IEOR 269, Spring 2010 Integer Programming and Combinatorial ...

IEOR269 notes, Prof. Hochbaum, 2010 1
Lec1
1 Introduction
Consider the general form of a linear program:
max
subject to
∑ n
i=1 c ix i
Ax ≤ b
In an integer programming optimization problem, the additional restriction that the x i must be
integer-valued is also present.
While at first it may seem that the integrality condition limits the number of possible solutions
and could thereby make the integer problem easier than the continuous problem, the opposite is
actually true. Linear programming optimization problems have the property that there exists an
optimal solution at a so-called extreme point (a basic solution); the optimal solution in an integer
program, however, is not guaranteed to satisfy any such property and the number of possible integer
valued solutions to consider becomes prohibitively large, in general.
While linear programming belongs to the class of problems P for which “good” algorithms exist
(an algorithm is said to be good if its running time is bounded by a polynomial in the size of the
input), integer programming belongs to the class of NP-hard problems for which it is considered
highly unlikely that a “good” algorithm exists. For some integer programming problems, such as
the Assignment Problem (which is described later in this lecture), efficient algorithms do exist.
Unlike linear programming, however, for which effective general purpose solution techniques exist
(ex. simplex method, ellipsoid method, Karmarkar’s algorithm), integer programming problems
tend to be best handled in an ad hoc manner. While there are general techniques for dealing with
integer programs (ex. branch-and-bound, simulated annealing, cutting planes), it is usually better
to take advantage of the structure of the specific integer programming problems you are dealing
with and to develop special purposes approaches to take advantage of this particular structure.
Thus, it is important to become familiar with a wide variety of different classes of integer
programming problems. Gaining such an understanding will be useful when you are later confronted
with a new integer programming problem and have to determine how best to deal with it. This
will be a main focus of this course.
One natural idea for solving an integer program is to first solve the “LP-relaxation” of the
problem (ignore the integrality constraints), and then round the solution. As indicated in the class
handout, there are several fundamental problems to using this as a general approach:
1. The rounded solutions may not be feasible.
2. The rounded solutions, even if some are feasible, may not contain the optimal solution. The
solutions of the ILP in general can be arbitrarily far from the solutions of the LP.
3. Even if one of the rounded solutions is optimal, checking all roundings is computationally
expensive. There are 2 n possible roundings to consider for an n variable problem, which
becomes prohibitive for even moderate sized n.