems in Multilevel Linear Programming

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.