The Computational Complexity of Multi-Level Bottleneck Programming Problems

Abstract

Multi-level programming problems are a simple class of sequenced-move games, in which the players are restricted in their moves by common constraints, and each player seeks to optimize a fixed criterion which depends on his own variables as well as those of the other players. Unlike in more traditional games, the players do not move simultaneously since the game proceeds according to a given hierarchical order.

In this paper we investigate the complexity of the (L + 1)-level bottleneck programming problem where all L + 1 players have a bottleneck objective function and the joint constraints are linear. We show that (L + 1)-level bottleneck programs are as hard as level L of the polynomial hierarchy. As a by-product, our proof method yields a simple alternative proof for the complexity of the (L + 1)-level linear programming problem which arises if all players have a linear objective function and all constraints are linear.