Integer Programs
CS 149 Staff
November 23, 2006
1 Introduction
Formulating a production problem as a linear program could lead to a recommen-
dation of producing 7.5 cars and 1.5 trucks. The customer that gets the front half
of a car and the back half of a truck is unlikely to be happy. To remedy this, it is
natural to add additional constraints to the linear program indicating that certain
variables must take only integer values. This is called an integer program (IP). So
how powerful are integer programs? It turns out that most NP hard optimization
problems can be naturally formulated as integer programs, including the knapsack
problem, satisfibility, graph bisection and traveling salesman problem.
Some problems, such as knapsack, can be translated into integer programs with
barely any work: max pTxsubject to wTx≤Cand x∈ {0,1}n. Many other
problems can be formulated as LPs with some effort. Here are some tricks typically
used when translating problems into integer programs.
1.1 Disjunction
Linear programs translating a conjunction into a linear program is easy, since all
constraints of a linear program must be satisfied. Many problems, such Boolean
satisfibility (SAT), contain disjunctions as well. Consider the constraint x1∨ ¬x2
is true. This can easily be translated into x1+ (1 −x2)≥1 with xi∈ {0,1}.
What about disjunctions of expressions, such as |x| ≥ 10, which is equivalent to
x≥10 ∨x≤ −10?
The solution here is to introduce a 0/1 slack variable sand rewrite the equations
as x+Ms ≥10 and x−(1 −s)M≤ −10, where Mis a very large number. If s= 0,
the first equation is in its original form and the second is trivially satisfied regardless
of the value of x. If s= 1, the first equation is trivially satisfied and the second is
not. Therefore this models disjunction.
1.2 Relying on the force of gravity
Consider the shortest path problem. One natural formulation is as a min cost flow
of 1 unit from sto twhere every edge has a flow of either zero or one. At first
glance this might seem to be insufficient since there could be cyclic flows in addition
to the path we’re looking for. If all edge weights are positive, any optimal solution
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