Price of Coupling in Multilevel Linear Programming

Multilevel programming is the standard framework for modeling hierarchical decision-making. In this paper, we characterize the computational complexity of deciding the existence of feasible and optimal solutions, as well as computing the optimal objective value in multilevel linear programming (LP). Our analysis considers various combinations of modeling assumptions, including the presence or absence of linking (coupling) constraints and whether all variables are bounded. In particular, we show the feasibility problem of $k$-level LP is $\Sigma^{p}_{k-1}$-complete for $k \ge 2$. Without linking constraints and unbounded variables, it is polynomial-time solvable for $k \le 4$ but becomes $\Sigma^{p}_{k-1}$-complete for $k \ge 5$, indicating a sharp jump in computational complexity assuming the polynomial hierarchy does not collapse. Combined with other results, one major implication is that no polynomial-time Turing machine can transform a bilevel LP instance with linking constraints into one without linking constraints while preserving feasibility unless P $=$ NP. In contrast, such machines exist for all $k \ge 5$. We observe similar phenomena with the decision of the existence of an optimal solution. In the bilevel case, feasibility and boundedness fully characterize the existence of an optimal solution, implying that the problem is DP-complete. However, these conditions are insufficient for $k \ge 3$ and the problem for $k \ge 3$ is $\Delta^{p}_k$-complete. Similar to the feasibility problem, the problem becomes polynomially solvable for $k=2,3$ without linking constraints and unbounded variables. However, the problem is $\Delta^{p}_k$-complete for $k \ge 4$, even with these simplifying assumptions. The computation of the optimal objective value is F$\Delta^{p}_k$-complete for any $k \ge 2$, even without linking constraints and unbounded variables.