Chapter V Polyhedral combinatorics

Abstract

Publisher Summary This chapter discusses the polyhedral combinatorics, which deals with the interactions between linear algebra, Euclidean geometry, linear programming, and combinatorial optimization. The polyhedral combinatorics provides min–max theorems that form an essential part of optimization algorithms.The role of polyhedral combinatorics in obtaining min–max relationships is discussed. The major methods and tools are also presented. The chapter describes two important ways of strengthening min–max relationships—facet characterizations and dual integrality. Polarity and blocking and antiblocking relations provides methods for reversing the roles of objects in min–max relations. A useful polyhedral property, dimension, and its uses are also covered in the chapter. All the relevant definitions and theorems of polyhedral theory and some approaches for obtaining polyhedral min–max theorems using extra variables, and ways of eliminating extra variables are also discussed in this chapter.