Complexity Issues in Bilevel Linear Programming

Abstract

In this chapter, we review some results with several co-authors for problems related to multiple level programming along three directions: complexity, polynomial algorithms for special cases, and potentially important application areas of multiple level programming techniques.Formulating practical problems in multiple level programming may not always imply their solutions can be obtained efficiently. A major set-back in this direction is the work [26] of Jeroslow that bilevel linear programming can not be solved or even approximately solved in polynomial time unless NP = P (also see [23, 33] for discussion on computational complexity). An immediate question is that, under what conditions, and for what classes of problems, multiple level programs can be solved or be approximately solved in polynomial time. Positive results for this question have been very limited. Liu and Spencer have shown that the problem can be solved in polynomial time when the number of control variables of the followers is bounded by a constant [30]. Deng, Wang and Wang [20] have presented a much simpler proof which allows for an extension to cases where several independent followers are at the lower level. To another direction, it would be helpful to study useful application areas of the multiple level programming formulation. We discuss an emerging methodology for decision/computing under incomplete information, the competitive analysis method, and its formulation as multiple level programming We believe interplays of these two related disciplines will be fruitful and advantageous to both.