Combinatorial Optimization: An Introduction

Abstract

Combinatorial optimization problem arise typically in the form of a mixed-integer linear program

$$ \max \left\{ {cx + dy:Ax + Dy \leqslant b,x \geqslant 0 and integer, y \geqslant 0} \right\}, $$

(MIP)

where A is any m x n matrix of reals, b is a column vector with m real components and c and d are real row vectors of length n and p, respectively. If n = 0 then we have a linear program. If p = 0 then we have a pure integer program. The variables x that must be integer-valued are the integer variables of the problem, the variables y the real or flow variables. There are frequently explicit upper bounds on either the integer or real variables or both. In many applications the integer variables model yes/no decisions, i.e., they assume only the values of zero or one. In this case the problem (MIP) is a mixed zero-one or a zero-one linear program depending on p > or p = 0